Area of circles

Part ofMathsPerimeter, Area, Volume

Key points

A equals pi r squared. Written below: pi r squared equals pi multiplied by r squared, equals pi multiplied by r multiplied by r. The third r squared, and the r multiplied by r, are coloured blue.
The formula for finding the area of a circle is 𝑨 = π𝒓²

  • 𝑨 = π𝒓² is the for finding the area of a circle. π is a and 𝒓 is the of the circle.

  • The of a circle is halved to give the radius. The value of π may be rounded to 3۰142 or 3۰14. The π button on a calculator is used in calculations. The area is given in including cm² and m².

  • A scientific calculator may be set to show answers in notation. Understanding how to use a calculator to convert between surd notation and the decimal equivalent is important.

  • The ability to round a number to a number of decimal places or to a number of significant figures is essential if answers are to be written to the correct degree of accuracy asked for in a question.

  • It may also be helpful to understand π (pi) and the circumference of a circle.

A equals pi r squared. Written below: pi r squared equals pi multiplied by r squared, equals pi multiplied by r multiplied by r. The third r squared, and the r multiplied by r, are coloured blue.
The formula for finding the area of a circle is 𝑨 = π𝒓²
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Calculating the area of a circle

To illustrate the formula for the area of a circle:

  1. A circle can be cut into .

  2. The sectors can be rearranged to form an rectangle shape. The more sectors the closer to a rectangle the shape will become.

  3. The length of the rectangle is half the , π𝒓. The width of the rectangle is the radius of the circle, 𝒓. The area of the rectangle is length × width, π𝒓 × 𝒓 = π𝒓².

  4. The formula for the area of the circle is 𝑨 = π𝒓².

To work out the area of a circle:

  1. Find the value of the radius. If the diameter is given, this must be halved.

  2. Substitute the value of the radius into the formula for the area of a circle.

  3. Multiply π by the square of the radius. This is the same as π × 𝒓 × 𝒓.

  4. If necessary, round to the .

  5. If using a scientific calculator, use the S

    ⇔

    D button to convert a value given in terms of π to a decimal.

Example

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  1. The image shows two shapes. The first shape is a circle that has been subdivided into sectors. Seven sectors are eighths of the circle. Four are coloured orange and three are coloured blue. The final eighth has been split into two sixteenths. These two sectors are coloured green. The second diagram shows the same nine sectors rearranged into an approximate rectangle. The direction of the sectors alter nate, up and down. They have been placed in the following order, based on their colour: green, orange, blue, orange, blue, orange, blue, orange, green.

    A circle can be cut into sectors and rearranged into an approximate rectangle.

Question

Find the area of the circle, using π = 3۰142. Round the answer to three significant figures.

An image of a circle. The diameter has been labelled as forty centimetres. Written top left: Use pi equals three point one four two. The circle is coloured blue.

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Calculating the radius and diameter of a circle, when given its area

  • The operations used to find the area are squaring the radius, then multiplying by π. The of multiply by π is divide by π. The inverse of square is .

  • To find the radius from the area of a circle, use the inverse operations:

    • Divide the area by π.

    • Find the square root.

  • To find the diameter, multiply the radius by 2

  • It may be useful to look at changing the subject.

Example

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  1. A equals pi r squared. Written right: A flow diagram representing the function. Reading left to right: r, right arrow, square, right arrow, multiplied by pi, right arrow, A. Written below: A flow diagram representing the inverse function. Reading right to left: A, left arrow, divide by pi, left arrow, square root, left arrow, r. Written beneath: r, equals, A over pi, square rooted. The r, and the r squared are coloured blue.

    To find the area of a circle, the radius is squared and multiplied by π. The inverse of this is to divide by π and find the square root.

Question

The circle has an area of 56 mm². Find the radius of the circle and give the answer to two decimal places.

An image of a circle. Inside the circle, the area has been labelled as fifty six millimetres squared. Written top left: Use pi equals three point one four two or the pi button on a calculator. The circle is coloured blue.

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Practise finding the area of a circle

Practise finding the area of a circle with this quiz. You may need a pen and paper to help you.

Quiz

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