Circle Calculator
Find area, circumference, diameter, and radius — enter any one value.
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Circle Formulas
A circle is a perfectly round shape where every point on its boundary is the same distance from its center. The key measurements are the radius (r), diameter (d), circumference (C), and area (A), all related by the constant π (pi ≈ 3.14159).
Key Relationships
Diameter = 2 × radius (d = 2r). Circumference = π × diameter = 2πr. Area = π × r². These four values are all linked — knowing any one lets you calculate the other three.
What Is Pi (π)?
Pi is the ratio of a circle's circumference to its diameter: π = C/d. It is the same for every circle, regardless of size: π ≈ 3.14159265. Pi is irrational — it never ends and never repeats. The first 10 digits are 3.1415926535.
Sector and Arc
A sector is a "pie slice" of a circle with central angle θ (in degrees). Sector area = (θ/360) × πr². Arc length = (θ/360) × 2πr.
Real-World Applications
Circle calculations appear in engineering (pipe cross-sections, wheel rotations), architecture (dome footprints), physics (circular motion, wave patterns), and everyday life (pizza size, clock faces, coins, tires).
Worked Example: Finding All Values from Circumference
Suppose you measure a tree trunk and find the circumference is 94.25 cm. Here is how to find all other measurements step by step:
Step 1 — Find the radius: r = C / (2π) = 94.25 / (2 × 3.14159) = 94.25 / 6.28318 ≈ 15 cm.
Step 2 — Find the diameter: d = 2r = 2 × 15 = 30 cm.
Step 3 — Find the area: A = π × r² = 3.14159 × 15² = 3.14159 × 225 ≈ 706.86 cm².
So a tree with a 94.25 cm circumference has a radius of 15 cm, a diameter of 30 cm, and a cross-sectional area of about 706.86 cm².
Quick Reference: Common Circle Values
| Radius (r) | Diameter (d) | Circumference (C) | Area (A) |
|---|---|---|---|
| 1 | 2 | 6.283 | 3.142 |
| 3 | 6 | 18.850 | 28.274 |
| 5 | 10 | 31.416 | 78.540 |
| 7 | 14 | 43.982 | 153.938 |
| 10 | 20 | 62.832 | 314.159 |
Frequently Asked Questions
A = π×5² = 25π ≈ 78.54 square units.
C = πd = 10π ≈ 31.42 units.
r = C/(2π). If C=31.42: r ≈ 31.42/(2×3.14159) ≈ 5.
r = √(A/π). If A=78.54: r = √(78.54/π) ≈ 5.
d = 2r. The diameter is a chord that passes through the center, equal to twice the radius.
A semicircle is half of a full circle, so its area is (π × r²) / 2. For radius 6: area = (π × 36) / 2 ≈ 56.55 square units. Its perimeter (straight edge + curved edge) = 2r + πr = r(2 + π).
Sector area = (θ / 360) × π × r², where θ is the central angle in degrees. For a 90° sector with radius 10: area = (90/360) × π × 100 = 25π ≈ 78.54 sq units. This is exactly one quarter of the full circle area.
Doubling the radius quadruples the area. Since A = π r², if you replace r with 2r you get π(2r)² = 4πr². For example, a circle with r=5 has area ≈ 78.54, but a circle with r=10 has area ≈ 314.16 — exactly four times as large.