The area of a sector of a circle of radius 4 cm is 25.6 cm 2. What is the radian measure of the arc of the sector?
3.2
Calculating Radian Measure of a Circle Sector
This problem asks us to find the radian measure of the arc (or central angle) of a sector of a circle, given its area and radius. We will use the formula relating the area of a sector to its radius and central angle in radians.
Understanding Circle Sectors
A sector of a circle is like a slice of pizza. It is bounded by two radii and the arc connecting their endpoints. The area of a sector depends on the radius of the circle and the central angle of the sector.
The formula for the area of a sector (\(A\)) of a circle with radius (\(r\)) and a central angle (\(\theta\)) measured in radians is:
\(A = \frac{1}{2} r^2 \theta\)
In this problem, we are given the area and the radius and need to find the angle (\(\theta\)) in radians.
Given Information
- Radius of the circle, \(r = 4\) cm
- Area of the sector, \(A = 25.6\) cm\(^2\)
Finding the Radian Measure
We need to rearrange the area formula to solve for \(\theta\):
\(A = \frac{1}{2} r^2 \theta\)
Multiply both sides by 2:
\(2A = r^2 \theta\)
Divide both sides by \(r^2\):
\(\theta = \frac{2A}{r^2}\)
Now, substitute the given values for \(A\) and \(r\) into this formula:
\(\theta = \frac{2 \times 25.6 \text{ cm}^2}{(4 \text{ cm})^2}\)
Calculate the values:
- \(2 \times 25.6 = 51.2\)
- \((4 \text{ cm})^2 = 16 \text{ cm}^2\)
Substitute these back into the equation for \(\theta\):
\(\theta = \frac{51.2 \text{ cm}^2}{16 \text{ cm}^2}\)
The units cm\(^2\) cancel out, leaving a dimensionless quantity, which is appropriate for radians.
\(\theta = \frac{51.2}{16}\)
Performing the division:
\(\theta = 3.2\)
The radian measure of the arc of the sector is 3.2 radians.
Comparing with Options
Let's compare our calculated radian measure with the given options:
- Option 1: 2.3
- Option 2: 3.2
- Option 3: 3.3
- Option 4: 3.4
Our calculated value, 3.2 radians, matches Option 2.
| Quantity | Value | Units |
|---|---|---|
| Radius (\(r\)) | 4 | cm |
| Area of Sector (\(A\)) | 25.6 | cm\(^2\) |
| Radian Measure (\(\theta\)) | 3.2 | radians |
Conclusion
Using the formula for the area of a circle sector, \(A = \frac{1}{2} r^2 \theta\), and rearranging it to solve for the angle \(\theta\), we found that the radian measure of the arc is 3.2 radians.
Revision Table: Circle Sector Formulas
Here are some key formulas related to circle sectors:
| Concept | Formula (Angle \(\theta\) in Radians) | Formula (Angle \(N\) in Degrees) |
|---|---|---|
| Area of Sector (\(A\)) | \(A = \frac{1}{2} r^2 \theta\) | \(A = \frac{N}{360^{\circ}} \pi r^2\) |
| Arc Length (\(L\)) | \(L = r \theta\) | \(L = \frac{N}{360^{\circ}} 2\pi r\) |
| Perimeter of Sector | Perimeter \( = r + r + L = 2r + r\theta\) | Perimeter \( = 2r + \frac{N}{360^{\circ}} 2\pi r\) |
Additional Information: Radians vs. Degrees
It's important to understand the two common units for measuring angles: degrees and radians.
- Degrees: A full circle is divided into 360 degrees (360\(^{\circ}\)). This unit is commonly used in geometry and everyday measurements.
- Radians: A radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle. A full circle is \(2\pi\) radians. This unit is preferred in calculus and many areas of physics and engineering because it simplifies formulas (like the sector area and arc length formulas).
The conversion between radians and degrees is:
- \(\pi\) radians = 180\(^{\circ}\)
- 1 radian \(= \frac{180^{\circ}}{\pi} \approx 57.3^{\circ}\)
- 1\(^{\circ}\) \(= \frac{\pi}{180}\) radians \(\approx 0.01745\) radians
In this specific problem, the formula provided for the sector area \(A = \frac{1}{2} r^2 \theta\) is specifically for when \(\theta\) is in radians. Using the correct units for the angle is crucial for accurate calculations.
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