Math

Circle Area, Perimeter, and Sector Calculator

Q: What's the circle area formula?

**Area = π × r²**, where r is the radius. If you know the diameter (d), radius is d/2, so: Area = π × (d/2)² = π × d² / 4. For a radius-10 circle: Area = π × 100 ≈ **314.16**.

Q: What's the difference between radius and diameter?

The **radius** is the distance from the center to the circle edge. The **diameter** is the distance from one side to the other passing through the center = **2 × radius**. A radius-5 circle has diameter 10.

Q: How do I calculate a circular sector area?

**Sector area = (angle/360) × π × r²**. For example, a 45° sector in a radius-10 circle: (45/360) × π × 100 = (1/8) × 314.16 = **39.27**. It's calculating what fraction of the full circle the sector occupies.

Q: What is arc length?

The **portion of circumference** corresponding to a given angle. **Arc length = (angle/360) × 2πr**. For a 90° arc in a radius-10 circle: (90/360) × 2π×10 = (1/4) × 62.83 = **15.71**.

Q: Can I calculate the radius if I know the area?

**Yes**: r = √(Area / π). If area is 100: r = √(100 / 3.1416) = √31.83 ≈ **5.64**. And if you know the perimeter: r = Perimeter / (2π).

Q: Why does area grow with the square of the radius?

Because area is a **two-dimensional** measure. Doubling the radius expands in 2 directions simultaneously (width and height), so area multiplies by 2² = 4. A radius-10 circle has 4 times the area of a radius-5 circle.

Q: How many pizza slices come from a 45° sector?

A 45° sector is **1/8 of the circle** (360°/45° = 8). So from a round pizza you get **8 slices** if you cut 45° sectors. Each slice's area = π × r² / 8.

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The circle is the most perfect geometric shape, and its properties appear everywhere: from wheels and gears to planetary orbits and wave signals. This calculator takes the radius and an optional angle (for sectors) and returns the area, perimeter (circumference), sector area, and arc length. If you don't enter an angle (or enter 360°), it calculates the complete circle. Ideal for geometry problems, design, engineering, and any calculation involving circles.

Last reviewed: May 12, 2026 Verified by Hacé Cuentas Team Source: Wolfram MathWorld — Circle, Khan Academy — Circle Geometry 100% private

When to use this calculator

You need to calculate the area of a circle for a geometry problem.
You want to know the circumference of a wheel or circular part.
You're calculating the area of a circular sector (a slice of pizza).
You need the arc length for design or engineering.
You want to verify manual geometry calculations with circle formulas.

Example: circle of radius 5 with 90° sector

Radius: 5. Angle: 90°.
Total area: π × 5² = π × 25 = 78.54.
Perimeter: 2 × π × 5 = 31.42.
Sector area (90°): (90/360) × 78.54 = 19.63.
Arc length: (90/360) × 31.42 = 7.85.

Result: A circle of radius 5 has area 78.54 and perimeter 31.42. The 90° sector (a quarter of the circle) has area 19.63 and arc 7.85.

How it works

1 min read

Fundamental Circle Formulas

Area

The area enclosed by a circle of radius r:

Area = π × r²

Where π (pi) ≈ 3.14159265358979...

Perimeter (circumference)

The total length of the circle's border:

Perimeter = 2 × π × r = π × d

Where d = 2r is the diameter.

Circular Sector Area

A sector is the 'slice' of the circle between two radii forming angle θ:

Sector Area = (θ/360°) × π × r²

For θ in radians: Sector Area = (θ/2) × r².

Arc Length

The portion of circumference corresponding to the sector:

Arc Length = (θ/360°) × 2 × π × r

For θ in radians: Arc Length = θ × r.

The Number π

π is the ratio between a circle's circumference and diameter. It's an irrational number (infinite non-repeating decimals) and transcendental (not the root of any polynomial with integer coefficients).

First 20 digits: 3.14159265358979323846...

For practical calculations, π ≈ 3.1416 is enough.

Quick Reference Table

Radius	Area	Perimeter
1	3.14	6.28
2	12.57	12.57
5	78.54	31.42
10	314.16	62.83
50	7,853.98	314.16
100	31,415.93	628.32

Area grows with the square of the radius (doubling radius quadruples area), while perimeter grows linearly.

Sector Applications

1. Pie charts: each sector represents a proportion of the total.
2. Pizza slices: calculating slice area based on number of cuts.
3. Engineering: bridge arches, cams, gears.
4. Astronomy: observation angles, orbital swept areas (Kepler's 2nd law).
5. Surveying: curved terrain area calculations.

Degrees vs. Radians

360° = 2π radians
1° = π/180 radians ≈ 0.01745 rad
1 radian = 180°/π ≈ 57.296°

In sector formulas, you can use degrees or radians as long as you're consistent. This calculator uses degrees because it's more intuitive.

Frequently asked questions

What's the circle area formula?

Area = π × r², where r is the radius. If you know the diameter (d), radius is d/2, so: Area = π × (d/2)² = π × d² / 4. For a radius-10 circle: Area = π × 100 ≈ 314.16.

What's the difference between radius and diameter?

The radius is the distance from the center to the circle edge. The diameter is the distance from one side to the other passing through the center = 2 × radius. A radius-5 circle has diameter 10.

How do I calculate a circular sector area?

Sector area = (angle/360) × π × r². For example, a 45° sector in a radius-10 circle: (45/360) × π × 100 = (1/8) × 314.16 = 39.27. It's calculating what fraction of the full circle the sector occupies.

What is arc length?

The portion of circumference corresponding to a given angle. Arc length = (angle/360) × 2πr. For a 90° arc in a radius-10 circle: (90/360) × 2π×10 = (1/4) × 62.83 = 15.71.

Can I calculate the radius if I know the area?

Yes: r = √(Area / π). If area is 100: r = √(100 / 3.1416) = √31.83 ≈ 5.64. And if you know the perimeter: r = Perimeter / (2π).

Why does area grow with the square of the radius?

Because area is a two-dimensional measure. Doubling the radius expands in 2 directions simultaneously (width and height), so area multiplies by 2² = 4. A radius-10 circle has 4 times the area of a radius-5 circle.

How many pizza slices come from a 45° sector?

A 45° sector is 1/8 of the circle (360°/45° = 8). So from a round pizza you get 8 slices if you cut 45° sectors. Each slice's area = π × r² / 8.