Sector Area

A sector is the pizza-slice region between two radii and their arc. Its area works like the arc length from Arc Length, but uses the area of the whole circle, $\pi r^2$, from Area of a Circle.

Why it works

A central angle of $\theta$ degrees takes $\frac{\theta}{360}$ of the full turn, so its sector holds $\frac{\theta}{360}$ of the circle's area. In radians the full turn is $2\pi$, so the fraction is $\frac{\theta}{2\pi}$ and the area is $\frac{\theta}{2\pi} \cdot \pi r^2 = \frac{1}{2}r^2\theta$.

The red arc is a length; the blue sector is an area. Both are the same fraction of their whole.

Key idea

$$\text{sector area} = \frac{\theta}{360} \cdot \pi r^2 \quad (\theta \text{ in degrees})$$ $$\text{sector area} = \frac{1}{2} r^2 \theta \quad (\theta \text{ in radians})$$

Worked example: A sector

Find the area of a $60^\circ$ sector of a circle with radius $9$.

$\frac{60}{360} = \frac{1}{6}$ of the circle, whose area is $81\pi$.
$\frac{1}{6} \cdot 81\pi = \frac{27\pi}{2} \approx 42.4$.

Answer: $\frac{27\pi}{2}$

Worked example: Arc length and area together

A sector has radius $10$ and a $72^\circ$ central angle. Find its arc length and its area.

Fraction: $\frac{72}{360} = \frac{1}{5}$.
Arc length: $\frac{1}{5} \cdot 2\pi \cdot 10 = 4\pi$ (units of length).
Area: $\frac{1}{5} \cdot \pi \cdot 100 = 20\pi$ (square units).

Answer: Arc $4\pi$; area $20\pi$

Worked example: A circular segment

Find the area between a $90^\circ$ arc of a circle with radius $6$ and the chord that joins its ends.

Sector: $\frac{90}{360} \cdot \pi \cdot 36 = 9\pi$.
Triangle formed by the two radii: $\frac{1}{2} \cdot 6 \cdot 6 = 18$.
Segment = sector minus triangle: $9\pi - 18 \approx 10.27$.

Answer: $9\pi - 18$

Two related questions come up often. The perimeter of a sector is two radii plus the arc: a $90^\circ$ sector of radius $4$ has perimeter $4 + 4 + 2\pi = 8 + 2\pi$. A circular segment, the region between a chord and its arc, is the sector minus the triangle formed by the two radii and the chord. For that same sector, the segment's area is $4\pi - 8 \approx 4.57$.

Common mistake

Mixing up sector area and arc length. Arc length uses $2\pi r$ and is a length (cm); sector area uses $\pi r^2$ and is an area (cm$^2$). Check the units, and check that a bigger radius makes the area grow faster than the arc (by $r^2$ versus $r$).

Sector Area

Practice

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