Topic 6 of 7

Circles: circumference and area

Step 9 · Proportional reasoning topics

Name the radius and diameter, see why pi is the same for every circle, and use C = πd and A = πr² to find circumference and area.

Radius, diameter and pi

The radius is the distance from the center of a circle to its edge. The diameter goes right across through the center, so it is twice the radius.

A circle with radius 4: the diameter, 8, goes right across through the center.

Pi, written π, is how many times the diameter fits around the circle. It is the same for every circle: about 3.14, or roughly 22/7.

Circle	Around	Across	Around ÷ across
Coin	7.5 cm	2.4 cm	about 3.1
Plate	81.7 cm	26 cm	about 3.14
Wheel	188.5 cm	60 cm	about 3.14

Circumference

Because π is the circumference divided by the diameter, the circumference is π times the diameter.

C = \pi d = 2\pi r

Example: From the diameter

A wheel has a diameter of 50 cm. How far does it roll in one turn? Use 3.14.

C = πd.
3.14 × 50 = 157.

Answer157 cm

Example: From the radius

A circle has a radius of 3 m. What is its circumference? Give it with π, then use 3.14.

C = 2πr, which is 2 × 3 = 6 lots of π: 6π.
6 × 3.14 = 18.84.

Answer6π m, about 18.84 m

Example: Working backwards

A circle's circumference is 31.4 cm. What is its diameter? Use 3.14.

d = C ÷ π.
31.4 ÷ 3.14 = 10.

Answer10 cm

Area of a circle

Cut a circle into thin slices and lay them head to tail. They make a shape close to a rectangle: half the circumference (πr) long and one radius high. So the area is πr × r.

A = \pi r^{2}

The slices of a circle of radius 4, laid head to tail: nearly a rectangle about 12.6 long (πr) and 4 high (r).

Example: Area from the radius

What is the area of a circle with a radius of 5 cm? Use 3.14.

A = πr², and 5² = 25, so A = 25π.
25 × 3.14 = 78.5.

Answer78.5 cm²

Example: Area from the diameter

A round rug has a diameter of 6 ft. What is its area? Use 3.14.

The radius is 6 ÷ 2 = 3.
3² = 9, so A = 9π.
9 × 3.14 = 28.26.

Answer28.26 ft²

Try it yourself

Work each one on paper first, then open the answer to check.

A circle has a diameter of 14 m. What is its radius?
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7 m
What is the circumference of a circle with a diameter of 20 cm? Use 3.14.
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62.8 cm
What is the circumference of a circle with a radius of 7 in? Use $\frac{22}{7}$ .
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44 in
What is the area of a circle with a radius of 10 m? Use 3.14.
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314 m²
What is the area of a circle with a diameter of 8 ft? Give it with π.
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16π ft²
A circle's circumference is 12.56 cm. What is its radius? Use 3.14.
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2 cm: the diameter is 12.56 ÷ 3.14 = 4.