Geometria

Circumference and Area of a Circle: Formulas and Solved Exercises

The formulas for circumference (C = 2πr) and area of a circle (A = πr²), how to work backwards to find the radius, and five solved exercises with verified step-by-step answers.

Recommended for: Grade 8 · Grade 9

The circumference of a circle is C = 2πr (or C = πd), and the area is A = πr². You only really need one number, the radius: halve the diameter if that is what you are given, or work backwards from the circumference or the area.

Circumference and area answer different questions

The circumference is the line, the border. The area is the surface that line encloses. The distinction sounds fussy until a problem asks you how much fencing a round flower bed needs versus how much soil fits inside it. The units are the reminder: cm for circumference, cm² for area.

The two formulas and their inverses

There are two forward formulas, but exam questions almost always need them backwards too:

  • C = 2πr → r = C / (2π)
  • A = πr² → r = √(A / π)
  • d = 2r → r = d / 2

When a problem gives you the area and asks for the circumference (or the other way round), the route is always the same: go back to the radius first, then move forward with the other formula. The radius is the bridge.

Working with π

π is irrational, with infinitely many non-repeating decimals. In middle school and early high school 3.14 is the usual approximation. One practical tip: carry π as a symbol through all your steps and substitute 3.14 only at the end, so rounding errors do not pile up.

Sectors and rings

A sector is a slice of the circle: its arc and its area are the same fraction of the whole. If the central angle is α, that fraction is α/360. An annulus is a ring instead, and its area is the big circle minus the small one, that is π(R² − r²).

The exercises below follow the same pattern at increasing difficulty: the radius is handed to you at first, then you need the inverse formulas and a rolling wheel, and finally a sector and a ring. Every result has been checked by redoing the arithmetic.

Solved exercises

1. A circle has a radius of 5 cm. Find its circumference and its area (use π ≈ 3.14). base

Show solution
  1. The circumference is C = 2πr.
  2. C = 2 × 3.14 × 5 = 31.4 cm.
  3. The area is A = πr², so first r² = 5 × 5 = 25.
  4. A = 3.14 × 25 = 78.5 cm².

Answer: C = 31.4 cm ; A = 78.5 cm²

2. A bicycle wheel has a diameter of 70 cm. How many metres does the bike travel in 100 full turns of the wheel? (π ≈ 3.14) base

Show solution
  1. In one full turn the wheel moves forward by exactly its circumference.
  2. Since the diameter is given, use C = πd.
  3. C = 3.14 × 70 = 219.8 cm.
  4. For 100 turns: 219.8 × 100 = 21,980 cm.
  5. Convert to metres by dividing by 100: 21,980 ÷ 100 = 219.8 m.

Answer: 219.8 m

3. The circumference of a circle is 43.96 cm. Find the radius and the area (π ≈ 3.14). intermedio

Show solution
  1. Start from C = 2πr and rearrange: r = C / (2π).
  2. 2π = 2 × 3.14 = 6.28.
  3. r = 43.96 ÷ 6.28 = 7 cm.
  4. Check: 6.28 × 7 = 43.96 cm, correct.
  5. Now the area: A = πr² = 3.14 × 7² = 3.14 × 49 = 153.86 cm².

Answer: r = 7 cm ; A = 153.86 cm²

4. In a circle of radius 12 cm, consider a sector with a central angle of 45°. Find the arc length and the area of the sector (π ≈ 3.14). intermedio

Show solution
  1. A 45° sector is a fraction of the whole circle: 45/360 = 1/8.
  2. Whole circumference: C = 2 × 3.14 × 12 = 75.36 cm.
  3. The arc is one eighth of it: 75.36 ÷ 8 = 9.42 cm.
  4. Whole area: A = 3.14 × 12² = 3.14 × 144 = 452.16 cm².
  5. The sector is one eighth of it: 452.16 ÷ 8 = 56.52 cm².

Answer: arc = 9.42 cm ; sector area = 56.52 cm²

5. An annulus (ring) is formed by an outer circle of radius 10 cm and an inner circle of radius 6 cm. Find the area of the ring and what percentage it is of the outer circle (π ≈ 3.14). avanzato

Show solution
  1. The area of the ring is the difference of the two areas: A = πR² − πr² = π(R² − r²).
  2. R² − r² = 100 − 36 = 64.
  3. A = 3.14 × 64 = 200.96 cm².
  4. Area of the outer circle: 3.14 × 100 = 314 cm².
  5. Percentage: 200.96 ÷ 314 = 0.64, that is 64%.

Answer: A = 200.96 cm² ; 64% of the outer circle

FAQ

What is the difference between circumference and area?

The circumference is the curved line around the circle, its boundary; the area is the whole surface enclosed by that line. That is why circumference is measured in cm and area in cm².

When should I use 3.14 and when should I leave π in the answer?

If the question asks for a numerical value, use 3.14 (or 3.1416 for more precision). If it asks for an exact answer, keep π: for example A = 25π cm² instead of 78.5 cm².

How do I find the radius if I know the area?

Rearrange A = πr² to get r = √(A/π). For example, if A = 78.5 cm², then 78.5 ÷ 3.14 = 25 and r = √25 = 5 cm.