Area of a Quarter Circle

The area of a quarter circle, also known as a quadrant, is a segment of a circle that represents one-fourth of its entire area and can be calculated using straightforward geometric principles. Understanding how to calculate the area of a quarter circle is essential in various fields such as geometry, engineering, and design. 

1.0Area of a Quarter Circle Definition

A quarter circle is created when a circle is divided into four equal parts. Each of these parts is a 90-degree sector, or one-fourth of the circle. The area of a quarter circle can be determined if you know either the radius or the diameter of the original circle. 

2.0What is 1/4 of a Circle Called?

One-fourth of a circle is called a quarter circle or a quadrant. This geometric shape is formed by dividing a full circle into four equal parts. Each part, representing a 90-degree sector, is a quadrant. 

Key Characteristics of a Quarter Circle:

• Angle: A quarter circle spans a 90-degree angle, which is one-fourth of the 360-degree angle of a full circle.
• Area: The area of a quarter circle is one-fourth of the area of the entire circle.
• Radius and Diameter: The quarter circle retains the same radius and half the diameter of the original circle.

3.0Area of a Quartile Circle Formulas

Calculating the area of a quarter circle, or a quadrant, involves using formulas derived from the area of a full circle. Here are the primary formulas based on whether you have the radius or the diameter of the circle.

Area of a Quarter Circle with Radius

The radius (r) is the distance from the center of the circle to any point on its circumference.

• Area of a Full Circle: $A_{\text{full }}=\pi r^2$
• Area of a Quarter Circle: $A_{\text{quarter }}=\frac{1}{4}\pi r^2$

Area of a Quarter Circle Using Diameter

The diameter (d) is the length of a line that passes through the center of the circle, connecting two points on its circumference. The radius is half of the diameter $\left(r=\frac{d}{2}\right)$ .

• Convert Diameter to Radius:

$r=\frac{d}{2}$

• Substitute Radius into Full Circle Formula:

$A_{\text{full }}=\pi {\left(\frac{d}{2}\right)}^2$

$A_{\text{full }}=\pi \frac{d^2}{4}$

• Area of a Quarter Circle:

$A_{\text{quarter }}=\frac{1}{4}\left(\pi \frac{d^2}{4}\right)$

$A_{\text{quarter }}=\frac{1}{16}\pi d^2$

4.0Solved Examples on Area of a Quarter circle

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

Solution:

${\mathrm{A}}_{\text{quarter }}=\frac{1}{4}\pi {\mathrm{r}}^2$

${\mathrm{A}}_{\text{quarter }}=\frac{1}{4}\pi (6{)}^2$

${\mathrm{A}}_{\text{quarter }}=\frac{1}{4}\pi (36)$

${\mathrm{A}}_{\text{quarter }}=9\pi$

Therefore, the area of the quarter circle is 9π square units.

Example 2: Find the area of a quarter circle with a diameter of 8 units.

Solution:

${\mathrm{A}}_{\text{quarter }}=\frac{1}{16}\pi {\mathrm{d}}^2$

${\mathrm{A}}_{\text{quarter }}=\frac{1}{16}\pi (8{)}^2$

${\mathrm{A}}_{\text{quarter }}=\frac{1}{16}\pi (64)$

$A_{\text{quarter }}=4\pi$

Therefore, the area of the quarter circle is 4π square units.

Example 3: A goat is tied to one corner of a rectangular field with dimensions 6 meters by 8 meters. The length of the rope θ is 5 meters. Calculate the area available for the goat to graze. 

Solution:

The area grazed by the goat includes a quarter- circle sector with radius 5 meters and possibly parts outside the rectangular field.

Area of the quarter circle = $\frac{1}{4}\pi r^2$

$A=\frac{1}{4}\pi (5{)}^2$

=$\frac{25}{4}\pi \approx$ 19.63 square meters

Therefore, grazing area is $\frac{25}{4}$ π square meters.

5.0Practice Questions on Area of a Quarter Circle

1. Find the area of a quarter circle with a radius of 5 units.
2. If the radius of a quarter circle is 3 units, what is its area?
3. Find the area of a quarter circle with a diameter of 10 units.
4. If the diameter of a quarter circle is 14 units what is its area?
5. A rectangle has dimensions such that a quarter circle can fit perfectly within it, with the quarter circle radius being 5 units. If the length of the rectangle is double its width, find the dimension of the rectangle and the area of the quarter circle.  

Answers:

1. $\frac{25\pi }{4}$
2. $\frac{9\pi }{4}$
3. $\frac{25\pi }{4}$
4. $\frac{49\pi }{4}$
5. Length = 10, width = 5, Area =$\frac{25\pi }{4}$.

6.0Sample Questions on Area of a Quartile Circle

1. How do you calculate the area of a quarter circle?

Ans: To calculate the area of a quarter circle, use the formula:$\text{ Area }=\frac{1}{4}\pi r^2$ where r is the radius of the circle. This formula is derived from the area of a full circle $\pi r^2$, divided by four.

2. What is the formula for the area of a quarter circle if the diameter is given?

Ans: If the diameter of the circle is given, first find the radius by dividing the diameter by 2. Then use the formula:$\text{ Area }=\frac{1}{4}\pi {\left(\frac{d}{2}\right)}^2$

where d represents the diameter of the circle.

3. How can I find the area of a quarter circle if I only know the circumference?

Ans: If you know the circumference C of the circle, first find the radius using the formula:

$r=\frac{C}{2\pi }$ Then, use the area formula for the quarter circle:$\text{ Area }=\frac{1}{4}\pi r^2$

4. How does the area of a quarter circle compare to the area of a full circle?

Ans: The area of a quarter circle is one-fourth the area of a full circle. If the area of a full circle is $\pi r^2$, then the area of a quarter circle is $\frac{1}{4}\pi r^2.$