Area of a sector: formula, calculator and worked example

What the symbols mean

$A$: the area of the sector
$r$: the radius of the circle
$θ$: the central angle, in degrees or radians (use the toggle)

When to use it

Use it for slices and wedges: a slice of pizza or cake, the area a windscreen wiper or a lawn sprinkler covers, a pie chart segment. Together with arc length it gives the perimeter of a sector ( $2r + s$ ).

Worked example

A cake of radius 6 cm is cut into a slice with an angle of 90°. What is the area of the top of the slice? (r = 6, θ = 90°)

Start from the formula with θ in degrees.
$A = \frac{θ}{360} π r^{2}$
Substitute $r$ = $6$ , $θ$ = $90 °$ and work it out.
$A = \frac{90}{360} π \cdot 6^{2}$
$= 0.25 π \cdot 36$
$\approx 28.2743$

Answer A ≈ 28.27 cm²

Why it works

A sector with angle θ is the fraction $θ/360$ of the whole circle, so its area is that fraction of $π r^{2}$ . In radians the fraction is $θ/2π$ , which simplifies to $12/r²θ$ .

Common mistakes

Using degrees in $12/r²θ$ . That formula needs radians.
Forgetting to square the radius.
Confusing the sector (the whole wedge) with the segment (the part between the chord and the arc).

Area of a sector

Calculator

See it move

What the symbols mean

When to use it

Worked example

Why it works

Common mistakes