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What Is the Area of a Circular Sector?

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Here you will learn about circular sectors and learn how to calculate how many degrees a circular sector has. You can combine this with the area of a circle and have what you need to calculate the area of a circular sector.

You already know that there are $360$ $^{\circ}$ in a circle. A circular sector is a portion of an entire circle. It is expressed using an angle, $v^{\circ}$ .

Formula

Area of a Circular Sector Expressed by Angle $v^{\circ}$

A = \frac{v^{\circ}}{360^{\circ}} \cdot π \cdot r^{2}

where $v^{\circ}$ is the angle, which tells you how large the part of circle’s area you’re calculating, is.

Example 1

What is the area of a circular sector at $60^{\circ}$ with a radius of $3 cm$ ?

We put $v^{\circ} = 60^{\circ}$ and $r = 3$ into the formula and calculate:

A = \frac{60^{\circ}}{360^{\circ}} \cdot π \cdot 3^{2} \approx \frac{1}{6} \cdot 3.14 \cdot 9 = 4.71 {cm}^{2}

Example 2

What is the area of a circular sector at $50^{\circ}$ with a radius of $4 cm$ ?

We put $v^{\circ} = 50$ and $r = 4$ into the formula and calculate:

A = \frac{50^{\circ}}{360^{\circ}} \cdot π \cdot 4^{2} \approx \frac{5}{36} \cdot 3.14 \cdot 16 \approx 6.98 {cm}^{2}

A part of a circle can also be expressed by a circle arc. If this arc length is known, you can calculate the area in this way:

Formula

Area of Circular Sector Expressed by Circular Arc

A = \frac{b \cdot r}{2},

where $b$ is the arc length.

The arc length $b$ is part of the circumference, which is $2 π r$ . By multiplying this part of the circumference by the formula of area, you get the area of a circular sector expressed by an arc length:

A = \frac{b}{2 \cdot π \cdot r} \cdot π \cdot r^{2} = \frac{b \cdot r}{2}

Example 3

What is the area of a circular sector with circular arc of $6 cm$ and a radius of $5 cm$ ?

We put in $b = 6$ and $r = 5$ into the formula and calculate:

A = \frac{6}{2 \cdot π \cdot 5} \cdot π \cdot 5^{2} = \frac{6 \cdot 5}{2} = 15 {cm}^{2}

Example 4

What is the area of a circular sector with circular arc of $2 cm$ and a radius of $7 cm$ ?

We put in $b = 2$ and $r = 7$ into the formula and calculate:

A = \frac{2 \cdot 7}{2} = 7 {cm}^{2}

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