Circumference and Area of Circles and Composite 2D Shapes

Radius, diameter and the boundary of a circle

A circle has a centre, and every point on its curved boundary is the same distance from that centre. That distance is the radius, $r$. The diameter, $d$, crosses the whole circle through its centre, so it contains two radii:

$$d=2r \qquad\text{and}\qquad r=\frac d2$$

This relationship matters because a question may give the diameter while the required formula uses the radius.

Circumference: the perimeter of a circle

For a polygon, perimeter is found by adding the lengths of its outside edges. A circle has no straight edges, so its perimeter has the special name circumference.

For every circle, dividing its circumference by its diameter gives the same constant, $\pi$. This explains the formula

$$C=\pi d.$$

Because $d=2r$, the same formula can also be written as

$$C=2\pi r.$$

Choose the version that matches the information given. For example, a circle of diameter $10$ cm has circumference $10\pi$ cm. The same circle has radius $5$ cm, so $2\pi r=2\pi\times5=10\pi$ cm as expected.

An answer such as $10\pi$ cm is exact. If a decimal is requested, use the calculator's $\pi$ button and round only after completing the calculation. Since circumference is a distance, its unit is a linear unit such as mm, cm or m.

The formula can also be rearranged. If $C$ is known, then

$$r=\frac{C}{2\pi}\qquad\text{or}\qquad d=\frac{C}{\pi}.$$

Area enclosed by a circle

Area measures the space inside a boundary. The area of a circle is

$$A=\pi r^2.$$

The radius is squared: $r^2$ means $r\times r$. If a circle has radius $4$ cm, then

$$A=\pi\times4^2=16\pi\text{ cm}^2.$$

This is approximately $50.3\text{ cm}^2$ to three significant figures. Area is measured in square units because it describes a two-dimensional region.

If the diameter is given, first use $r=d/2$. For a diameter of $24$ cm, the radius is $12$ cm, so the area is $\pi\times12^2=144\pi\text{ cm}^2$. Using would make the answer four times too large because .

The area formula can be reversed when the area is known:

$$r^2=\frac A\pi$$

and therefore

$$r=\sqrt{\frac A\pi}.$$

The positive square root is used because a radius is a length.

Parts of circles

A semicircle is half a circle. Its area is therefore

$$\frac12\pi r^2.$$

Its curved edge is half a circumference, so its length is $\pi r$. However, the complete perimeter of a semicircle also contains its straight diameter:

$$P=\pi r+2r.$$

A quarter-circle has area $\frac14\pi r^2$ and a curved edge of length $\frac14(2\pi r)=\frac12\pi r$. Its complete perimeter includes two radii, giving

$$P=\frac12\pi r+2r.$$

This distinction between an arc length and the whole perimeter is essential: a perimeter must form one complete trip around the boundary.

Perimeters of composite shapes

A composite shape is assembled from familiar shapes such as rectangles, triangles and parts of circles. To find its perimeter, imagine tracing a finger once around the outside. Add every exposed straight or curved section exactly once.

Edges where two component shapes have been joined are internal, so they do not belong to the outside perimeter. For example, when a semicircle is attached along one side of a rectangle, that shared diameter is hidden. The semicircular arc may be part of the perimeter, but the joined diameter is not.

Some side lengths may need to be deduced before adding. In a rectilinear shape, the total distance travelled to the right must equal the total travelled to the left, and the total distance upwards must equal the total downwards. These facts can supply missing lengths.

Areas of composite shapes

There are two main strategies.

Split and add: divide the region into non-overlapping familiar shapes, calculate each area and add the results.

Enclose and subtract: calculate the area of a larger simple shape and subtract a missing or unshaded part.

Choose divisions that create shapes whose dimensions can be found. If a rectangle has a semicircle of diameter $12$ cm attached to it, the semicircle's radius is $6$ cm, not $12$ cm. Its area is therefore $\frac{}{}$, which can then be added to the rectangle's area.

When a circular piece is cut out, subtract its area instead. The structure of a calculation should match the diagram: added material gives addition; a hole or removed region gives subtraction. Do not subtract a boundary length when the question asks for area, or add areas when it asks for perimeter.

Exact answers, decimals and checking

“Give your answer in terms of $\pi$” means leave $\pi$ in the answer, such as $36\pi\text{ cm}^2$. This is exact. If the question requests a number of decimal places or significant figures, evaluate using the $\pi$ button and round at the end.

Check that the answer has the correct type of unit and a sensible size. Rounding the measurements to one significant figure can give a quick estimate. A circle of radius about $10$ cm should have circumference about $2\times3\times10=60$ cm and about . An answer far from these estimates suggests that the may have been used as the radius or that part of a composite boundary has been missed.

Circle facts

$$d=2r,\qquad r=\frac d2$$ $$C=$$