How to find the perimeter of a quarter circle?
Understand the Problem
The question is asking how to calculate the perimeter of a quarter circle. The perimeter consists of the arc length and the two straight line segments that make up the radius of the circle.
Answer
The perimeter of a quarter circle with radius \( r \) is \( P = \frac{\pi r}{2} + 2r \).
Answer for screen readers
The perimeter of a quarter circle with radius ( r ) is given by $$ P = \frac{\pi r}{2} + 2r. $$
Steps to Solve
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Identify the components of the perimeter The perimeter of a quarter circle consists of the arc length and the two radii. If the radius of the circle is ( r ), then the total perimeter ( P ) can be calculated using both the arc length and the lengths of the radii.
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Calculate the arc length The formula for the arc length of a circle is given by $$ L = \frac{\theta}{360^\circ} \cdot 2\pi r $$ where ( \theta ) is the angle in degrees. For a quarter circle, ( \theta = 90^\circ ). Thus, the arc length becomes $$ L = \frac{90^\circ}{360^\circ} \cdot 2\pi r = \frac{1}{4} \cdot 2\pi r = \frac{\pi r}{2}. $$
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Calculate the total perimeter The total perimeter ( P ) of the quarter circle is the sum of the arc length and the lengths of the two radii (which is ( 2r )): $$ P = \text{Arc length} + \text{2 radii} = \frac{\pi r}{2} + 2r. $$
Combining these gives: $$ P = \frac{\pi r}{2} + 2r. $$
The perimeter of a quarter circle with radius ( r ) is given by $$ P = \frac{\pi r}{2} + 2r. $$
More Information
This formula combines the curved part (the arc) with the straight segments (the two radii). The arc length of a quarter circle is one-fourth of the circumference of the full circle, which explains why we divide and multiply by 2 in the derivation.
Tips
- Forgetting to add the lengths of the two radius segments to the arc length.
- Confusing the angle for a quarter circle, which is ( 90^\circ ), with other segments.
