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Circle Calculator

Calculate circle area, circumference, diameter, radius, arc length, and sector area with step-by-step geometric derivations.

Circle Calculations
Area
78.5398
Circumference
31.4159
Diameter
10
Sector Area (90°)
19.635
Step-by-Step Mathematical Deduction:
Given radius r = 5
Diameter: d = 2r = 2 × 5 = 10
Circumference: C = 2πr = 2 × π × 5 = 31.4159
Area: A = πr² = π × (5)² = 78.5398
For central angle θ = 90°:
Arc Length: s = (θ / 360°) × 2πr = (90 / 360) × 31.4159 = 7.854
Sector Area: A_sector = (θ / 360°) × πr² = (90 / 360) × 78.5398 = 19.635
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What Is a Circle Calculator?

A circle is the set of all points in a two-dimensional plane that are equidistant from a central point. The distance from the center to any point on the boundary is the radius (r), and the boundary distance is the circumference (C).

How to Use This Calculator

  1. Enter the circle radius (or adjust central angle for arc/sector calculations).
  2. Instantly view the calculated Area (A), Circumference (C), and Diameter (d).
  3. Review the step-by-step mathematical substitutions using π ≈ 3.14159265.

Circle Area and Circumference Formulas

A = \pi r^2, \quad C = 2\pi r = \pi d, \quad A_{\text{sector}} = \frac{\theta}{360^\circ}\pi r^2

Area is calculated by multiplying pi by radius squared. Circumference is 2 times pi times radius. Sector area is the proportion of the central angle theta over 360 degrees.

Worked Example

Scenario: Find area and circumference of a circle with radius r = 7 cm

Diameter: d = 2 × 7 = 14 cm

Circumference: C = 2 × π × 7 ≈ 43.9823 cm

Area: A = π × (7)² = 49π ≈ 153.938 cm²

Result: Area ≈ 153.94 cm², Circumference ≈ 43.98 cm

Tips & Key Notes

  • If you know diameter instead of radius, simply divide diameter by 2.
  • To find radius from circumference, divide circumference by 2π (approx. 6.28318).
  • To find radius from area, divide area by π, then take the square root.

Frequently Asked Questions

What is the relationship between diameter and radius?

The diameter is exactly twice the length of the radius: d = 2r, or r = d / 2.

How do you find the area of a circle sector?

Multiply the full circle area (πr²) by the fraction of the circle subtended by the central angle: (θ / 360°).

What is arc length?

Arc length is the curved distance along the circumference corresponding to a central angle: s = (θ / 360°) × 2πr.

Why is pi used in circle calculations?

Pi (π ≈ 3.14159) represents the universal ratio of any circle’s circumference to its diameter (C / d).

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