The perimeter of a quarter circle is 3.57 cm. What is the quarter circles area?
The Correct Answer and Explanation is:
To find the area of a quarter circle, we need to first understand how the perimeter of a quarter circle relates to its radius. The perimeter of a quarter circle consists of two parts:
- Arc length: The curved portion of the quarter circle, which is a quarter of the circumference of the full circle.
- Straight edges: These are the two radii of the circle, which form the right-angle sides of the quarter circle.
Given that the perimeter of the quarter circle is 3.57 cm, we can break it into its components:
- Arc length: This is a quarter of the total circumference of the circle. The formula for the circumference of a full circle is ( C = 2\pi r ), where ( r ) is the radius. The arc length of the quarter circle is ( \frac{1}{4} \times 2\pi r = \frac{\pi r}{2} ).
- Straight edges: These are simply the two radii, so their total length is ( 2r ).
Thus, the total perimeter ( P ) of the quarter circle is the sum of the arc length and the straight edges:
[
P = \frac{\pi r}{2} + 2r
]
We are told that the perimeter is 3.57 cm, so:
[
3.57 = \frac{\pi r}{2} + 2r
]
Now, solve for ( r ).
Step 1: Solve for ( r )
Rewrite the equation:
[
3.57 = \frac{\pi r}{2} + 2r
]
Factor out ( r ):
[
3.57 = r \left( \frac{\pi}{2} + 2 \right)
]
Numerically approximate ( \frac{\pi}{2} + 2 ) (where ( \pi \approx 3.14 )):
[
\frac{3.14}{2} + 2 = 1.57 + 2 = 3.57
]
So the equation simplifies to:
[
3.57 = r \times 3.57
]
Thus, ( r = 1 ) cm.
Step 2: Calculate the Area of the Quarter Circle
The area ( A ) of a quarter circle is given by:
[
A = \frac{1}{4} \pi r^2
]
Substitute ( r = 1 ) cm:
[
A = \frac{1}{4} \pi (1)^2 = \frac{1}{4} \pi \approx \frac{1}{4} \times 3.14 = 0.785 \, \text{cm}^2
]
Conclusion
The area of the quarter circle is approximately 0.785 cm².