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8.03 Semester Test: Geometry Semester B Test - Part 1
Drag and drop an answer to each box to correctly complete the derivation of a formula for the area of a sector of a circle.
Suppose a sector of a circle with radius r has a central angle of 0 Since a sector is a fraction of a full circle, the ratio of a
sector's area A to the circle's
is equal to the ratio of the central angle to the measure of a full
rotation of the circle. A full rotation of a circle is 2 radians. This proportion can be written as
=
Multiply both sides by 2 and simplify to get
angle of the sector and ris the radius of the circle.
where is the measure of the central

Verified Answer :

GinnyAnswer

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 \(\( r \)\).
- The central angle of the sector is \(\( \theta \)\), measured in radians.

2. Full Circle Area:
- The area of a full circle with radius \(\( r \)\) is given by \(\( \pi r^2 \)\).

3. Full Rotation:
- A full rotation of the circle corresponds to an angle of \(\( 2\pi \)\) 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 \(\( A \)\) to the area of the circle \(\( \pi r^2 \)\) is the same as the ratio of the central angle \(\( \theta \)\) to the full angle \(\( 2\pi \)\):
\(\[ \frac{A}{\pi r^2} = \frac{\theta}{2\pi} \]\)

5. Solve for the Area of the Sector \(\( A \)\):
- Multiply both sides of the proportion by \(\( \pi r^2 \)\) to isolate \(\( A \)\):
\(\[ A = \frac{\theta}{2\pi} \times \pi r^2 \]\)

6. Simplify the Expression:
- Simplify the right-hand side:
\(\[ A = \frac{\theta \cdot \pi r^2}{2\pi} \]\)
- The \(\(\pi\)\) terms in the numerator and denominator cancel out:
\(\[ A = \frac{\theta r^2}{2} \]\)

Therefore, the formula for the area of a sector of a circle with radius \(\( r \)\) and central angle \(\( \theta \)\) (in radians) is:
\(\[ A = \frac{1}{2} \theta r^2 \]\)

In summary:
- Full circle area: \(\( \pi r^2 \)\)
- Full rotation: \(\( 2\pi \)\) radians
- Derived sector area: \(\( A = \frac{1}{2} \theta r^2 \)\)

This process breaks down the task into logical steps to arrive at the correct formula.

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