How to Find the Area of a Quarter Circle?
A quarter circle has one fourth of the area of a full circle with the same radius. Its boundary includes the curved arc and two radii.
TL;DR: Slice a pizza into four equal pieces and grab one. That's a quarter circle. Since it's exactly one-fourth of the full circle, its area is just the regular circle area divided by 4. The perimeter follows the same one-fourth logic for the curved arc, then adds two straight radius edges to close the shape. Once you see it as "the whole thing, but divided by 4," you don't need a separate formula sheet for quarter circles ever again.
Key takeaways:
- Area of a quarter circle: \(A=\frac{1}{4}\pi r^2\) (one-fourth of \(\pi r^2\)).
- Arc length of a quarter circle: \(s=\frac{1}{4}(2\pi r)=\frac{\pi r}{2}\).
- Perimeter of a quarter circle: \(P=2r+\frac{\pi r}{2}\) (two straight edges plus the curved arc).
- The central angle of a quarter circle is \(90^\circ\) or \(\pi/2\) radians.
- Use \(\pi\approx 3.14\) for quick estimates; use the \(\pi\) button for precise answers.
When a circle is divided into \(4\) equal parts, \(4\) quadrants are formed, and each of these quarters is known as a quadrant. In this guide, you will learn more about the quarter circle and how to find its area.
A quarter-circle is one-fourth part of the whole circle. So, the area of a quarter circle is the fourth part of the whole area of a circle.
Type the problems from this lesson and check each step.
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A step-by-step guide to finding the area of a quarter circle
A circle is a collection of points that are at a constant distance from a fixed point. This fixed point and fixed distance are called “center” and “radius”, respectively. A quarter-circle is one-fourth. So the area of a quarter circle is exactly one-fourth of the area of a full circle.
Area of a quarter circle using radius
We know that the area of a circle is \(πr^2\). A quarter-circle is a one-fourth portion of a full circle, so its area is one-fourth of the area of the circle. Thus, the area of a quarter circle in terms of radius \(\color{blue}{= \frac{πr^2}{4}}\)
Area of a quarter circle using the diameter
Since we have \(d = 2r\), \(r= \frac{d}{2}\). By substituting it in the above formula, we can get the area of a quarter circle in diameter.
The area of a quarter circle \(\color{blue}{= \frac{π(\frac{d}{2})^2}{ 4 }= \frac{πd^2}{16}}\)
How to find the area of a quarter circle?
Here are the steps to find the area of a quarter circle:
- If the radius \((r)\) is given, immediately replace it with the formula \(\frac{πr^2}{4}\).
- If the diameter \((d)\) is given, solve \(d = 2r\) for \(r\) and use the formula \(\frac{πr^2}{4}\) (or) immediately replace the value of d in the formula \(\frac{πd^2}{16}\).
- When the circumference \((C)\) is given, solve \(C = 2πr\) for \(r\) and replace it in the formula \(\frac{πr^2}{4}\).
- When area \((A)\) is given, solve \(A = πr^2\) for \(r\) and replace it with the formula \(\frac{πr^2}{4}\) (or) simply find \(\frac{A}{4}\).
Finding Area of a Quarter Circle , Example 1:
A circle has a diameter of \(32\space cm\), find the area of a quarter circle. \((\pi =3.14)\)
Solution:
Diameter of circle \(= 32\space cm\)
Area of a quarter circle \(= \frac{πd^2}{16}\)
\(=\frac{3.14 × 32^2}{16}\)
\(=\frac{3.14 × 32×32}{16}\)
\(=\frac{3215.36}{16}\)
\(=200.96\space cm^2\)
Exercises for Finding Area of a Quarter Circle
- Find the area of the quadrant with a radius \(24\space cm\).
- Find the area of the quadrant with a diameter \(16 \space in\)
- \(\color{blue}{452.16\space cm^2}\)
- \(\color{blue}{50.24 \space in^2}\)
Recommended EffortlessMath Books
For a workbook that pairs every shape, formula, and proof with worked examples, the Geometry for Beginners walks you through every high-school geometry topic at your own pace. If you’re heading toward trig and pre-calc next, the Pre-Calculus for Beginners extends the same ideas into trigonometry and beyond.
Frequently Asked Questions
What is the formula to find the area of a quarter circle?
The area of a quarter circle is A = (1/4)πr², where r is the radius. It is one-fourth of a full circle’s area because the two straight edges meet at a 90° angle, which is one-fourth of 360°. For a radius of 2, A = (1/4)π(4) = π ≈ 3.14 square units. The same one-fourth idea works for the curved arc length too.
How do you find the area of a quarter circle with radius?
Square the radius, multiply by π, then divide by 4. For a quarter circle with radius 6 cm: 6² = 36, and 36π ÷ 4 = 9π, which is about 28.27 cm². The order does not matter, so you can also divide 36 by 4 first. Remember to write the answer in square units, since area measures the space inside the shape.
How do you find the area of a quarter circle with the diameter?
Halve the diameter to get the radius, then use A = (1/4)πr². With a diameter of 10 m, the radius is 5 m, so A = (1/4)π(25) = 25π/4 ≈ 19.63 m². You can also use A = πd²/16 directly. Plugging the diameter into the radius formula by mistake makes the answer four times too big.
How do you find the area of a quarter circle in terms of pi?
Work out (1/4)r² and leave π as a symbol instead of multiplying by 3.14. For a radius of 12, (1/4)(144) = 36, so the area is 36π square units. That exact form is what a question asking for an answer in terms of π wants. If it asks for a decimal instead, 36π ≈ 113.10 square units.
How do you find the perimeter of a quarter circle?
Add the curved arc to the two straight radii: P = 2r + πr/2. The arc is one-fourth of the circumference 2πr. For a radius of 6, the arc is 6π/2 = 3π ≈ 9.42, and the two radii add 12, so the perimeter is about 21.42 units. Students often forget the straight edges and give only the arc length.
How do you find the area of a three-quarter circle?
A three-quarter circle is the shape left after removing one quarter, so its area is (3/4)πr². With a radius of 4, that is (3/4)π(16) = 12π ≈ 37.70 square units. Another way is to find the full circle, 16π, and subtract one quarter, 4π, which also gives 12π. Both methods are a good check on each other.
Related EffortlessMath Lessons
If a topic on this page feels rusty, these short lessons go deeper:
- Area and circumference of circles (resource currently unavailable)
- How to find the area of a rectangle
- How to find the perimeter (resource currently unavailable)
- Secant angles
- Word problems involving area of quadrilaterals and triangles
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