Solved Examples on Quarter Circle

Example 1: Find the area of a quarter circle with a radius of 6 cm.

Solution:

The area of a quarter circle is given by the formula: (1/4)πr².

Given that the radius (r) is 6 cm, substitute this value into the formula:

Area = (1/4)π(6²)

= (1/4)π(36)

= (1/4) × 36 × π = 9π

So, the area of the quarter circle is 9π cm².

Example 2: The perimeter of a quarter circle is 10 cm. Find the radius of the quarter circle.

Solution:

The perimeter of a quarter circle is the sum of the length of the curved part and two radii.

Let’s denote the radius as r.

So, the perimeter is: Perimeter = (1/4)(2πr) + 2r

Given that the perimeter is 10 cm, we can set up the equation:

10 = (1/4)(2πr) + 2r

10 = (π/2)r + 2r

10 = (π/2 + 2)r

r = 10 / (π/2 + 2)

r = 20 / (π + 4)

So, the radius of the quarter circle is 20 / (π + 4) cm.

Example 3: A quarter circle is inscribed in a square with a side length of 8 cm. Find the area of the shaded region, which is the area outside the quarter circle but inside the square.

Solution:

First, let’s find the area of the square. Since the side length is 8 cm, the area is 8² = 64 square cm.

Next, let’s continue with our quest and calculate a quarter of this circle’s area. The radius or the quarter circle is half the side length of the square is 8/2=4cm. Using the formula for the area of a quarter circle, we have:

Area of quarter circle = (1/4)π(4²) = 4π square cm.

The shaded region’s area is the difference between the area of the square and the area of the quarter circle:

Shaded area = Area of square – Area of quarter circle = 64 – 4π square cm.

Example 4: A string is tightly wrapped around a quarter circle with a radius of 10 cm. Find the length of the string.

Solution:

The length of the string is the circumference of the quarter circle, which is just one-fourth of the circumference of the full circle with the same radius.

The formula for the circumference of a circle is 2πr, so the circumference of the quarter circle is (1/4) × 2πr = (1/2)πr.

Given that the radius is 10 cm, we have: Length of string = (1/2)π(10) = 5π cm.

So, the length of the string is 5π cm.

Quarter circle is an element of a circular shape that occupies one-fourth of the circle’s perimeter edge and has the same ratio in terms of the area, forming a right angle with the adjacent plane.

This article provides a background on the quarter circle by discussing its formulas and properties as well as real-life uses and gives examples about calculating the area and perimeter of the figure, and problems to solve for practice.