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English Medium / Class 7 / Math / Mensuration - Volume and Surface area, Cuboid, Cube, Cylinder, Cone, Sphere
Mensuration - Volume and Surface area, Cuboid, Cube, Cylinder, Cone, Sphere
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Introduction

Mensuration is the branch of mathematics that deals with the measurement of 3D objects. Every object around us, like a box or a ball, occupies space and has a surface. We measure this using Volume and Surface Area.

 

 

Explanation Step by Step

To calculate these values: 1. Identify the shape. 2. Note the given dimensions (length, radius, height). 3. Convert all units to be same. 4. Apply the formula.

Sub-topics

1. Cuboid

A cuboid is a box-like object with 6 rectangular faces. It is defined by Length (L), Breadth (B), and Height (H).

Examples

Example 1 (Real-life)
Find the volume of a mobile phone box with dimensions 15 cm x 8 cm x 5 cm.
Volume = 15 x 8 x 5 = 600 cubic cm.
Tricky Example
If the length and breadth are doubled but height remains same, how does volume change?
The volume becomes 4 times the original (2L x 2B x H = 4 LBH).

2. Cube

A cube is a special cuboid where all sides are equal (L=B=H=a).

Examples

Example 1
Find the surface area of a cube with side 10 cm.
Area = 6 * 102 = 600 sq cm.

3. Cylinder

A cylinder has a circular base and a fixed height. E.g., a pencil or a soda can.

Examples

Example 1
Calculate the curved surface area of a pipe with radius 3.5 cm and length 20 cm.
Area = 2πrh = 2 * (22/7) * 3.5 * 20 = 440 sq cm.

4. Cone

A cone narrows smoothly from a flat base to a point called the apex.

Examples

Example 1
A conical tent is 12 m high and its radius is 5 m. Find the slant height.
Slant height = √122 + 52 = 13 m.

5. Sphere

A sphere is a perfectly round geometrical object in 3D space.

Examples

Example 1
Find the volume of a football with radius 7 cm.
Volume = (4/3) πr3 = (4/3) * (22/7) * 73 ≈ 1437.33 cubic cm.

Tricks and Shortcuts

  • For a Cube, if you know the diagonal d, the Volume is (d3) / (3 √3).
  • The surface area of a sphere is exactly the same as the curved surface area of a cylinder that just fits around it.

Common Mistakes

  • Confusing radius with diameter. Always divide diameter by 2 before putting it in formulas.
  • Mixing units (e.g., using cm for radius and meters for height).

Practice Questions

Easy Questions

  1. Find the volume of a cube with side 2 cm.
  2. A cylinder has radius 7 cm and height 1 cm. Find its volume.
  3. (Tricky) How many vertices does a sphere have?

Medium Questions

  1. Calculate the total surface area of a cylinder with radius 7 cm and height 10 cm.
  2. (Tricky Real-life) How much gift wrap (area) is needed to cover a cube of side 5 cm?
  3. Find the volume of a cone if its base area is 154 sq cm and height is 9 cm.

Hard Questions

  1. (Tricky) A sphere is melted to form a cone of the same radius. Find the ratio of the height of the cone to the radius.
  2. Find the cost of painting a hemispherical bowl of radius 7 cm at 5 per sq cm.
  3. If a cuboid of 10x8x6 cm is cut into 1 cm cubes, how many such cubes are formed?

Revision Summary

Volume is space inside; Surface Area is the outer skin. Remember the relationship between Cylinder and Cone (1:3 ratio) and keep your units consistent!

If you double the side of a cube, how much will its volume be?

A
    
2 times
B
    
4 times
C
    
 8 times
D
    
16 times
Explaination

The volume of a cuboid is 1200cm³. If its length is 15 cm and breadth is 10 cm, what is its height?

A
    
12 cm
B
    
10 cm
C
    
8 cm
D
    
15 cm
Explaination
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