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.
What is a Quarter Circle?
Quarter circle is a geometric shape representing one-fourth of a complete circle, bounded by a curved arc and two perpendicular radii. It resembles a quarter of a pie or pizza slice. Commonly used in geometry, engineering, and design, it encapsulates a quarter of the area and one-quarter of the circumference of a full circle.
Below is a list of all the formulas related to the quarter circle:
Property | Formula |
|---|
Arc Length (s) | ¼ × Circumference (s = ¼ × πr) |
|---|
Area (A) | ¼ × πr² |
|---|
Central Angle (θ) | 90° (in degrees) or π/2 radians |
|---|
Chord Length (c) | √(2r²) |
|---|
Sector Area (A_sec) | ½ × r² × θ |
|---|
Segment Area (A_seg) | Sector Area - Area of triangle formed by radii and chord |
|---|
Perimeter (P) | Arc Length + 2 × r |
|---|
Area of a Quarter Circle
Area of a quarter circle is defined as the amount of space enclosed by one-fourth of a full circle. For a full circle, its area is equal to pi times the square of the radius (Area =πr2 ).Therefore, the formula for the area of a quarter circle is:
Area of Quarter Circle = 1/4 πr2
The area of a quarter circle can also be calculated using different formulas based on the diameter, Since the diameter is twice the radius (d = 2r), the formula can be written as:
Area = (1/16) x π x d2
Example
Calculate the are of the quarter circle if the radius (r) is 4 units.
Solution:
Area of quarter circle = 1/4 πr2
- Square the radius: 4 x 4 = 16
- Multiply by π: 16 x π ≈ 16 x 3.14159 ≈ 50.27
- Divide by 4: 50.27 / 4 ≈ 12.57
So, the area of the quarter circle is approximately 12.57 square units.
Perimeter of Quarter Circle
To find the perimeter of a quarter circle, you add the length of the curved part (arc) and the two straight sides. One-fourth of the circumference of a full circle (since a quarter circle is a fourth of a full circle) plus the length of the radius twice. The formula for the perimeter of a quarter circle with radius "r" is:
Perimeter = Half the circumference (πr/2) + 2 times the radius (2r)
Perimeter of Quarter Circle = πr/2 + 2r
Centroid of quarter circle
The centroid of a quarter circle circle with a radius of ( r ) is at (4r/3π, 4r/3π). From here, the calculation reveals that the centroid is 4r/3π units in both the x and the y direction from the origin; the origin is the point where the two radii are perpendicular to each other.
Properties of Quarter Circle
A quarter circle is a two-dimensional geometric shape consisting of one-fourth of a complete circle, with properties including:
- A quarter circle subtends a right angle (90 degrees) at its center.
- The length of the arc of a quarter circle is equal to one-fourth of the circumference of the full circle from which it is derived.
- A quarter circle has rotational symmetry of order four. This means that it looks the same after a rotation of 90, 180, or 270 degrees about its center.
- A quarter circle divides the coordinate plane into four quadrants. It lies entirely in the first quadrant if the center of the circle is at the origin.
- The perimeter of a quarter circle includes the length of the arc plus the lengths of the two radii.
Applications of Quarter Circle
Even though they have a simple physical shape, quarter circles are useful in many different areas. Here are a few specific instances:
- Architecture and Design: Quarter circles are commonly used in architectural design for creating rounded corners, arches, and decorative elements. They add aesthetic appeal to buildings and structures.
- Engineering: In engineering, quarter circles are utilized in the design of curved components, such as gears, cam profiles, and curved piping systems. They help optimize space utilization and improve functionality.
- Geometry and Mathematics: Quarter circles serve as fundamental geometric shapes used in teaching and learning geometry. They are often employed in exercises and problems involving angles, area, perimeter, and arc length calculations.
- Manufacturing: Quarter circles are frequently encountered in manufacturing processes, such as machining, metal fabrication, and woodworking. They are used in the production of curved parts, molds, and templates.
- Roads and Transportation: Quarter circles are utilized in road design for creating curved sections, such as roundabouts and curved roadways. They help improve traffic flow, reduce vehicle speeds, and enhance safety.
Read More:
Solved Examples on Quarter Circle
Example 1: Find the area of a quarter circle with a radius of 6 cm.
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.
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.
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.
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.
Practice Problems on Quarter Circle- Unsolved
1. Find the area of a quarter circle with a radius of 12 cm.
2. The perimeter of a quarter circle is 16π cm. Find the radius of the quarter circle.
4. If the length of a string wrapped around a quarter circle is 3π cm, what is the radius of the quarter circle?
5. A quarter circle is cut from a circular cardboard with a radius of 8 cm. Find the area of the remaining cardboard.
6. The circumference of a quarter circle is 12 cm. Determine the radius of the quarter circle.
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