Sector Area Formula

A sector is a “pie slice” of a circle bounded by two radii and an arc. Its area is A = (θ/360)·πr² for a central angle θ in degrees, or A = ½·r²·θ in radians — the fraction of the whole circle’s area.

Sector Area formulas

Sector area (degrees): $A = \dfrac{\theta}{360}\cdot \pi r^2$
Sector area (radians): $A = \tfrac{1}{2} r^2 \theta$
Arc length: $L = \dfrac{\theta}{360}\cdot 2\pi r$

where r = radius, θ = central angle (degrees), A = sector area

Sector area (degrees)

A = \dfrac{\theta}{360}\cdot \pi r^2

Worked example

A quarter sector of a radius-6 circle has area 9π ≈ 28.27 square units.

A = \dfrac{90}{360}\cdot \pi\cdot 6^2 = \tfrac14\cdot 36\pi = 9\pi \approx 28.27

Sector area (radians)

A = \tfrac{1}{2} r^2 \theta

Arc length

L = \dfrac{\theta}{360}\cdot 2\pi r

Interactive sector area explorer

Drag a slider (or type a value, or use the arrow keys) and watch the diagram, the results, and the full step-by-step substitution update live.

Radius r

Central angle θ θ

Sector area A 28.27units²

Arc length L 9.42units

Step-by-step substitution

A = (θ/360) · πr² = (90/360) · π · 6² = 28.27
arc L = (θ/360) · 2πr = 9.42

Classroom use: sector area formulas

Who it's for: Grades 9–12 · High-school geometry students learning sector area, typically straight after arc length.

Learning objectives

After using this page, students will be able to:

Compute sector area from a radius and central angle.
Relate sector area to total circle area by the fraction θ/360.
Distinguish sector area from arc length, including their differing units.

Suggested classroom activity

Ask for the area of a quarter sector of radius 6 before showing the explorer, then check the 9π result together. Running arc length and sector area side by side is what stops students conflating the two — one is a length, the other an area.

Standards alignment

Aligned to the Common Core State Standards for Mathematics. Codes are given exactly as published so they can be matched against state crosswalks.
Code	Standard
`HSG.C.B.5`	Derive the arc-length and sector-area relationships using similarity.
`7.G.B.4`	Know and use the formulas for the area and circumference of a circle.

Free to use and adapt in class under the terms on our license page — no account, no sign-up, no ads. Content last reviewed August 2026.

Related & next steps

Add sector area to your formula sheet →

Frequently asked questions

What is the formula for the area of a sector?

A sector’s area is A = (θ/360)·πr² with θ in degrees. It is the fraction θ/360 of the full circle area πr². A 90° sector is one quarter of the circle.

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