Area of a Sector

The area of a sector can be calculated in two different ways. If you know the radius (r) of the circle and the arc length (l), use the formula $$ A_{sector} = \frac{l \cdot r}{2} $$ If the central angle (α) is known and expressed in radians, you can use the equivalent formula $$ A_{sector} = \frac{\alpha \cdot r^2}{2} $$ Both formulas give exactly the same result.

Worked Example

Let's calculate the area of a sector in a circle with radius r = 3 and central angle α = 28.44°.

The first step is to convert the angle from degrees to radians.

$$ \alpha = 28.44^\circ \cdot \frac{\pi}{180^\circ} = 0.5 \text{ rad} $$

Important. The formulas for the area of a sector require the angle to be measured in radians. If you use degrees, the result will be incorrect.

Now calculate the arc length AB.

$$ AB = \alpha \cdot r $$

$$ AB = 0.5 \cdot 3 = 1.5 $$

So the arc length is l = 1.5.

Once you know the arc length and the radius, finding the area is straightforward.

$$ A_{sector} = \frac{l \cdot r}{2} $$

$$ A_{sector} = \frac{1.5 \cdot 3}{2} = 2.25 $$

Therefore, the area of the sector is 2.25 square units.

Note. You can also calculate the area directly from the central angle and the radius:

$$ A_{sector} = \frac{\alpha \cdot r^2}{2} = \frac{0.5 \cdot 3^2}{2} = \frac{4.5}{2} = 2.25 $$

As expected, both formulas produce the same result.

Why Does the Formula Work?

The formula for the area of a sector comes from a simple geometric observation.

A sector occupies the same fraction of the circle's area as its central angle occupies of the full angle around the center. Since a complete circle corresponds to an angle of 2π radians, we can write:

$$ A_{sector} : A_{circle} = \alpha : 2\pi $$

Or, in fraction form:

$$ \frac{A_{sector}}{A_{circle}} = \frac{\alpha}{2\pi} $$

Solving for the area of the sector gives:

$$ A_{sector} = \frac{\alpha}{2\pi} \cdot A_{circle} $$

Since the area of a circle is πr^2:

$$ A_{sector} = \frac{\alpha}{2\pi} \cdot \pi r^2 $$

After canceling π, we obtain:

$$ A_{sector} = \frac{\alpha r^2}{2} $$

This is the standard formula used when the central angle is known in radians.

To derive the alternative form, recall the definition of a radian:

$$ \alpha = \frac{l}{r} $$

Substituting this expression into the previous formula gives:

$$ A_{sector} = \frac{1}{2} \cdot \frac{l}{r} \cdot r^2 $$

Simplifying the expression, we obtain:

$$ A_{sector} = \frac{1}{2} \cdot l \cdot r $$

which is usually written as:

$$ A_{sector} = \frac{l \cdot r}{2} $$

This shows that the two formulas are equivalent. You can use whichever form is more convenient, depending on the information available.