Circle Formula: A = πr², C = 2πr (13 Formulas + Calculator)

Area of a circle = π × r² (A = πr²), where r is the radius. The circumference (also called the perimeter) of a circle = 2π × r = π × d, where d is the diameter. These two — plus d = 2r — are the three formulas every circle problem reduces to. Below are all 13 circle formulas you'll need: area, perimeter / circumference, radius, diameter, arc length, sector area, chord length, segment area, the inscribed-angle theorem, and the analytic-geometry equation of a circle. Each comes with a worked example.

The Formulas

Name	Formula	Notes
Area (from radius)	`A = π × r²`	r = radius. The classic "area of a circle formula".
Area (from diameter)	`A = π × d² / 4`	Use when you only know the diameter. Derived from A = πr² with r = d/2.
Circumference	`C = 2π × r = π × d`	Sometimes called the circle perimeter formula — both names refer to this.
Perimeter of a Circle	`P = 2π × r`	Identical to circumference. "Perimeter" + "circumference" are synonyms for circles.
Diameter	`d = 2 × r`	Twice the radius. Also d = C/π if you know circumference.
Radius (from area)	`r = √(A / π)`	Inverse of A = πr². Useful when area is given.
Radius (from circumference)	`r = C / (2π)`	Inverse of C = 2πr. Common in real-world measurements.
Sector Area	`A_s = ½ × r² × θ`	θ in radians. For degrees: A_s = (θ°/360) × πr².
Arc Length	`L = r × θ`	θ in radians. For degrees: L = (θ°/360) × 2πr.
Chord Length	`c = 2r × sin(θ/2)`	θ = central angle subtending the chord. Useful for inscribed shapes.
Segment Area	`A_seg = ½ × r² × (θ − sin θ)`	θ in radians. The region between a chord and the arc.
Inscribed Angle	`∠inscribed = ½ × ∠central`	An inscribed angle is half the central angle subtending the same arc.
Equation of a Circle	`(x − h)² + (y − k)² = r²`	Center at (h, k), radius r. The analytic-geometry standard form.

Worked Examples

Example 1: Find area + circumference of a circle with radius 5 cm

Area: A = π × 5² = 25π ≈ 78.54 cm²
Circumference (= perimeter): C = 2π × 5 = 10π ≈ 31.42 cm
Diameter: d = 2 × 5 = 10 cm

Example 2: Find the radius from area = 50 cm²

Start from A = π × r² → r² = A / π = 50 / π ≈ 15.915
r = √15.915 ≈ 3.99 cm
Then diameter = 2 × 3.99 ≈ 7.98 cm, and circumference = 2π × 3.99 ≈ 25.07 cm

Example 3: Find the area when you only know the diameter (d = 12 cm)

Use the diameter formula A = π × d² / 4
A = π × 12² / 4 = π × 144 / 4 = 36π
A ≈ 113.10 cm²

Example 4: Find the arc length and sector area for a 60° sector in a circle with r = 10 cm

Convert to radians: θ = 60° × (π/180) = π/3 ≈ 1.0472 rad
Arc length: L = r × θ = 10 × π/3 ≈ 10.47 cm
Sector area: A_s = ½ × r² × θ = ½ × 100 × π/3 ≈ 52.36 cm²
Cross-check via degrees: A_s = (60/360) × π × 10² = (1/6) × 100π ≈ 52.36 cm² ✓

Frequently Asked Questions

What is the formula for the area of a circle?

The area of a circle is A = π × r², where r is the radius. If you only know the diameter d, use the equivalent form A = π × d² / 4. π (pi) is approximately 3.14159.

Is the perimeter of a circle the same as its circumference?

Yes. For a circle, "perimeter" and "circumference" mean the same thing — the distance around the outside. Both equal 2π × r (or π × d). The word "perimeter" is more common in school textbooks; "circumference" is the technical term used in geometry.

How do you find the radius of a circle from the circumference?

Divide the circumference by 2π: r = C / (2π). For example, a circle with circumference 31.42 cm has radius ≈ 31.42 / 6.2832 ≈ 5 cm.

How do you find the area of a circle from the diameter?

Use A = π × d² / 4. The diameter is squared, multiplied by π, and divided by 4. Alternatively, halve the diameter to get the radius and use A = πr². Both formulas give the same answer.

What does π (pi) mean in circle formulas?

π is the ratio of any circle's circumference to its diameter (π = C/d ≈ 3.14159). This ratio is identical for every circle, no matter the size, which is why every circle formula contains π.

How do you find the area of a sector of a circle?

Sector area = (θ/360) × π × r² when θ is the central angle in degrees, or A_s = ½ × r² × θ when θ is in radians. For example, a 90° sector in a circle of radius 4 has area = (90/360) × π × 16 = 4π ≈ 12.57.

Circle Formulas