Circumference and Area of Circles
Grade 7 · Math · Study Guide
Use pi to calculate circumference and area of circles from radius or diameter measurements.
A circle is the set of all points that are the same distance from the center. That distance from center to edge is the radius (r). The diameter (d) is the distance across the circle through the center, so d = 2r. The perimeter of a circle has a special name — circumference — and it is calculated using the number pi (π ≈ 3.14159).
The circumference formula is C = πd or C = 2πr. Pi is the ratio of any circle's circumference to its diameter — this ratio is always the same no matter how big or small the circle. Pi is an irrational number, meaning its decimal goes on forever without repeating, but we usually approximate it as 3.14 or use the fraction 22/7.
The area of a circle is the amount of space inside it, measured in square units. The formula is A = πr². To use it, you square the radius first, then multiply by π. For example, a circle with radius 5 cm has area A = π × 5² = π × 25 ≈ 78.54 cm².
A common mistake is mixing up radius and diameter. Always identify which one you are given before plugging into a formula. If you are given the diameter, divide by 2 to get the radius before using A = πr².
Circles show up everywhere: wheels, pipes, coins, planets, pizza, eyes, and sports fields. Every time an engineer designs a circular pipe or a baker cuts a round cake, these formulas are doing the work. Understanding circles also leads directly to volume formulas for cylinders and spheres that come up in 8th grade.
| Measurement | Formula | What You Need |
|---|---|---|
| Circumference | C = πd | Diameter |
| Circumference | C = 2πr | Radius |
| Area | A = πr² | Radius |
| Diameter from radius | d = 2r | Radius |
| Radius from diameter | r = d ÷ 2 | Diameter |
| Given | Find | Work | Answer |
|---|---|---|---|
| r = 6 cm | Circumference | C = 2π(6) = 12π ≈ 37.70 cm | ≈ 37.70 cm |
| d = 10 m | Circumference | C = π(10) = 10π ≈ 31.42 m | ≈ 31.42 m |
| r = 4 ft | Area | A = π(4²) = 16π ≈ 50.27 ft² | ≈ 50.27 ft² |
| d = 14 cm | Area | r = 7; A = π(7²) = 49π ≈ 153.94 cm² | ≈ 153.94 cm² |
Key terms
- Circle
- The set of all points in a plane equidistant from a fixed center point
- Radius (r)
- The distance from the center of a circle to any point on its edge
- Diameter (d)
- The distance across a circle through the center; d = 2r
- Circumference (C)
- The distance around a circle; C = πd or C = 2πr
- Pi (π)
- The ratio of circumference to diameter for any circle; approximately 3.14 or 22/7
- Area (A)
- The amount of space inside a circle; A = πr²
Practice questions
A circle has a radius of 7 cm. What is its circumference? (Use π ≈ 3.14)
- 21.98 cm
- 43.96 cm
- 153.86 cm
- 78.50 cm
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Answer: 43.96 cm
The formula for the area of a circle is A = πd.
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Answer: False
A circle has a diameter of 20 cm. Its radius is ___ cm.
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Answer: 10
A circle has a radius of 5 m. What is its area? (Use π ≈ 3.14)
- 15.70 m²
- 31.40 m²
- 78.50 m²
- 157.00 m²
Show answer
Answer: 78.50 m²
Pi (π) is the ratio of a circle's circumference to its diameter.
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Answer: True
Fun facts
- Pi has been calculated to over 100 trillion decimal places by computers — it never ends and never repeats, making it one of the most fascinating numbers in all of mathematics.
- The ancient Egyptians estimated pi as (16/9)² ≈ 3.1605 over 3,800 years ago, getting within 1% of the true value using only rope and stakes.
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