Maharashtra Board Class 7 Maths part 2 Chapter 11 Circle PDF Download

Circle

Let's Recall

Identify the radii, chords and diameters in the circle alongside and write their names in the table below.

Circumference Of A Circle

Activity I

Place a cylindrical bottle on a paper and trace the outline of its base. Use a thread to measure the circumference of the circle.

Activity II

Measure the circumference of a bangle with the help of a thread.

Activity III

Measure the circumference of any cylindrical object using a thread.

Teacher's Note

You can use any round object like a plate or a glass. Your students can measure around it with a string and compare with the diameter.

Exam Trick

Remember: Circumference is the distance around a circle. It is like the perimeter of a circle.

Points to Remember

Circumference means the distance around a circle.
We use a thread to measure the circumference.
We can use any round object to do this activity.

Let's Learn

Relationship Between Circumference And Diameter

Measure the circumference and diameter of the objects given below and enter the ratio of the circumference to its diameter in the table.

Examine the ratio of the circumference to the diameter. What do we see?

The ratio of the circumference of any circle to its diameter is a little over 3 and remains constant. This constant is represented by the Greek letter \(\pi\). Great mathematicians have proved through hard work that this number is not a rational number. In practice, the value of \(\pi\) is taken to be \(\frac{22}{7}\) or 3.14. If the value of \(\pi\) has not been given in a problem, it is taken to be \(\frac{22}{7}\).

If radius is 'r', diameter 'd' and circumference 'c', then \(\frac{\text{circumference(c)}}{\text{diameter(d)}} = \pi\) so \(c = \pi d\).

But \(d = 2r\) therefore \(c = \pi \times 2r\) or \(c = 2\pi r\).

Example 1

The diameter of a circle is 14 cm. Find its circumference.

Solution: Diameter: \(d = 14\) cm

Circumference \(= \pi d\)

\(c = \frac{22}{7} \times 14\)

Circumference of the circle = 44 cm

Example 2

The radius of a circle is 35 cm. Find its circumference.

Solution: Radius of the circle \(r = 35\) cm

Circumference \(= 2\pi r\)

\(c = 2 \times \frac{22}{7} \times 35\)

Circumference of the circle = 220 cm

Example 3

The circumference of a circle is 198 cm. Find its radius and diameter.

Solution: Circumference \(c = 2\pi r\)

\(198 = 2 \times \frac{22}{7} \times r\)

\(r = 198 \times \frac{1}{2} \times \frac{7}{22}\)

Radius = 31.5 cm

Therefore Diameter \(= 2 \times 31.5 = 63\) cm

Example 4

The circumference of a circle is 62.80 cm. Taking \(\pi = 3.14\), find its diameter.

Solution: Circumference \(c = \pi d\)

\(62.80 = 3.14 \times d\)

\(\frac{62.80}{3.14} = d\)

\(20 = d\)

Therefore Diameter = 20 cm

Teacher's Note

In real life, you use circumference when you buy a belt or a circular mat. You need to know the distance around it.

Exam Trick

Remember: If you have radius, use \(c = 2\pi r\). If you have diameter, use \(c = \pi d\). Both give the same answer.

Points to Remember

\(\pi = \frac{22}{7}\) or 3.14
Circumference formula with radius: \(c = 2\pi r\)
Circumference formula with diameter: \(c = \pi d\)
The ratio of circumference to diameter is always \(\pi\).

Example 5

The radius of a circular plot is 7.7 metres. How much will it cost to fence the plot with 3 rounds of wire at the rate of 50 rupees per metre?

Solution: Circumference of circular plot \(= 2\pi r = 2 \times \frac{22}{7} \times 7.7 = 48.4\) m

Length of wire required for one round of fencing = 48.4 m.

Cost of one round of fence = length of wire \(\times\) cost per metre.

\(= 48.4 \times 50 = 2420\) rupees.

Cost of 3 rounds of fencing \(= 3 \times 2420 = 7260\) rupees

When finding the ratio of like terms, their units must be the same. 22 km \(= 22 \times 1000 = 22000\) m.

Example 6

The radius of the wheel of a bus is 0.7 m. How many rotations will a wheel complete while travelling a distance of 22 km?

Solution: Circumference of circle \(= \pi d\)

\(= \frac{22}{7} \times 0.7\)

\(= 2.2\) m

When the wheel completes one rotation it crosses a distance of 2.2 m., (1 rotation = 1 circumference)

Total number of rotations \(= \frac{\text{distance}}{\text{circumference}} = \frac{22000}{2.2} = \frac{220000}{22} = 10000\)

A wheel completes 10000 rotations to cover the distance of 22 km.

Teacher's Note

When a wheel moves, it travels the distance equal to its circumference in one rotation. This is why bigger wheels move faster than smaller wheels.

Exam Trick

Remember: One rotation = one circumference. Just divide total distance by circumference to get the number of rotations.

Points to Remember

Circumference is the distance around a circle.
One rotation of a wheel = one circumference.
Number of rotations = total distance divided by circumference.
Always change km to m when comparing measurements.

Practice Set 42

1. Complete the table below.

2. If the circumference of a circle is 176 cm, find its radius.

3. The radius of a circular garden is 56 m. What would it cost to put a 4-round fence around this garden at a rate of 40 rupees per metre?

4. The wheel of a bullock cart has a diameter of 1.4m. How many rotations will the wheel complete as the cart travels 1.1 km?

Let's Recall

Arc Of The Circle

A plastic bangle is shown alongside. Suppose it breaks at points A and B. What is each of these pieces called as a part of a circle?

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