Determining the angle of a circular sector when the area and radius are known is a useful skill in geometry. This step-by-step guide will walk you through the process using formulas, explanations, and real number examples to make it easy to understand and follow.
Step 1: Show the Formula
To find the angle of a circular sector, we use the following formula:
\[ \theta = \frac{2A}{r^2} \]
where:
- \( \theta \) is the central angle in radians.
- \( A \) is the area of the sector.
- \( r \) is the radius of the circle.
Step 2: Explain the Formula
The formula \( \theta = \frac{2A}{r^2} \) is derived from the area formula of a sector \( A = \frac{1}{2} r^2 \theta \). By rearranging this formula to solve for \( \theta \), we get \( \theta = \frac{2A}{r^2} \).
Step 3: Use Actual Numbers as an Example
Let's assume we have a sector with an area (\( A \)) of 20 square units and a radius (\( r \)) of 5 units.
Step 4: Calculate the Angle
Now, substitute the given values into the formula:
\[ \theta = \frac{2 \times 20}{5^2} \]
\[ \theta = \frac{40}{25} \]
\[ \theta = 1.6 \text{ radians} \]
Step 5: Convert the Angle to Degrees
To convert the angle from radians to degrees, use the conversion factor \( 180^\circ / \pi \):
\[ \theta_{\text{degrees}} = 1.6 \times \frac{180^\circ}{\pi} \]
Using \(\pi \approx 3.14159\):
\[ \theta_{\text{degrees}} = 1.6 \times \frac{180}{3.14159} \]
\[ \theta_{\text{degrees}} \approx 1.6 \times 57.2958 \]
\[ \theta_{\text{degrees}} \approx 91.67^\circ \]
Final Value
The central angle of the circular sector with an area of 20 square units and a radius of 5 units is \( 1.6 \) radians or approximately \( 91.67^\circ \).
