Maharashtra Board Class 10 Maths Part II Chapter 7 Mensuration PDF Download

Mensuration

Mixed Examples On Surface Area And Volume Of Different Solid Figures

Last year we studied surface area and volume of some three dimensional figures. Let us recall the formulae to find the surface areas and volumes.

Teacher's Note

These formulas help us find how much space is inside shapes and how much paper we need to cover them. For example, a water tank at home is a cylinder, so we use the cylinder formula to find how much water it can hold.

Exam Trick

Remember the word "CBSCH" - Cuboid, Box (cube), Sphere, Cylinder, Hemisphere. Learn one formula for each shape and you can solve any problem.

Points To Remember

Surface area tells us how much area is on the outside of a shape.
Volume tells us how much space is inside a shape.
Lateral surface area is only the curved or side part, not the top and bottom.
Slant height is different from height in cones - slant height is along the slope.
For 3D shapes, always check if you need area or volume before solving.

Solve The Following Examples

Ex. (1) The length, breadth and height of an oil can are 20 cm, 20 cm and 30 cm respectively as shown in the adjacent figure. How much oil will it contain? (1 litre = 1000 cm³)

Ex. (2) The adjoining figure shows the measures of a Joker's cap. How much cloth is needed to make such a cap?

Let's Think

As shown in the adjacent figure, a sphere is placed in a cylinder. It touches the top, bottom and the curved surface of the cylinder. If radius of the base of the cylinder is 'r',

(1) What is the ratio of the radii of the sphere and the cylinder?

(2) What is the ratio of the curved surface area of the cylinder and the surface area of the sphere?

(3) What is the ratio of the volumes of the cylinder and the sphere?

Teacher's Note

When a sphere fits perfectly inside a cylinder touching all sides, it shows how different shapes can be related. You can see this in toys and decorative items at home.

Exam Trick

If a sphere is inside a cylinder and touches it, the diameter of sphere = diameter of cylinder. The height of cylinder = diameter of sphere. Use this to find ratios quickly.

Points To Remember

A sphere inside a cylinder makes two shapes equal in one direction.
When shapes are related like this, their ratios have a pattern.
Radius of sphere = radius of cylinder when sphere touches the curved surface.
Height of cylinder = 2 times radius when sphere touches top and bottom.
Use these facts to solve ratio problems faster.

Activity

As shown in the above figures, take a ball and a beaker of the same radius as that of the ball. Cut a strip of paper of length equal to the diameter of the beaker. Draw two lines on the strip dividing it into three equal parts. Stick it on the beaker straight up from the bottom. Fill water in the beaker upto the first mark of the strip from the bottom. Push the ball in the beaker slowly so that it touches its bottom. Observe how much the water level rises.

You will notice that the water level has risen exactly upto the total height of the strip.

Try to understand how we get the formula for the volume of a the sphere. The shape of the beaker is cylindrical.

Therefore, the volume of the part of the beaker upto height 2r can be obtained by the formula of volume of a cylinder. Let us assume that the volume is V.

\(V = \pi r^2 h = \pi \times r^2 \times 2r = 2\pi r^3\) \((\because h = 2r)\)

But V = volume of the ball + volume of the water which was already in the beaker.

= volume of the ball + \(\frac{1}{3} \times 2\pi r^3\)

\(\therefore\) volume of the ball = \(V - \frac{1}{3} \times 2\pi r^3 = 2\pi r^3 - \frac{2}{3}\pi r^3 = \frac{6\pi r^3 - 2\pi r^3}{3} = \frac{4\pi r^3}{3}\)

Hence we get the formula of the volume of a sphere as \(V = \frac{4}{3}\pi r^3\)

(Now you can find the answer of the question number 3 relating to the examples given above)

Teacher's Note

This activity shows a real experiment. When you pour water in a glass with a ball, the water level rises by the volume of the ball. This is how we measure volume of irregular shapes.

Exam Trick

The water displacement method is the easiest way to find volume. If water rises by 100 ml when you put a ball in it, the ball's volume is 100 ml. Remember this for any object.

Points To Remember

Volume of sphere = \(\frac{4}{3}\pi r^3\) - learn this formula by heart.
A cylinder with height 2r and radius r can contain exactly one sphere of radius r and another \(\frac{2}{3}\) of it.
Water displacement shows volume practically.
The formula comes from the relationship between cylinder and sphere volumes.
This method works for finding volume of any irregular object.

This is a preview of the first 3 pages. To get the complete book, click below.