How to Find the Area of a Quarter Circle?

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.

Related Topics

• How to Find the Area and Perimeter of the Semicircle?

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

1. Find the area of the quadrant with a radius \(24\space cm\).
2. Find the area of the quadrant with a diameter \(16 \space in\)

1. \(\color{blue}{452.16\space cm^2}\)
2. \(\color{blue}{50.24 \space in^2}\)

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Frequently Asked Questions

What’s a quarter circle?

A quarter circle is one-fourth of a full circle – the region bounded by two perpendicular radii and the connecting arc. It looks like a slice of pie with a \(90^\circ\) corner. The curved edge is the arc; the two straight edges are radii.

What’s the area formula?

\(A=\frac{1}{4}\pi r^2\). Take the full-circle area (\(\pi r^2\)) and divide by 4 because a quarter circle is one-fourth of the whole. For \(r=4\), area is \(\frac{1}{4}\pi(16)=4\pi\approx 12.57\) square units. For \(r=10\), area is \(25\pi\approx 78.54\) square units. Keep the answer in terms of \(\pi\) when the problem allows exact answers.

How do I find the perimeter?

Add the two straight sides and the curved arc: \(P=2r+\frac{\pi r}{2}\). For \(r=4\), perimeter is \(2(4)+\frac{\pi(4)}{2}=8+2\pi\approx 14.28\) units. Don’t forget the straight sides – some students only compute the arc.

What’s the arc length of a quarter circle?

\(s=\frac{1}{4}(2\pi r)=\frac{\pi r}{2}\). The full circle has circumference \(2\pi r\), so a quarter is one-fourth of that. For \(r=10\), arc length is \(\frac{\pi(10)}{2}=5\pi\approx 15.71\) units. The two straight sides each have length \(r\), so the total perimeter is \(2r + \frac{\pi r}{2}\).

What’s the angle of a quarter circle?

The central angle is \(90^\circ\) or \(\pi/2\) radians. That’s literally what makes it a “quarter” – it spans one-fourth of a full \(360^\circ\) rotation. The two straight edges meet at this \(90^\circ\) angle.

Walk through a worked example?

A quarter-circle garden has radius 8 ft. Area: \(A=\frac{1}{4}\pi(8)^2=\frac{64\pi}{4}=16\pi\approx 50.27\) ft\(^2\). Arc length: \(s=\frac{\pi(8)}{2}=4\pi\approx 12.57\) ft. Perimeter (if you fence the whole boundary): \(2(8)+4\pi=16+4\pi\approx 28.57\) ft. If grass seed costs \(\$2\) per square foot, the garden takes about \(\$100.54\) of seed.

What if the problem gives me the diameter?

Divide by 2 to get the radius first. If the diameter is 14, the radius is 7. Then \(A=\frac{1}{4}\pi(7)^2=\frac{49\pi}{4}\approx 38.48\). Mistaking diameter for radius is a common slip – quadruples the area.

Can I leave answers in terms of \(\pi\)?

Yes – many problems prefer exact answers like \(9\pi\) over decimal approximations like 28.27. “In terms of \(\pi\)” means don’t multiply out; just leave the symbol. If the problem says “to the nearest hundredth,” then plug in \(\pi\approx 3.14159\) and round.

How does a quarter circle compare to a semicircle?

A semicircle is half a circle; a quarter circle is one-fourth. Semicircle area is \(\frac{1}{2}\pi r^2\); quarter-circle area is \(\frac{1}{4}\pi r^2\) – exactly half of the semicircle. Their arc lengths follow the same pattern.

Where do quarter-circle problems show up on tests?

Middle-school and high-school geometry tests, the SAT, ACT, GED, HiSET, and most state tests. Common formats: find the area of a quarter circle, find the area of a region made of a quarter circle inside a square, find the perimeter of a shape combining quarter circles and straight edges.

Related EffortlessMath Lessons

If a topic on this page feels rusty, these short lessons go deeper:

• Area and circumference of circles
• How to find the area of a rectangle
• How to find the perimeter
• Secant angles
• Word problems involving area of quadrilaterals and triangles