What is a quarter circle?

A quarter circle is one-fourth of a full circle. It is formed when a circle is divided into four equal parts/sections by two perpendicular lines that intersect at the circle's center. The shape resembles a segment of a pie or an arc with a right-angle sector.

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Circle

The circle, a perfect shape, boasts infinite symmetry and boundless continuity. Its every point equidistant from the center, symbolizing unity and completeness.

Area of Sector

The Area of a Sector of a Circle is the measure of the region enclosed by the two radii and the corresponding arc.

Limits

Limits are crucial in analyzing the behavior of functions, determining continuity, and evaluating derivatives and integrals.

Hyperbola

The hyperbola is a conic section formed by the intersection of a plane with both halves of a double cone. It consists of two...

Parabola

The parabola, a conic section, is a symmetric U-shaped curve. It appears in everyday scenarios and physics, showcasing its versatility.

Conic Sections

Conic Sections are the curves obtained by an interesting double-napped right circular cone by a plane.

Coplanar Vectors

Coplanar vectors are vectors that lie within the same plane in a three-dimensional space.

Continuity

Continuity of a function is a fundamental concept in mathematics that describes the connectedness of the graph of the function.

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

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 $(r = \frac{d}{2})$ .

Convert Diameter to Radius:

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

Substitute Radius into Full Circle Formula:

$A_{full} = π (\frac{d}{2})^{2}$

$A_{full} = π \frac{d ^{2}}{4}$

Area of a Quarter Circle:

$A_{quarter} = \frac{1}{4} (π \frac{d ^{2}}{4})$

$A_{quarter} = \frac{1}{16} π d^{2}$

Image showing the area of a quarter circle

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:

$A_{quarter} = \frac{1}{4} π r^{2}$

$A_{quarter} = \frac{1}{4} π (6)^{2}$

$A_{quarter} = \frac{1}{4} π (36)$

$A_{quarter} = 9 π$

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:

$A_{quarter} = \frac{1}{16} π d^{2}$

$A_{quarter} = \frac{1}{16} π (8)^{2}$

$A_{quarter} = \frac{1}{16} π (64)$

$A_{quarter} = 4 π$

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:

Example problem on area of a quarter circle

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 = $\frac{1}{4} π r^{2}$

$A = \frac{1}{4} π (5)^{2}$

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

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

5.0Practice Questions on Area of a Quarter Circle

Find the area of a quarter circle with a radius of 5 units.
If the radius of a quarter circle is 3 units, what is its area?
Find the area of a quarter circle with a diameter of 10 units.
If the diameter of a quarter circle is 14 units what is its area?
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:

$\frac{25 π}{4}$
$\frac{9 π}{4}$
$\frac{25 π}{4}$
$\frac{49 π}{4}$
Length = 10, width = 5, Area = $\frac{25 π}{4}$ .

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: $Area = \frac{1}{4} π r^{2}$ where r is the radius of the circle. This formula is derived from the area of a full circle $π r^{2}$ , 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: $Area = \frac{1}{4} π (\frac{d}{2})^{2}$

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:

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

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 $π r^{2}$ , then the area of a quarter circle is $\frac{1}{4} π r^{2} .$

Related Video:

Join ALLEN!

(Session 2026 - 27)

Name

Mobile Number

Class

Choose class

Goal

Choose your goal

Preferred Programs

Preferred Mode

State

Choose State

I agree to

I authorise ALLEN Career Institute Pvt Ltd to send me regular updates via Phone calls, Whatsapp, SMS, Robocalls (Automated Calls), Emails, or on postal address.

Circle

The circle, a perfect shape, boasts infinite symmetry and boundless continuity. Its every point equidistant from the center, symbolizing unity and completeness.

Area of Sector

The Area of a Sector of a Circle is the measure of the region enclosed by the two radii and the corresponding arc.

Limits

Limits are crucial in analyzing the behavior of functions, determining continuity, and evaluating derivatives and integrals.

Hyperbola

The hyperbola is a conic section formed by the intersection of a plane with both halves of a double cone. It consists of two...

Parabola

The parabola, a conic section, is a symmetric U-shaped curve. It appears in everyday scenarios and physics, showcasing its versatility.

Conic Sections

Conic Sections are the curves obtained by an interesting double-napped right circular cone by a plane.

Coplanar Vectors

Coplanar vectors are vectors that lie within the same plane in a three-dimensional space.

Continuity

Continuity of a function is a fundamental concept in mathematics that describes the connectedness of the graph of the function.

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

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 $(r = \frac{d}{2})$ .

Convert Diameter to Radius:

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

Substitute Radius into Full Circle Formula:

$A_{full} = π (\frac{d}{2})^{2}$

$A_{full} = π \frac{d ^{2}}{4}$

Area of a Quarter Circle:

$A_{quarter} = \frac{1}{4} (π \frac{d ^{2}}{4})$

$A_{quarter} = \frac{1}{16} π d^{2}$

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:

$A_{quarter} = \frac{1}{4} π r^{2}$

$A_{quarter} = \frac{1}{4} π (6)^{2}$

$A_{quarter} = \frac{1}{4} π (36)$

$A_{quarter} = 9 π$

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:

$A_{quarter} = \frac{1}{16} π d^{2}$

$A_{quarter} = \frac{1}{16} π (8)^{2}$

$A_{quarter} = \frac{1}{16} π (64)$

$A_{quarter} = 4 π$

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 = $\frac{1}{4} π r^{2}$

$A = \frac{1}{4} π (5)^{2}$

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

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

5.0Practice Questions on Area of a Quarter Circle

Find the area of a quarter circle with a radius of 5 units.
If the radius of a quarter circle is 3 units, what is its area?
Find the area of a quarter circle with a diameter of 10 units.
If the diameter of a quarter circle is 14 units what is its area?
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:

$\frac{25 π}{4}$
$\frac{9 π}{4}$
$\frac{25 π}{4}$
$\frac{49 π}{4}$
Length = 10, width = 5, Area = $\frac{25 π}{4}$ .

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: $Area = \frac{1}{4} π r^{2}$ where r is the radius of the circle. This formula is derived from the area of a full circle $π r^{2}$ , 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: $Area = \frac{1}{4} π (\frac{d}{2})^{2}$

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:

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

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 $π r^{2}$ , then the area of a quarter circle is $\frac{1}{4} π r^{2} .$

Related Video:

Frequently Asked Questions