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Mensuration

Definition of Mensuration Maths :

Mensuration is a division of mathematics that studies geometric figure calculation and its parameters such as area, length, volume, lateral surface area, surface area, etc. It outlines the principles of calculation and discusses all the essential equations and properties of various geometric shapes and figures.

What is Mensuration?

Mensuration is a subject of geometry. Mensuration deals with the size, region, and density of different forms both 2D and 3D. Now, in the introduction to Mensuration, let’s think about 2D and 3D forms and the distinction between them.

What is a 2D Shape?

A 2D diagram is a shape laid down on a plane by three or more straight lines or a closed segment. Such forms do not have width or height; they have two dimensions-length and breadth and are therefore called 2D shapes or figures. Of 2D forms, area (A) and perimeter (P) is to be determined.

What is a 3D Shape?

A 3D shape is a structure surrounded by a variety of surfaces or planes. These are also considered robust types. Unlike 2D shapes, these shapes have height or depth; they have three-dimensional length, breadth and height/depth and are thus called 3D figures. 3D shapes are made up of several 2D shapes. Often known as strong forms, volume (V), curved surface area (CSA), lateral surface area (LSA) and complete surface area (TSA) are measured for 3D shapes.

Difference between 2D and 3D shapes

Introduction to Menstruation: Important Terms

Until we switch to the list of important formulas for measurement, we need to clarify certain important terms that make these measurement formulas:

The area is called the surface occupied by a defined closed region. It is defined by the letter A and expressed in a square unit.

Perimeter (P):

The total length of the boundary of a figure is called its perimeter. Perimeter is determined by only two-dimensional shapes or figures. It is the continuous line along the edge of the closed vessel. It is represented by P and measures are taken in a square unit.

Volume (V):

The width of the space contained in a three-diMensional closed shape or surface, such that, the area by a room or cylinder. Volume is denoted by the alphabet V and the SI unit of volume is the cubic meter.

Curved Surface Area (CSA):

The curved surface area is the area of the only curved surface, ignoring the base and the top such as a sphere or a circle. The abbreviation for the curved surface area is CSA.

Lateral Surface Area (LSA):

The total area of all of a given figure’s lateral surfaces is called the Lateral Surface Area. Lateral surfaces are the layers covering the artefact. The acronym for the lateral surface area is LSA.

Total Surface Area (TSA):

The calculation of the total area of all surfaces is called the Cumulative Surface Region in a closed shape. For example, we get its Total Surface Area in a cuboid by adding the area of all six surfaces. The acronym for the total surface area is TSA.

Square Unit (/):

One square unit is simply the one-unit square area. When we quantify some surface area, we relate to the sides of one block square to know how many of these units will fit in the figure given.

Cube Unit (/):

One cubic unit is the one-unit volume filled by a side cube. When we calculate the volume of any number, we refer to this cube of one unit and how many these component cubes will fit in the defined closed form.

Tools Require for Mensuration

Calliper - A tool for measuring the diameter.

Try Square - A tool for determining the squareness and flatness of a surface.

Meter Stick - A measuring device with a one-meter length.

Compass - An arc and circle drawing tool.

List of Mensuration Formulas for 2D shapes:

As our introduction to Mensuration and the relevant words are over, let’s switch to the equations for Mensuration, as this is a discussion focused on an equation. The 2D figure has a list of formulas of measurement that define a relationship between the various parameters. Let’s look into detail about the estimation equations of some kinds.

Area = \[(side)^{2}\] sq. units.

Perimeter = \[(4 \times sides)\] units.

Diagonal = \[\sqrt{2 \times side}\]units.

Area = \[(length \times breadth) \]sq. units.

Perimeter = 2(length + breadth) units.

Diagonal, D = \[\sqrt{length^{2} + breath^{2}}\]units.

Scalene Triangle:

Area , \[A = \frac{height \times base}{2} \]sq. units

Perimeter = (side a + side b + side c) units.

Equilateral Triangle:

Area = \[ \frac{\sqrt{3}}{4} \times side^{2} \] sq. units.

Perimeter = \[(3 \times side)\] units.

Isosceles Triangle:

Area = \[A = \frac{height \times base}{2} \]sq. units

Perimeter = \[(2 \times side \] + base) units.

Right Angled Triangle:

Area = \[A = \frac{leg_{a}\times leg_{b}}{2} \]sq. units

Perimeter = \[leg_{a} + leg_{b} + \sqrt {leg_{a}^{2} + leg_{b}^{2}}\] units

Hypotenuse = \[\sqrt {leg_{a}^{2} + leg_{b}^{2}}\] units

Area = \[ π \times radius^{2} \] sq. units.

Circumference =\[ 2π \times radius\] units.

Diameter, D = \[2 \times radius \] units.

List of Mensuration Formulas for 3D shapes:

The 3D figure has a list of formulas for measurement that define a relationship between the various parameters. Let’s look into the details about the estimation equations of some kinds.

Volume = \[ side^{3} \] cubic units.

Lateral Surface Area = \[4 \times side^{2}\] sq. units.

Total Surface Area = \[6 \times side^{2}\] sq. units.

Diagonal Length d = \[\sqrt {length^{2} + width^{2} + height^{2}}\] units.

Volume = (length + width + height) cubic units.

Lateral Surface Area =\[ 2 \times height (length + width) \]sq. units.

Total Surface Area = \[2(length \times width + length \times height + height \times width)\] sq. units.

Diagonal Length = \[ length^{2} + breadth^{2} + height^{2}\] units.

Volume = \[ \frac{4}{3} \prod \times side^{2} \] cubic units.

Surface Area = \[4\prod \times radius^{2} \]sq. Units.

Hemisphere:

Volume = \[ \frac{2}{3} \prod \times side^{2} \] cubic units.

Total Surface Area =\[3\prod \times radius^{2} \]sq. Units.

Volume = \[\prod \times radius^{2} \times height \] cubic units.

Curved Surface Area (excluding the areas of the top and bottom circular regions) = \[(2 \times \prod \times R \times h)\] sq. units.

Where, R = radius

Total Surface Area = \[(2 \times \prod \times R \times h) + (2 \times \prod \times R^{2}) \] sq. units

Volume = \[ \frac{1}{3} \prod \times radius^{2} \times height \] cubic units.

Curved Surface Area =\[ 𝜋 \times radius \times height\] sq. units.

Total Surface Area = 𝜋 x radius(length + height) sq. units

Slant Height of Cone =\[ \sqrt{height^{2} + base radius^{2}}\] sq. units.

Using these above formulas for the Mensuration, most of the Mensuration problems can be solved.

FAQs on Mensuration

1. In Mathematics, what is Mensuration?

Mensuration is an area of mathematics that explores the measurements of various geometrical shapes and their areas, perimeters, and volumes, among other things. These shapes can be found in two or three diMensions. It also investigates the calculation of 2D and 3D geometric figures. Mensuration comprises the use of mathematical formulas and algebraic equations to do calculations. Thus, Mensuration refers to the branch of geometry involved with measuring lengths and volumes. It serves as the foundation for computing and describes the fundamentals and properties of various figures and forms. The ancient Egyptians were the first to apply mathematical methods for land mapping and levelling. The length of fabric needed for sewing, the size of a wall to be painted, or the amount of water needed to fill the tank are all examples of Mensuration.

2. What are some of the ways that Mensuration is used in our daily lives?

Mensuration is used in a range of contexts in real life:

When buying or selling land, it is necessary to measure the floor and site spaces.

The total distance around a circular racetrack.

Fencing is required around the perimeter of a garden.

Agriculture field measurement.

We might wish to check the weather before leaving the house. The temperature monitored with thermometers will help us decide what to wear.

It is necessary to measure the surface areas of a house to estimate the cost of painting.

To find the volume of water in the river and tanks.

Volumes needed to package milk, liquids, and solid edible food items are measured.

To calculate the measure of carpet needed to cover a specific room.

The amount of soil required to fill a hole in the ground.

3. What are the types of Mensuration?

There are two types of Mensuration: 2D Mensuration and 3D Mensuration.

The 2D figures have only two diMensions, which are primarily length and width. There is no height or depth to the two-diMensional figure. 2D shapes have areas but not volume. The sides of the 2D shapes are made up of straight lines. X-axes and Y-axes are used to draw 2D forms. The various 2-diMensional figures are – Square, Rectangle, Circle, Triangles, P arallelogram.

The three diMensions of 3D figures are length, breadth, and height or depth. More objects, such as books, pencils, cylinders, balls etc., found in our daily lives are examples of three-diMensional figures. 3D shapes do have Areas and Volume too since they occupy space. X-axes, Y-axes, and Z-axes are used to draw 3D shapes. The general three-diMensional figures are – Cube, Cuboid, Cylinder, Sphere, Cone.

4. What is the difference between geometry and Mensuration?

Mensuration is the calculation of numerous parameters of forms such as perimeter, area, volume, and so on, whereas geometry is the study of the properties and relationships of points and lines of different shapes. Geometry is a subject section of mathematics concerned with the shapes, angles, diMensions, and sizes of a wide range of objects that we see in daily life. Mensuration is a geometry concept. Mensuration is concerned with the size, region, and density of various 2D and 3D forms.

5. Can I get study material for Mathematics for my exam on Vedantu?

Yes, students can get all kinds of study material (Syllabus, Sample Paper, Revision notes etc.) for Mathematics on Vedantu’s website, and which can be downloaded in free PDF Format. These study tools are created by our in-house subject matter specialists, all of whom have years of teaching experience. These notes cover nearly all the mathematical concepts for the respective classes in an engaging manner, ensuring that learning is both enjoyable, Downloading the finest and most up-to-date study material from Vedantu and doing well in your exams.

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Mathematics Presentation for Class 9

Mathematics, chapter 1: number systems, chapter 2: polynomials, chapter 3: coordinate geometry, chapter 4: linear equations in two variables, chapter 5: introduction to eucluid's geometry, chapter 6: lines and angles, chapter 7: triangles, chapter 8: quadrilaterals, chapter 9: areas of parallelograms and triangles, chapter 10: circle, chapter 11: constructions, chapter 12: heron's formula, chapter 13: surface areas and volumes, chapter 14: statistics, chapter 15: probability.

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Mensuration

Mensuration is the branch of mathematics that studies the measurement of the 2D and 3D figures on parameters like length, volume, shape, surface area, etc. In other words, it is the process of measurement based on algebraic equations and mathematical formulas. Let us learn more about the concept of mensuration, the formulas and solve a few examples to understand it better.

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What is Mensuration?

Mensuration can be explained as an act of measurement. We live in a three-dimensional world. The concept of measurement plays an important role in primary as well as secondary school mathematics. Moreover, measurement has a direct connection to our everyday lives. When learning to measure objects we learn to do so for both 3D shapes and 2D shapes . Objects or quantities can be measured using both standard and nonstandard units of measurement. For example, a non-standard unit of measuring length would be handspans. You can even do an activity on it by asking children to measure the length of objects using handspans. Let children notice that while measuring objects using non-standard units there will always be a scope of a discrepancy. Hence the need for standard units of measurement. To measure parameters like length, weight, and capacity we now have units like kilometer, meter, kilogram, gram, liter, milliliter, etc.

3D Shapes Definition

A shape or a solid that has three dimensions is called a 3D shape that has faces, edges, and vertices. They have a surface area that includes the area of all their faces. The space occupied by these shapes gives their volume. Some examples of 3D shapes are cube, cuboid, cone, cylinder and some real-world examples are a book, a birthday hat, and, a coke tin.

2D Shapes Definition

In geometry, 2D shapes can be defined as plane figures that are completely flat and have only two dimensions – length and width. They do not have any thickness and can be measured only by the two dimensions.

Uses of Mensuration

Mensuration is an important topic with high applicability in real-life scenarios. Given below are some of the scenarios.

Measurement of agricultural fields, floor areas required for purchase/selling transactions.
Measurement of volumes required for packaging milk, liquids, solid edible food items.
Measurements of surface areas required for estimation of painting houses, buildings, etc.
Volumes and heights are useful in knowing water levels and amounts in rivers or lakes.
Optimum cost packaging sachets for milk etc. like tetra packing.

Important Mensuration Terms

Mensuration deals with the measurement of plane shapes and solid shapes. Let us see some of the important terms used:


	Area is the amount of space occupied by a two-dimensional figure. It is expressed in square units.
	Perimeter is the total distance around the shape or the length of the boundary of any closed shape. It is expressed in square units.
	Volume is the amount of space occupied by a 3D shape. It is expressed in cubic meter.
	Surface Area is the total area occupied by the surfaces of a 3D object. They are classified into two - Curved or Lateral Surface Area and Total Surface Area.

Mensuration Formulas

Mensuration formulas involve both 3D and 2D shapes. The most commonly used formula is the surface area and volume of these shapes. However, let us learn all the formulas for these shapes.

3D Shape Formulas

The following table shows different 3D shapes and their formulas.


	Diameter = 2 × r; (where 'r' is the radius) Surface Area = 4πr Volume = (4/3)πr
	Total Surface Area = 2πr(h+r); (where 'r' is the radius and 'h' is the height of the cylinder) Volume = πr h
	Curved Surface Area = πrl; (where 'l' is the slant height and l = √(h + r )) Total Surface Area = πr(l + r) Volume = (1/3)πr h
	Lateral Surface Area = 4a ; (where 'a' is the side length of the cube) Total Surface Area = 6a Volume = a
	Lateral Surface Area = 2h(l + w); (where 'h' is the height, 'l' is the length and 'w' is the width) Total Surface Area = 2 (lw + wh + lh) Volume = (l × w × h)
	Surface Area = [(2 × Base Area) + (Perimeter × Height)] Volume = (Base Area × Height)
	Surface Area = Base Area + (1/2 × Perimeter × Slant Height) Volume = [(1/3) × Base Area × Altitude]

2D Shape Formulas

The following table shows the formulas that are used to calculate the area and perimeter of a few common 2D shapes:

2D Shape	Area Formula	Perimeter Formula
	A = π × r , where 'r' is the radius of the circle and 'π' is a constant whose value is taken as 22/7 or 3.14	Circumference (Perimeter) = 2πr
	Area = ½ (Base × height)	Perimeter = Sum of the three sides
	Area = Side	Perimeter = 4 × side
	Area = Length × Width	Perimeter = 2 (Length + Width)

Important Notes:

Mensuration and measurement are learned together when an object is measured using non-standard unit of measurement.
Solid shapes and nets help in measuring an object. Solid shapes will help understand faces, edges, and vertices. Nets will help in visualizing the structure of 3D shapes.

☛ Related Topics:

Measuring Length
Measuring Weight
Measuring Capacity

Mensuration Examples

Example 1: Find the area of a square with a side of 5 cm.

Area of a square = side × side. Here, side = 5 cm

Substituting the values, 5 × 5= 25.

Therefore, the area of the square = 25 square cm.

Example 2: Find the surface area of a cuboid of length 4 units, width 5 units, and height 6 units.

Given that, length of the cuboid = 4 units, width of the cuboid = 5 units, height of the cuboid = 6 units

Surface area of the cuboid is 2 × (lw + wh + lh) square units

= 2 × (lw + wh + lh)

= 2[(4 × 5) + (5 × 6) + (4 × 6)]

= 2(20 + 30 + 24)

= 148 square units.

Therefore, the surface area of the cuboid is 148 square units.

Example 3: Find the area of a circle whose radius is 6 cm.

Solution: Yes, a circle comes under the category of 2D shapes. The area of a circle = π × r 2 ; where 'r' is the radius of the circle and π is a constant whose value is 22/7 or 3.14.

Area of the circle = π × r 2

= 3.14 × 6 2

= 3.14 × 36

Therefore, the Area of the circle = 113.04 square cm.

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Practice Questions on Mensuration

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FAQs on Mensuration

Who introduced mensuration.

Archimedes is remembered as the greatest mathematician of the ancient era. He contributed significantly in geometry regarding the area of plane figures and areas as well as volumes of curved surfaces.

What is Mensuration in Math?

Mensuration in maths deals with the geometric qualities of 2D and 3D shapes such as the area, volume, and perimeter. In other words, the study of measurements of these shapes is known as mensuration.

What is the Difference Between Mensuration and Geometry?

Mensuration refers to the calculation of various parameters of shapes like the perimeter, area, volume, etc. Whereas, geometry deals with the study of properties and relations of points and lines of various shapes.

What are 2D and 3D Mensuration?

2D mensuration is the calculation of 2D shapes on different parameters such as area and perimeter. Whereas 3D mensuration is the study of volume and lateral surface area of 3D shapes.

What is Mensuration Formula?

Mensuration formulas involve the formulas used to calculate the different parameters of 2D and 3D shapes such as the area, perimeter, and volume. You can read about the detailed list of formulas of both these shapes in the previous section of this article.

School Guide
Mathematics
Number System and Arithmetic
Trigonometry
Probability
Mensuration
Maths Formulas
Integration Formulas
Differentiation Formulas
Trigonometry Formulas
Algebra Formulas
Mensuration Formula
Statistics Formulas
Trigonometric Table

Mensuration in Maths | Formulas for 2D and 3D Shapes, Examples

Mensuration is a branch of mathematics concerned with the calculation of geometric figures and their parameters such as weight, volume, form, surface area, lateral surface area, and so on.

Let’s learn about all the mensuration formulas in maths.

Mensuration Meaning

Mensuration is the branch of mathematics that deals with the measurement of various geometric figures and shapes. This includes calculating areas, volumes, and perimeters of two-dimensional shapes like squares, rectangles, circles, and triangles, as well as three-dimensional figures like cubes, cylinders, spheres, and cones.

These shapes can exist in 2 ways:

Two-Dimensional Shapes – circle, triangle, square, etc.
Three-Dimensional Shapes – cube, cuboid, cone, etc.

Difference Between 2D and 3D Shapes

2-Dimensional vs 3-Dimensional Shapes
2D Shape	3D Shape
Any shape is 2D if it is bound by three or more straight lines in a plane.	A shape is a three-dimensional shape if there are several surfaces or planes around it.
There is no height or depth in these shapes.	In contrast to 2D forms, these are sometimes known as solid shapes and have height or depth.
These shapes just have length and width as their dimensions.	Since they have depth (or height), breadth, and length, they are referred to as three-dimensional objects.
We can calculate their perimeter and area.	Their volume, curved surface area, lateral surface area, or total surface area can all be calculated.

Mensuration Terminologies

Here is the list of terms you will come across in mensuration class. We have provided the term, it’s abbreviation, unit and definition for easy understanding.

Abbreviation	Unit	Definition
A	m or cm	The surface that the closed form covers is known as the area.
P	cm or m	A perimeter is the length of the continuous line that encircles the specified figure.
V	cm or m	A 3D shape’s space is referred to as its volume.
CSA	m or cm	The overall area is known as a Curved surface area if there is a curved surface. Example: Sphere
LSA	m or cm	The term “Lateral Surface area” refers to the combined area of all lateral surfaces that encircle the provided figure.
TSA	m or cm	The total surface area is the total of all the curved and lateral surface areas.
–	m or cm	A square unit is the area that a square of side one unit covers.
–	m or cm	The space taken up by a cube with a single side.

Mensuration Formula For 2D Shapes

The following table provides a list of all mensuration formulas for 2D shapes :


	a	4a
	l × b	2 (l + b)
	πr	2 π r
	√[s(s−a)(s−b)(s−c)], Where, s = (a+b+c)/2	a+b+c
	½ × b × h	2a + b
	(√3/4) × a	3a
	½ × b × h
	½ × d1 × d2	4 × side
	b × h	2(l+b)
	½ h(a+c)	a+b+c+d

Learn More:

Area of Trapezium
Area of Polygons
Area of General Quadrilateral
Heron’s Formula
Applications of Heron’s Formula
Area of 2D Shapes
Perimeter of circular figures
Areas of sector and segment of a circle
Areas of combination of plane figures

Mensuration Formula for 3D Shapes

The following table provides a list of all mensuration formulas for 3D shapes :


a	LSA = 4 a	6 a
l × b × h	LSA = 2h(l + b)	2 (lb +bh +hl)
(4/3) π r	4 π r	4 π r
(⅔) π r	2 π r	3 π r
π r h	2π r h	2πrh + 2πr
(⅓) π r h	π r l	πr (r + l)

Learn More :

Surface Area of Cube, Cuboid, and Cylinder
Volume of Cube, Cuboid, and Cylinder
Volume and Capacity
Surface Area of 3D Shapes
Volumes of Cubes and Cuboids
Surface Areas and Volumes
Volumes of a combination of solids
Conversion of solids
Frustum of a Cone
Section of a Cone
Conic Section

Solved Problems on Mensuration

Let’s solve some example problems on mensuration.

Problem 1: Find the volume of a cone if the radius of its base is 1.5 cm and its perpendicular height is 5 cm.

Radius of the cone, r = 1.5 cm Height of the cone, h = 5 cm ∴ Volume of the cone, V = 13πr 2 h=13×227×(1.5) 2 ×5= 11.79 cm 3 Thus, the volume of the cone is 11.79 cm 3 .

Problem 2: The dimensions of a cuboid are 44 cm, 21 cm, 12 cm. It is melted and a cone of height 24 cm is made. Find the radius of its base.

The dimensions of the cuboid are 44 cm, 21 cm and 12 cm. Let the radius of the cone be r cm. Height of the cone, h = 24 cm It is given that cuboid is melted to form a cone. ∴ Volume of metal in cone = Volume of metal in cuboid ⇒(1/3)πr 2 h=44×21×12 (Volume of cuboid=Length×Breadth×Height) ⇒(1/3)×(22/7)×r 2 ×24=44×21×12 ⇒r= √(44×21×12×21) / (22×24) =21 cm Thus, the radius of the base of cone is 21 cm.

Problem 3: The radii of two circular ends of frustum shape bucket are 14 cm and 7 cm. The height of the bucket is 30 cm. How many liters of water can it hold? (1 litre = 1000 cm 3 ).

Radius of one circular end, r1 = 14 cm Radius of other circular end, r2 = 7 cm Height of the bucket, h = 30 cm ∴ Volume of water in the bucket = Volume of frustum of cone =(1/3)πh(r 1 2 +r 1 r 2 +r 2 2 ) =13×22/7×30×(142+14×7+72) =13×22/7×30×343=10780 cm 3 =107801000=10.780 L Thus, the bucket can hold 10.780 litres of water.

FAQs On Mensuration

What is mensuration.

Mensuration deals with the calculation of geometric figures and their parameters such as weight, volume, form, surface area, lateral surface area, and so on.

What are 2D and 3D Shapes?

Any shape is considered to be 2D if it is bound by three or more straight lines in a plane whereas a shape is a three-dimensional shape if there are several surfaces or planes around it.

What is Area of Cylinder Formula?

Lateral or Curved Surface area of a cylinder = 2π r h Total Surface Area of a cylinder = 2πrh + 2πr 2

What is TSA (Total Surface Area) of Sphere Formula?

Area of Sphere is given by the following formula : A= 4 π r 2

What is Volume of Cone Formula?

Volume of Cone is given by the following formula : V= (⅓) π r 2 h

What is Area of Triangle Formula?

Area of Triangle is given by the following formula : A= 1/2 ×b ×h

What is Area of Circle Formula?

Area of Circle is given by the following formula : A= π r 2

What is Volume of Cylinder Formula?

Volume of Cylinder is given by the following formula : V= π r 2 h

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Mensuration Questions

Mensuration questions with answers are available for students. The problems have been solved in an elaborate manner to make each and every student understand the concept easily. The questions here are based on Class 8 and Class 10 Maths syllabus. The problems have been prepared, as per the latest NCERT guidelines.

Also, read : Mensuration

Area of Rectangle = Length x Breadth

Area of Square = Side

Area of Triangle = ½ Base x Height

Area of Parallelogram = Base x Height

Area of circle = πr

Perimeter of polygons = Sum of their sides

Circumference of circle = 2πr

Questions on Mensuration with Solutions

1. A circle has a radius of 21 cm. Find its circumference and area. (Use π = 22/7)

Solution: We know,

Circumference of circle = 2πr = 2 x (22/7) x 21 = 2 x 22 x 3 = 132 cm

Area of circle = πr 2 = (22/7) x 21 2 = 22/7 x 21 x 21 = 22 x 3 x 21

Area of circle with radius, 21cm = 1386 cm 2

2. If one side of a square is 4 cm, then what will be its area and perimeter?

Solution: Given,

Length of side of square = 4 cm

Area = side 2 = 4 2 = 4 x 4 = 16 cm 2

Perimeter of square = sum of all its sides

Since, all the sides of the square are equal, therefore;

Perimeter = 4+4+4+4 = 16 cm

= ½ d (h +h )

Where d is the diagonal of quadrilateral dividing it into two triangles

h and h are the heights of two triangles falling on the same base (diagonal of quad.)

= ½ d d

Where d and d are diagonals of rhombus

= ½ h (a+b)

Where a and b are the two parallel sides of trapezium

h is the distance between a and b.

Also, read:

Mensuration Class 8
Important Questions Class 8 Maths Chapter 11 Mensuration

3. Suppose a quadrilateral having a diagonal of length 10 cm, which divides the quadrilateral into two triangles and the heights of triangles with diagonals as the base, are 4 cm and 6 cm. Find the area of the quadrilateral.

Diagonal, d = 10 cm

Height of one triangle, h 1 = 4cm

Height of another triangle, h 2 = 6cm

Area of quadrilateral = ½ d(h 1 +h 2 ) = ½ x 10 x (4+6) = 5 x 10 = 50 sq.cm.

4. A rhombus having diagonals of length 10 cm and 16 cm, respectively. Find its area.

Solution: d 1 = 10 cm

d 2 = 16 cm

Area of rhombus = ½ d 1 d 2

A = ½ x 10 x 16

5. The area of a trapezium shaped field is 480 m 2 , the distance between two parallel sides is 15 m and one of the parallel sides is 20 m. Find the other parallel side.

Solution: One of the parallel sides of the trapezium is a = 20 m, let another parallel side be b, height h = 15 m.

The given area of trapezium = 480 m 2

We know, by formula;

Area of a trapezium = ½ h (a+b)

480 = ½ (15) (20+b)

20 + b = (480×2)/15

b = 64 – 20 = 44 m

TSA of Cuboid = 2(lb + bh + hl)

TSA of Cube = 6l

TSA of Cylinder = 2πr (r + h)

6. The height, length and width of a cuboidal box are 20 cm, 15 cm and 10 cm, respectively. Find its area.

Solution: Total surface area = 2 (20 × 15 + 20 × 10 + 10 × 15)

TSA = 2 ( 300 + 200 + 150) = 1300 cm 2

7. If a cube has its side-length equal to 5cm, then its area is?

Area = 6l 2 = 6 x 5 x5 = 150 sq.cm

8. Find the height of a cylinder whose radius is 7 cm and the total surface area is 968 cm 2 .

Solution: : Let height of the cylinder = h, radius = r = 7cm

Total surface area = 2πr (h + r)

TSA = 2 x (22/7) x 7 x (7+h) = 968

V of cube = l

Volume of cuboid = l × b × h

Volume of cylinder = πr h

Check: Mensuration Formulas Class 10

9. Find the height of a cuboid whose volume is 275 cm 3 and base area is 25 cm 2 .

Solution: Volume of cuboid = l × b × h

Base area = l × b = 25 cm 2

275 = 25 × h

h = 275/25 = 11 cm

10. A rectangular piece of paper 11 cm × 4 cm is folded without overlapping to make a cylinder of height 4 cm. Find the volume of the cylinder.

Solution: Length of the paper will be the perimeter of the base of the cylinder and width will be its height.

Circumference of base of cylinder = 2πr = 11 cm

2 x 22/7 x r = 11 cm

Volume of cylinder = πr 2 h = (22/7) x (7/4) 2 x 4

= 38.5 cm 3

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Mensuration

Jan 06, 2020

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Mensuration. Try this. 1. Consider the following situations. In each find out whether you need volume or area and why?. ( i ). Quantity of water inside a bottle. Here we need Volume 3-Dimension shape. ( ii ) Canvas need for making a tent.

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Mensuration Try this 1. Consider the following situations. In each find out whether you need volume or area and why? ( i ). Quantity of water inside a bottle Here we need Volume 3-Dimension shape ( ii ) Canvas need for making a tent Here we need area – Lateral Surface Area or Total Surfae Area ( iii ) Number of bags inside the lorry Here we need Volume 3-Dimension shape ( iv ) Gas filled in a cylinder Here we need Volume 3-Dimension shape ( v ) Number of match sticks that can be put in the match box Here we need Volume 3-Dimension shape

Try this Compute 5 more such examples and ask your friends to choose what they need ? ( i ) Quantity of Ice cream in Ice Cream Cone ( ii ) Quantity of Whitewash required to whitewash the wallls of room ( iii ) Quatity of rice bags in a heap of rice ( iv ) Quantity of water required to store in a steel pot ( v ) Number of Cloths to store in a suitcase

Try this 1.Break the pictures in the following figures into Solids of Known shapes Cylinder , Hemisphere Cubiod, cylinder Cuboid, Cylinder Sphere, Cylinder Sphere,Sphere Cylinder , Cylinder Sphere Cylinder , Cone , Cone

Try this 2. Think of 5 more things around you that can be seen as a Combination of shapes. Name the shapes that combine to make them . Ring Cuboid Cylinder Cylinder Sphere Hemisphere Cuboid Sphere Cylinder Cuboid Cuboid Cylinder

Try this 2. Think of 5 more things around you that can be seen as a Combination of shapes. Name the shapes that combine to make them .

Figure Lateral / Curved Surface area Total Surface Area Noman Clature Volume Name of the Solid l : length b:breadth h: height lbh Cuboid 2h ( l+b ) 2(lb+bh+hl ) Cube 4a2 6a2 a3 a : side of the cube Lateral Surfave area +2(area of the end Surface) Right Prism Area of base x height Perimeter of base x height r: radius of the base h: height Regular Circular Cylinder Lateral Surface area + area of the base (area of the base ) X height (perimeter of base ) x Slant height Right Pyramid Right Circular cone r : radius of base h: height l : Slant height Sphere r : radius r : radius Hemisphere

Example:1 The radius of a conical tent is 7 meters and its height is 10 meters . Calculate the length of canvas used in making the tent if width of convas is 2 meters . Solution:Radius of of conical tent Height Slant height of the conical Tent Lateral Surface Area of conical Tent Area of the canvas used in making the tent Breadth of canvas Area of the canvas = length of canvas X breadth of the canvas 268.4 = length of the canvas x 2 Length of the canvas

Example : 2 An Oil drum is in the shape of a cylinder having the following dimensions . Diameter is 2 m. and height is 7 meters . The painter charges Rs.3 per m2 to paint the drum . Find the total charges to be paid to the painter for 10 drums . Solution : It is given that diameter of the cylinderical oil drum = 2m 7 m Radius of cylinder = r = d / 2 = 2 / 2 = 1m Height of the Oil drum = h = 7 m 1m Total surface area of the cylinderical oil drum = The painter charges Rs.3 per 1 Sq. m Total charges to be paid to the painter for 1 drum = 3x50.28 = Rs.150.84 Total charges to be paid to the painter for 10 drums = 10x150.84 = Rs.1508.40

Example : 3 A sphere , a cylinder and a cone are of the same radius and same height . Find the ratio of their curved surface areas ? Solution : Let r be the common radius of a sphere , a cone and a cylinder Height of the sphere (h) = Diameter of the sphere = 2r The height of the cone = height of cylinder = height of sphere = 2r Slant height of the cone S1 = Curved Surface Area of Sphere = S2 = Curved surface Area of the cylinder S3 = Curved Surface Area of Cone Ratio of curved Surface area as

Example : 4 A company wanted to manufacture 1000 hemispherical basins from a thin steel sheet . If the radius of hemispherical basins is 21 cm. Find the required area of steel sheet to manufacture the above hemispherical basins ? Solution : Radius of the hemispherical basin Lateral Surface area of hemishperical basin The steel sheet required for on e basin Total area of steel sheet required for 1000 basins

Example : 5 A right circular cylinder has base radius 14 cm and height 21 cm. Find (i) Area of base or area of each end (ii) Curved Surface area (iii) Total Surface area and (iv) Volume of the right circular cylinder Solution : Radius of the cylinder Height of the cylinder (i) Area of base or area of each end of cylinder (ii) Curved Surface area of the right circular cylinder (iii) Total Surface area of the right circular cylinder area of the base Curved Surface area (iv) Volume of the right circular cylinder = Area of the base X height

Example : 6 Find the volume and Surface area of a sphere of radius 2.1 cm Solution : Radius of Sphere Surface area of sphere Volume of sphere

Example : 7 Find the Volume and the total Surface area of a hemisphere of radius 3.5 cm . Solution : Radius of hemisphere Total Surface area of hemisphere Volume of hemisphere

Exercise - 10.1 1. A Joker’s cap is in the form of right circular cone whose base radius is 7 cm and height is 24 cm . Find the area of the sheet required to make 10 such caps . Solution : base radius of the right circular cone shape Joker’s cap Height Slant height of cone The area of the sheet require to make one cap = Curved surface area of the right circular cone The area of the sheet require to make 10 such caps

Exercise - 10.1 2. A sports company was ordered to prepare 100 Paper cylinders for shuttle cocks. The required dimensions of the cylinder are 35 cm length / height and its radius is 7 cm. Find the required area of thin paper sheet needed to make 100 cylinders ? Solution : Radius of the cylinder Height Required area of thin paper sheet needed to make one cylinder = The Total Surface area of the Cylinder Required area of thin paper sheet needed to make 100 cylinders

Exercise – 10.1 3. Find the volume of right circular cone with radius 6 cm and height 7 cm. Solution : Radius of right circular cone Height volume of right circular cone

Exercise - 10.1 4. The lateral surface area of a cylinder is equal to the curved surved surface area of a cone . If the radius be the same . Find the ratio of the height of the cylinder and slant height of the cone . Solution : The radius of cylinder and cone be same . Let r. Let height of the cylinder be h and slant height of the cone be l Lateral surface area of the cylinder = Curved surface area of the cone The lateral surface area of a cylinder is equal to the curved surved surface area of a cone .

Exercise - 10.1 5. A Self help group wants to manufacture joker’s cap ( Conical caps) of 3cm radius and 4 cm height . If the available color paper sheet is 1000 cm2 , then how many caps can be manufactured from that paper sheet ? Solution: Radius of Joker,s cap ( Conical cap ) Height Slant height = Curved Surface area of conical Cap Colour paper required to manufacture one Joker;s conical cap = 47.14cm2 Number of caps can be manufactured from 1000 cm2 colour paper sheet

Exercise - 10.1 6. A Cylinder and cone have bases of equal radii and arc of equal heights . Show that their volumes are in the ratio of 3:1 Solution : Given that a cylinder and cone have bases of equal radii Let radii is r Also given that a cylinder and cone have equal heights. Let heights be h Volume of a cylinder = Volume of a cone = The ratio of their volumes

Exercise - 10.1 7. A Solid Iron rod has a cylinderical shape. Its height is 11 cm and base diameter is 7 cm . Then find the total volume of 50 rods ? Solution : Height of the cylinderical shape Solid Iron rod Base diameter of a cylinderical shape solid Iron rod Radius of Cylinderical shape solid Iron rod Volume of cylinderical shape solid Iron rod = Volume of 50 cylinderical shape solid Iron rods =

Exercise - 10.1 8. A heap of rice is in the form of a cone of diameter 12 m and height 8 m . Find its volume ? How much canvas cloth is required to cover the heap ? Solution : diameter of a heap of rice which is in the form of cone Radius Height of a heap of rice Slant height Volume of a heap of rice = Lateral Surface area of the heap of rice which is in the form of cone

Exercise - 10.1 9. The curved surface area of a cone is 4070 cm2 and its diameter is 70 cm . What is its slant height ? Solution : diameter of a cone Radius Let Slant height of the cone be l Curved surface area of cone

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