To derive the formula for the area of a sector of a circle, let's follow these steps:
1. Understand the Circle's Properties:
- The radius of the circle is [tex]\( r \)[/tex].
- The central angle of the sector is [tex]\( \theta \)[/tex], measured in radians.
2. Full Circle Area:
- The area of a full circle with radius [tex]\( r \)[/tex] is given by [tex]\( \pi r^2 \)[/tex].
3. Full Rotation:
- A full rotation of the circle corresponds to an angle of [tex]\( 2\pi \)[/tex] radians.
4. Set Up the Proportion:
- Since a sector is just a portion of a full circle, the ratio of the area of the sector [tex]\( A \)[/tex] to the area of the circle [tex]\( \pi r^2 \)[/tex] is the same as the ratio of the central angle [tex]\( \theta \)[/tex] to the full angle [tex]\( 2\pi \)[/tex]:
[tex]\[
\frac{A}{\pi r^2} = \frac{\theta}{2\pi}
\][/tex]
5. Solve for the Area of the Sector [tex]\( A \)[/tex]:
- Multiply both sides of the proportion by [tex]\( \pi r^2 \)[/tex] to isolate [tex]\( A \)[/tex]:
[tex]\[
A = \frac{\theta}{2\pi} \times \pi r^2
\][/tex]
6. Simplify the Expression:
- Simplify the right-hand side:
[tex]\[
A = \frac{\theta \cdot \pi r^2}{2\pi}
\][/tex]
- The [tex]\(\pi\)[/tex] terms in the numerator and denominator cancel out:
[tex]\[
A = \frac{\theta r^2}{2}
\][/tex]
Therefore, the formula for the area of a sector of a circle with radius [tex]\( r \)[/tex] and central angle [tex]\( \theta \)[/tex] (in radians) is:
[tex]\[
A = \frac{1}{2} \theta r^2
\][/tex]
In summary:
- Full circle area: [tex]\( \pi r^2 \)[/tex]
- Full rotation: [tex]\( 2\pi \)[/tex] radians
- Derived sector area: [tex]\( A = \frac{1}{2} \theta r^2 \)[/tex]
This process breaks down the task into logical steps to arrive at the correct formula.
