1. Understanding a Sector
A sector of a circle is the region enclosed by two radii and the arc between them. It looks like a slice of a pizza or a piece of a pie. The size of a sector depends on two things:
- the radius (r) of the circle
- the central angle (θ) formed by the two radii
If the angle is large, the sector area is large; if the angle is small, the sector area is small.
1.1. Central Angle
The central angle \(\theta\) is the angle between the two radii that form the sector. This angle determines what fraction of the circle the sector represents.
For example, a \(90^\circ\) sector is one-fourth of the whole circle because:
\( \dfrac{90^\circ}{360^\circ} = \dfrac{1}{4} \)
2. Area Formula (Angle in Degrees)
If the radius of the circle is \(r\) and the central angle is \(\theta\) degrees, then the area of the sector is:
\( A = \dfrac{\theta}{360^\circ} \times \pi r^2 \)
2.1. Why the Formula Works
The area of the whole circle is:
\( A = \pi r^2 \)
The angle \(\theta\) tells us what fraction of the circle the sector represents. So we multiply the total area by \( \dfrac{\theta}{360^\circ} \).
3. Area Formula (Angle in Radians)
Sometimes angles are given in radians. If the angle is \(\theta\) radians, then the area of the sector is:
\( A = \dfrac{1}{2} r^2 \theta \)
3.1. When to Use This Formula
This version is used mainly in higher classes or when working with trigonometry. Radian measure makes many formulas simpler.
4. Examples
These examples show how to apply the sector area formulas in different situations.
4.1. Example 1 (Angle in Degrees)
Find the area of a sector of radius \(r = 6\,\text{cm}\) and central angle \(\theta = 60^\circ\).
\( A = \dfrac{60}{360} \times \pi \times 6^2 \)
\( A = \dfrac{1}{6} \times 36\pi = 6\pi \)
If \(\pi = 3.14\):
\( A = 6 \times 3.14 = 18.84\,\text{cm}^2 \)
4.2. Example 2 (Angle in Radians)
Find the area of a sector with radius \(r = 10\,\text{cm}\) and central angle \(\theta = 1.2\,\text{radians}\).
\( A = \dfrac{1}{2} r^2 \theta = \dfrac{1}{2} \times 10^2 \times 1.2 \)
\( A = \dfrac{1}{2} \times 100 \times 1.2 = 60\,\text{cm}^2 \)