In this lesson, we will learn how to find the area and the circumference of a circle given its radius or diameter and how to relate both the area and circumference to solve various problems.

Students will be able to

Q1:

A circle has an area of 40 cm^{2}.

Calculate the radius of the circle accurate to three decimal places.

Use this value to calculate the circumference of the circle, giving your answer accurate to one decimal place.

Q2:

With circles 𝑀 and 𝑁 touching at 𝐴, the shaded area is 331 cm^{2}. Given that 𝑀𝑁=3.7cm, determine the sum of the two radii correct to the nearest hundredth.

Q3:

Emma is interested in finding the formula for the area of a circle. She has studied how to find the circumference of a circle and knows the formula 𝑐=2𝜋𝑟. She has seen a picture of a series of concentric circles and wants to use that to work out the area of a circle. She has a series of circular rings made of string and organizes them as seen in the given picture. She then cuts them at a single point and places them, keeping the distances between the rings the same as they were in the circle, to form the triangle shown.

What can be said about the radius of the orginal circle and the height of the triangle?

The base of the triangle must be equal to the circumference of the circle. If the circle has radius 𝑟, what is the length of the base of the triangle?

In terms of 𝑟 and 𝜋, work out the area of the triangle. Fully simplify your result.

After this experiment, what formula do you think Emma decided to use to work out the area of a circle?

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