Area of a Quarter Circle

A quarter circle is one-fourth of a full circle. It resembles a slice of pie, that is divided into four equal parts. It includes a 90° angle and a curved edge, forming a right-angled sector. Calculating the area of a quarter circle is simple and useful in various applications. This article will guide you through the steps to determine the area, making it easy to understand and apply.

A quarter circle is a geometric shape that represents one-fourth of a full circle. It is formed by dividing a circle into four equal parts.

To visualize a quarter circle, imagine a pizza divided into four equal slices where the tip of each slice is at the center of the circle and the curved edge is part of the circumference. Each slice represents one-fourth of the entire pizza, so we can say that each slice is a quarter circle.

Properties of a Quarter Circle

Some of the common properties of quarter circle are:

• Quater circle represents one of the four quadrants of the full circle.
• Radius of quarter circle is same as the parent circle (circle from which quarter circle is cut out).
• The two straight sides (radii) form a right angle (90°) at the center of quater circle where they meet.
• The perimeter of a quarter circle includes the length of the arc plus the lengths of the two radii.
• Quater circles shows order 4 rotational symmetry.

Formula for the area of a quarter circle with radius r is:

A = (πr²) / 4

Derivation of the Formula

Area of a Circle: The formula for the area of a circle with radius r is given by, πr².

Quarter of a Circle: Since a quarter circle is one-fourth of a full circle, the area of a quarter circle will be one-fourth of the area of a full circle.

Area of Quarter Circle = (1/4) × Area of Circle

Thus, Area of quarter = (πr²) / 4.

So, the formula for the area of a quarter circle with radius r is: (πr²) / 4.

Calculating the Area of a Quarter Circle

We can use the formula discussed above i.e., Area of Quarter Circle = (πr²) / 4

Let’s consider an example for better understanding.

Example: Find the area of Quarter Circle with radius 6 units.

Solution:

Given: radius = 6 units

Area of Quarter Circle = (πr²) / 4

Thus, Area = (π × 6²) / 4 = 36π/4 = 9π square units.

Thus, the area of Quarter Circle with radius 6 unit is 9π square units.

The perimeter (P) of a quarter circle consists of:

• The arc length of the quarter circle.
• The two radii that form the straight sides.

As quarter circle is one fourth of a circle, thus its arc length is also one fourth of circumference of circle.

Thus, arc length of quarter circle = 2πr/4 = πr/2

Perimeter of quarter circle = πr/2 + 2r

In conclusion, the quarter circle is a geometric object made up of a 90° angle in the middle of a circle, two straight edges, and an arc. Its area is 1/4 of a full circle’s, and the formula for it is Area = (πr²) / 4. There are uses for this idea in many areas, such as planning, building, and math.

Read More,

• Circle
• Circumference of Circle
• Arc of Circle
• Sector of Circle

Example 1: What is the area of a quarter circle in square meters if its radius is 8 meters?

Solution

If the radius of a quarter circle is 8 meters, the area of the quarter circle can be calculated using the formula: Area = (1/4) πr².

Substituting the given value, we get

Area = (1/4) x π x 8² = (1/4) x π x 64 = 16π.

Therefore, the area of the quarter circle is 16π square meters.

Example 2: In a yard, there is a flower bed that is 5-feet across and made like a quarter circle. How many square feet does this flower bed cover?

Solution:

The area of a quarter-circle shaped flower bed with a radius of 5 feet can be calculated similarly.

Using the formula Area = (1/4) πr², we get

Area = (1/4) x π x 5² = (1/4) x π x 25 = 6.25π.

Therefore, the area of the flower bed is 6.25π square feet.

Example 3: An artist is painting a picture that is 10 meters across and in the shape of a quarter circle. How big will the painting be in square meters?

Solution:

For the mural with a radius of 10 meters, the area can be found using the formula

Area = (1/4) πr².

Substituting the given value, we get

Area = (1/4) πr² = (1/4) x π x 100 = 25π.

Thus, the area of the mural is 25π square meters.

Problem 1: Find the perimeter of a quarter circle with a radius of 4 cm.

Problem 2: A quarter circle has a radius of 7 meters. Calculate its perimeter.

Problem 3: Determine the perimeter of a quarter circle with a radius of 10 inches.

Problem 4: The radius of a quarter circle is 3.5 feet. What is the perimeter?

Problem 5: Find the perimeter of a quarter circle whose radius is 8.2 meters.

What is a quarter circle?

A quarter circle is a geometric shape that forms when an arc joins the ends of two radii, creating a right angle at the center of a circle. It has an arc as its curved side and two radii as its straight sides.

How is the area of a quarter circle calculated?

The area of a quarter circle is calculated using the formula: Area = (πr²) / 4, where ‘r’ is the radius of the quarter circle.

What are the properties of a quarter circle?

• A quarter circle has several properties:
• It forms a 90-degree center angle.
• Its area is half that of a complete circle.
• It has an arc-formed curving border.
• It has two right-angled intersecting radii.
• It exhibits order 4 rotational symmetry.
• It finds applications in engineering, architectural design, and geometric computations.

Where is the concept of quarter circles applied?

Quarter circles are applied in various fields including engineering, architectural design, and geometric calculations involving circles and sectors.

What are some examples of calculating the area of quarter circles?

Examples include determining the area of quarter circle flower beds, murals, portions of pool bottoms, or specialty pizzas, given their respective radii or diameters.