Circumference and Area of Circles

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The circumference and area of a circle come from its radius, r. The circumference is the distance around the circle: C = 2\pi r, or C = \pi d using the diameter. The area is the space inside: A = \pi r^2. The number \pi (pi) is about 3.14, so a circle with a radius of 5 has a circumference of about 31.4.

Before this lesson: You should know perimeter, the area of a parallelogram and exponents. It also helps to know that pi is a irrational number.

Parts of a Circle

  • The center is the point in the middle. Every point on the circle is the same distance from it.
  • The radius (r) is the distance from the center to the edge.
  • The diameter (d) goes all the way across, through the center. It is twice the radius: d = 2r.
  • The circumference (C) is the distance around the circle. It is the circle’s perimeter.
Left: a circle with its center marked, a radius r from the center to the edge, a diameter d across the whole circle, and the circumference C labeled around the outside. Right: the circle cut into 8 wedges laid side by side, tips up and down, forming a shape close to a parallelogram that is about πr long and r tall.
The radius goes from the center to the edge, and the diameter is twice as long. Cut into wedges, a circle becomes almost a parallelogram πr long and r tall.

What Pi Means

For every circle, big or small, the circumference divided by the diameter is the same number. That number is called pi, written \pi.

Pi is about 3.14159. Its decimal never ends or repeats. In this lesson we use \pi \approx 3.14 unless a problem says otherwise. You can also leave answers “in terms of \pi,” like 10\pi. That is the exact answer.

Circumference and Area of a Circle Formulas

FindFormulaUnits
Circumference from the radiusC = 2\pi rlength (cm, ft)
Circumference from the diameterC = \pi dlength (cm, ft)
AreaA = \pi r^2square units (cm², ft²)

Why is the area \pi r^2? Cut a circle into thin wedges and line them up, tips up and down, as in the picture above. They make a shape close to a parallelogram. Its height is r, and its length is half the circumference, \pi r. So the area is \pi r \times r = \pi r^2.

Worked Examples

Example 1: Circumference from the radius

A circle has a radius of 5 cm. Find its circumference.

Step 1. Use C = 2\pi r = 2 \times \pi \times 5 = 10\pi.

Step 2. Use \pi \approx 3.14: 10 \times 3.14 = 31.4.

Answer: C = 10\pi \approx 31.4 cm

Example 2: Circumference from the diameter

A bike wheel has a diameter of 12 inches. Find its circumference.

Step 1. Use C = \pi d = 12\pi.

Step 2. 12 \times 3.14 = 37.68.

Answer: C = 12\pi \approx 37.68 in

Example 3: Area from the radius

A circle has a radius of 6 m. Find its area.

Step 1. Square the radius first: 6^2 = 36.

Step 2. Multiply by pi: A = 36\pi \approx 36 \times 3.14 = 113.04.

Answer: A = 36\pi \approx 113.04 m²

Example 4: Area from the diameter

A round table has a diameter of 10 ft. Find its area.

Step 1. Find the radius: r = 10 \div 2 = 5 ft.

Step 2. A = \pi r^2 = \pi \times 5^2 = 25\pi.

Step 3. 25 \times 3.14 = 78.5.

Answer: A = 25\pi \approx 78.5 ft²

Example 5: Word problem with pizza sizes

Is one 12-inch pizza more food than two 8-inch pizzas? Pizza sizes give the diameter.

Step 1. The 12-inch pizza has a radius of 6 in. Its area is \pi \times 6^2 = 36\pi \approx 113.04 in².

Step 2. Each 8-inch pizza has a radius of 4 in. Its area is \pi \times 4^2 = 16\pi. Two of them make 32\pi \approx 100.48 in².

Step 3. Compare: 113.04 - 100.48 = 12.56 in².

Answer: The one 12-inch pizza is more food, by about 12.56 in².

Common Mistakes with Circles

MistakeExample of the errorFix
Using the diameter in \pi r^2A 10 ft table: \pi \times 10^2 = 100\piHalve the diameter first: r = 5, so A = 25\pi.
Doubling instead of squaring\pi \times 6^2 = \pi \times 126^2 = 6 \times 6 = 36, so A = 36\pi.
Mixing up the formulasUsing 2\pi r to find areaCircumference is around (2\pi r). Area is inside (\pi r^2).
Squaring after multiplying by pi(\pi \times 6)^2Square only the radius: \pi \times 6^2.
Wrong unitsWriting the area as 113.04 mArea uses square units: 113.04 m². Circumference uses length units.

Working Backward from Circumference or Area

If you know the circumference, divide by pi to get the diameter: d = C \div \pi. A circle with C = 31.4 cm has d = 31.4 \div 3.14 = 10 cm.

If you know the area, divide by pi, then take the square root: r = \sqrt{A \div \pi}. A circle with A = 78.5 ft² has r^2 = 25, so r = 5 ft.

Practice Problems

  1. Find the circumference of a circle with radius 3 cm. Use 3.14 for pi.
  2. Find the circumference of a circle with diameter 20 in.
  3. Find the area of a circle with radius 4 m.
  4. Find the area of a circle with diameter 14 ft.
  5. Find the area of a circle with radius 2.5 cm. Round to the nearest hundredth.
  6. A circle has a circumference of 43.96 in. Find its diameter.
  7. A circle has an area of 28.26 m². Find its radius.
  8. Find the area of a half circle with radius 10 cm.
  9. A wheel has a diameter of 26 in. How far does it roll in 100 full turns?
  10. A round pond has a radius of 8 m. Fencing costs $5 per meter. How much does it cost to fence around the pond?
Show answers
  1. C = 2\pi(3) = 6\pi \approx 18.84 cm
  2. C = 20\pi \approx 62.8 in
  3. A = \pi(4^2) = 16\pi \approx 50.24 m²
  4. r = 7, so A = 49\pi \approx 153.86 ft²
  5. A = \pi(2.5^2) = 6.25\pi \approx 19.63 cm²
  6. d = 43.96 \div 3.14 = 14 in
  7. r^2 = 28.26 \div 3.14 = 9, so r = 3 m
  8. A half circle is half the area: \frac{1}{2} \times \pi \times 10^2 = 50\pi \approx 157 cm²
  9. One turn is 26\pi in. 100 turns: 2600\pi \approx 8164 in.
  10. C = 2\pi(8) = 16\pi \approx 50.24 m, and 50.24 \times 5 = 251.20. It costs $251.20.

Circumference and Area of a Circle FAQ

What is the formula for the area of a circle?

The area of a circle is A = \pi r^2. Square the radius, then multiply by pi.

What is the formula for circumference?

The circumference is C = 2\pi r, or C = \pi d if you know the diameter. Both give the same answer, because d = 2r.

What is the difference between circumference and area?

Circumference is the distance around the circle, in length units. Area is the space inside the circle, in square units.

Should I use 3.14 or the π button?

Use what your teacher or problem asks for. The \pi button gives a more exact answer, so it may differ slightly from an answer made with 3.14.

Summary

  • The radius goes from the center to the edge, and the diameter is twice the radius.
  • Circumference: C = 2\pi r = \pi d.
  • Area: A = \pi r^2. Halve a diameter before you square it.
  • \pi \approx 3.14. An answer like 25\pi is exact.
  • Circumference uses length units. Area uses square units.

Previous lesson: Area of Trapezoids

Next lesson: Area of Composite Figures

Back to the Pre-Algebra Index

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