Area of a Sector of a Circle

Q: How do you find the area of a sector when given arc length instead of the angle?

If you know the arc length $s$ and the radius $r$, first find the central angle in radians using $\theta = \frac{s}{r}$. Then substitute into $A = \frac{1}{2} r^2 \theta$. This simplifies to $A = \frac{1}{2} r s$, which is a useful shortcut.

Q: Why does the radian formula for sector area look like $\frac{1}{2} r^2 \theta$?

A full circle has an angle of $2\pi$ radians and an area of $\pi r^2$. The sector is the fraction $\frac{\theta}{2\pi}$ of the full circle. Multiplying gives $\frac{\theta}{2\pi} \cdot \pi r^2 = \frac{1}{2} r^2 \theta$. The $\pi$ cancels neatly, which is one reason radians are so convenient.

Area of a Sector of a Circle

The formula is given below.

Sector of a circle diagram with radius r and angle θ. Area = (1/2)r²θ (radians) = (θ/360)πr² (degrees).

See also

Sector of a circle, radians, degrees

Key Formula

A = \frac{\theta}{360} \cdot \pi r^2 \quad \text{(degrees)} \qquad \text{or} \qquad A = \frac{1}{2} r^2 \theta \quad \text{(radians)}

Where:

$A$ = Area of the sector
$r$ = Radius of the circle
$\theta$ = Central angle of the sector (in degrees for the first formula, in radians for the second)
$\pi$ = The constant pi, approximately 3.14159

Worked Example

Problem: Find the area of a sector with a central angle of 90° and a radius of 10 cm.

Step 1: Identify the given values. The central angle is 90° and the radius is 10 cm.

\theta = 90°, \quad r = 10 \text{ cm}

Step 2: Write the sector area formula for degrees.

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

Step 3: Substitute the values into the formula.

A = \frac{90}{360} \cdot \pi (10)^2 = \frac{1}{4} \cdot 100\pi

Step 4: Simplify to get the exact area, then approximate.

A = 25\pi \approx 78.54 \text{ cm}^2

Answer: The area of the sector is

25\pi \approx 78.54

cm².

Another Example

This example uses the radian version of the formula, whereas the first example used degrees. It demonstrates that you must match the formula to the angle unit.

Problem: Find the area of a sector with a central angle of

\frac{\pi}{3}

radians and a radius of 6 m.

Step 1: Identify the given values. The angle is in radians, so use the radian formula.

\theta = \frac{\pi}{3}, \quad r = 6 \text{ m}

Step 2: Write the sector area formula for radians.

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

Step 3: Substitute the values into the formula.

A = \frac{1}{2} (6)^2 \cdot \frac{\pi}{3} = \frac{1}{2} \cdot 36 \cdot \frac{\pi}{3}

Step 4: Simplify the expression.

A = \frac{36\pi}{6} = 6\pi \approx 18.85 \text{ m}^2

Answer: The area of the sector is

6\pi \approx 18.85

m².

Frequently Asked Questions

What is the difference between the area of a sector and the area of a segment?

A sector is the 'pie slice' shape bounded by two radii and an arc. A segment is the region between a chord and the arc it cuts off. To find a segment's area, you calculate the sector area and then subtract the area of the triangle formed by the two radii and the chord.

How do you find the area of a sector when given arc length instead of the angle?

If you know the arc length

s

and the radius

r

, first find the central angle in radians using

\theta = \frac{s}{r}

. Then substitute into

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

. This simplifies to

A = \frac{1}{2} r s

, which is a useful shortcut.

Why does the radian formula for sector area look like

\frac{1}{2} r^2 \theta

A full circle has an angle of

2\pi

radians and an area of

\pi r^2

. The sector is the fraction

\frac{\theta}{2\pi}

of the full circle. Multiplying gives

\frac{\theta}{2\pi} \cdot \pi r^2 = \frac{1}{2} r^2 \theta

. The

\pi

cancels neatly, which is one reason radians are so convenient.

Area of a Sector vs. Arc Length of a Sector

	Area of a Sector	Arc Length of a Sector
What it measures	The two-dimensional region inside the sector (in square units)	The length of the curved edge of the sector (in linear units)
Formula (degrees)	$A = \frac{\theta}{360} \cdot \pi r^2$	$s = \frac{\theta}{360} \cdot 2\pi r$
Formula (radians)	$A = \frac{1}{2} r^2 \theta$	$s = r\theta$
Units	Square units (cm², m², etc.)	Linear units (cm, m, etc.)
Relationship	$A = \frac{1}{2} r s$	$s = \frac{2A}{r}$

Why It Matters

You encounter sector area problems throughout geometry, trigonometry, and calculus courses, especially when working with circles and polar coordinates. In real life, sector area applies to situations like calculating the area a sprinkler covers when it sweeps through a fixed angle, or determining how much material you need for a fan-shaped piece of fabric or metal. It also builds the foundation for understanding integration in polar form, where you compute areas using

\frac{1}{2}\int r^2\,d\theta

Common Mistakes

Mistake: Using the degree formula with an angle measured in radians (or vice versa).

Correction: Always check the unit of your angle first. If the angle is in degrees, use

\frac{\theta}{360} \cdot \pi r^2

. If it is in radians, use

\frac{1}{2} r^2 \theta

. Mixing these up gives an answer that is off by a large factor.

Mistake: Forgetting to square the radius in the formula.

Correction: The sector area formula involves

r^2

, not

r

. Using

r

alone gives you a quantity related to arc length, not area. Double-check that you are squaring the radius before multiplying.

Related Terms

Sector of a Circle — The shape whose area this formula computes
Radian — Angle unit used in the radian form of the formula
Degree — Angle unit used in the degree form of the formula
Arc Length — Length of the curved boundary of a sector
Area of a Circle — Sector area is a fraction of the full circle area
Central Angle — The angle that determines the sector's size
Formula — General term for mathematical equations like this one
Segment of a Circle — Related region found by subtracting a triangle from a sector