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 given its perimeter, we need to first understand the formula for the perimeter of a quarter circle and use it to find the radius, then use the radius to compute the area.
Step 1: Understand the Perimeter Formula
The perimeter PP of a quarter circle includes:
- One-quarter of the circumference of a full circle: 14⋅2πr=πr2\frac{1}{4} \cdot 2\pi r = \frac{\pi r}{2}
- Two straight sides: the radius and another radius
So the formula for perimeter is: P=πr2+2rP = \frac{\pi r}{2} + 2r
We are told that the perimeter is 3.57 cm: πr2+2r=3.57\frac{\pi r}{2} + 2r = 3.57
Step 2: Solve for rr
Use π≈3.1416\pi \approx 3.1416: 3.1416r2+2r=3.57\frac{3.1416 r}{2} + 2r = 3.57 1.5708r+2r=3.571.5708r + 2r = 3.57 3.5708r=3.573.5708r = 3.57 r≈3.573.5708≈1.0 cmr \approx \frac{3.57}{3.5708} \approx 1.0 \, \text{cm}
Step 3: Use the Radius to Find the Area
The area AA of a full circle is: A=πr2A = \pi r^2
So the area of a quarter circle is: A=14πr2A = \frac{1}{4} \pi r^2
Substitute r=1r = 1 cm: A=14⋅3.1416⋅(1)2=3.14164=0.7854 cm2A = \frac{1}{4} \cdot 3.1416 \cdot (1)^2 = \frac{3.1416}{4} = 0.7854 \, \text{cm}^2
Final Answer:
0.7854 cm2\boxed{0.7854 \, \text{cm}^2}
Explanation
To find the area of a quarter circle with a given perimeter of 3.57 cm, we begin by recalling the formula for the perimeter of a quarter circle. A quarter circle consists of two straight sides (both equal to the radius) and a curved arc, which is one-fourth of the full circle’s circumference. The total perimeter is thus made up of the arc length and the two radii.
Mathematically, the perimeter PP of a quarter circle is given by: P=πr2+2rP = \frac{\pi r}{2} + 2r
We substitute the known value of the perimeter, 3.57 cm, into this equation and solve for rr, the radius. Using the approximate value of π\pi as 3.1416, we get: 1.5708r+2r=3.57⇒3.5708r=3.571.5708r + 2r = 3.57 \Rightarrow 3.5708r = 3.57
Solving for rr, we find that the radius is approximately 1 cm.
Next, we use the formula for the area of a circle: A=πr2A = \pi r^2
To find the area of a quarter circle, we multiply this by 14\frac{1}{4}. Substituting r=1r = 1 cm, the area becomes: 14⋅3.1416⋅12=0.7854 cm2\frac{1}{4} \cdot 3.1416 \cdot 1^2 = 0.7854 \, \text{cm}^2
This calculation shows that when the perimeter of a quarter circle is 3.57 cm, the radius is 1 cm, and the area is approximately 0.7854 square centimeters.
