Lesson Worksheet: Areas and Circumferences of Circles Mathematics

In this worksheet, we will practice finding the area and the circumference of a circle given its radius or diameter and relating both the area and circumference to solve various problems.


A circle has an area of 40 cm2.

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.


With circles 𝑀 and 𝑁 touching at 𝐴, the shaded area is 331 cm2. Given that 𝑀𝑁=3.7cm, determine the sum of the two radii correct to the nearest hundredth.


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?

  • AThe height is half the radius.
  • BThe radius is longer than the height.
  • CThey are equal.
  • DThe radius is shorter than the height.
  • EThe height is twice the radius.

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?

  • A2𝑟𝜋
  • B𝑟𝜋
  • C2𝜋𝑟
  • D2𝜋𝑟
  • E𝜋𝑟

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

  • A𝐴=𝜋𝑟
  • B𝐴=𝜋𝑟
  • C𝐴=𝜋𝑟+2
  • D𝐴=(𝜋+1)𝑟
  • E𝐴=2𝜋𝑟

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

  • A𝐴=2𝜋𝑟
  • B𝐴=𝜋𝑟+2
  • C𝐴=(𝜋+1)𝑟
  • D𝐴=𝜋𝑟
  • E𝐴=𝜋𝑟


Given that the area of a circle is 49,896 cm2, find its radius. (Taking 𝜋=227).


Find the area between two concentric circles of radii 9 cm and 13 cm respectively. Express you answer in terms of 𝜋.

  • A250𝜋 cm2
  • B81𝜋 cm2
  • C88𝜋 cm2
  • D169𝜋 cm2


Using 3.14 as an approximation to 𝜋, calculate the area of the shaded region.


A goat is tethered by a 10-meter long rope to the corner of a barn. What area of the field can the goat reach?

  • A3254𝜋 m2
  • B(100𝜋−60) m2
  • C75𝜋 m2
  • D100𝜋 m2
  • E80𝜋 m2


The bull’s eye on the given archery target has a radius of 3 in. The entire target has a radius of 15 in. Determine the area of the target outside of the bull’s eye rounded to the nearest square inch.


Using 3.14 in place of 𝜋, determine the radius of a circle whose area is 153.86.


A circle has a circumference of 90 cm.

Work out the square of the circumference.

  • A8,100
  • B180
  • C16,200
  • D810
  • E90

Divide the square of the circumference by 4𝜋, giving your answer accurate to two decimal places.

  • A1,289.16
  • B14.32
  • C644.58
  • D64.46
  • E7.16

Work out the radius of the circle to three decimal places.

Work out the area of the circle. Round your answer to two decimal places.

  • A644.58 cm2
  • B481.68 cm2
  • C4,050.00 cm2
  • D340.65 cm2
  • E1,289.16 cm2

This lesson includes 10 additional questions and 123 additional question variations for subscribers.

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