Area of a Quadrant: Formula, Definition, Examples

What Is a Quadrant of a Circle?

A quadrant of a circle is one-fourth of the circle, the region cut off by two radii that meet at the centre at a right angle, together with the arc between them. Because two perpendicular radii split a full 360° circle into four equal 90° pieces, each piece is a quadrant, and all four are identical in size. A quadrant is also called a quarter circle, and the two names mean the same thing.

A quadrant is a special sector of a circle, the sector whose central angle is exactly 90°. Every quadrant is a sector, but only the 90° sector is a quadrant. You meet quadrants in quarter-round windows, 90° pipe bends, and the curved corner of a running track.

For an Example: A corner sprinkler waters exactly one quarter of a circle, and its reach sets the area.

Push the sprinkler into the corner of a lawn and it sweeps 90°, spraying a clean quarter-circle of grass. Change its reach and the watered area changes with the square of that reach, not in proportion to it. That quarter-circle region has a name in geometry, the quadrant, and one short formula tells you exactly how much ground it covers.

How Do You Find the Area of a Quadrant?

Since a quadrant is one of four equal parts of a circle, its area is simply the area of the circle divided by 4.

$$\text{Area of a quadrant} = \frac{1}{4}\times \pi r^2 = \frac{\pi r^2}{4}.$$

Variable glossary: $r$ is the radius of the circle and $\pi \approx 3.14159$ (often taken as $\tfrac{22}{7}$ for exam arithmetic).

If you are given the diameter $d$ instead of the radius, substitute $r = \dfrac{d}{2}$:

$$\text{Area} = \frac{\pi}{4}\left(\frac{d}{2}\right)^2 = \frac{\pi d^2}{16}.$$

How Is the Quadrant Area Related to the Full Circle?

The formula is not a rule to memorise blindly; it comes straight from the whole circle. A full circle spans a central angle of 360° and has area $\pi r^2$. A quadrant spans 90°, which is $\tfrac{90}{360} = \tfrac{1}{4}$ of the turn. Area scales with the fraction of the turn, so:

$$\text{Area of quadrant} = \frac{90^\circ}{360^\circ}\times \pi r^2 = \frac{1}{4}\times \pi r^2 = \frac{\pi r^2}{4}.$$

The four quadrants of a circle add back to the whole: $4 \times \dfrac{\pi r^2}{4} = \pi r^2$. Seeing the formula as "one-quarter of the turn" also explains why a semicircle is $\tfrac{\pi r^2}{2}$ and a full circle is $\pi r^2$, all the same idea at different fractions.

What Is the Perimeter of a Quadrant?

The perimeter of a quadrant is a common follow-up question, and it is not the same as its area. The boundary is made of three parts: the two straight radii and the curved arc.

The arc is one-quarter of the full circumference, so the arc length is $\tfrac{1}{4}(2\pi r) = \tfrac{\pi r}{2}$. Adding the two radii:

$$\text{Perimeter} = r + r + \frac{\pi r}{2} = 2r + \frac{\pi r}{2}.$$

The two straight radii are easy to forget, and leaving them out is a frequent slip covered in the mistakes section below.

Examples of Area of a Quadrant

The examples run from a direct substitution to a real-world problem that mixes area and perimeter.

Example 1

Find the area of a quadrant whose radius is 7 cm. Use $\pi = \dfrac{22}{7}$.

$$\text{Area} = \frac{\pi r^2}{4} = \frac{\frac{22}{7}\times 7^2}{4} = \frac{\frac{22}{7}\times 49}{4} = \frac{22 \times 7}{4} = \frac{154}{4} = 38.5 \text{ cm}^2.$$

Final answer: 38.5 cm².

Example 2

A student finds the area of a quadrant of radius 10 cm as $\dfrac{\pi (10)^2}{2}$. What went wrong, and what is the correct area? Use $\pi = 3.14$.

Watch the wrong path first. The student divided by 2 instead of 4, which is the formula for a semicircle, not a quadrant. A quadrant is one of four parts, so the divisor is 4.

$$\text{Area} = \frac{\pi r^2}{4} = \frac{3.14 \times 100}{4} = \frac{314}{4} = 78.5 \text{ cm}^2.$$

The wrong method gives 157 cm² (a half circle); the correct quadrant area is exactly half of that.

Final answer: 78.5 cm².

Example 3

The diameter of a circle is 32 cm. Find the area of one quadrant. Use $\pi = 3.14$.

Use the diameter form, or first halve to get $r = 16$ cm.

$$\text{Area} = \frac{\pi d^2}{16} = \frac{3.14 \times 32^2}{16} = \frac{3.14 \times 1024}{16} = 3.14 \times 64 = 200.96 \text{ cm}^2.$$

Final answer: 200.96 cm².

Example 4

Find the perimeter of a quadrant of radius 14 cm. Use $\pi = \dfrac{22}{7}$.

$$\text{Perimeter} = 2r + \frac{\pi r}{2} = 2(14) + \frac{\frac{22}{7}\times 14}{2} = 28 + \frac{44}{2} = 28 + 22 = 50 \text{ cm}.$$

Final answer: 50 cm.

Example 5

A quadrant has area 154 cm². Find its radius. Use $\pi = \dfrac{22}{7}$.

Solve the area formula for $r$.

$$\frac{\pi r^2}{4} = 154$$

$$r^2 = \frac{154 \times 4}{\pi} = \frac{616 \times 7}{22} = 196$$

$$r = 14 \text{ cm}.$$

Final answer: the radius is 14 cm.

Example 6

A quarter-circle flower bed has radius 3.5 m. How much turf covers it, and how much edging fences its curved arc only? Use $\pi = \dfrac{22}{7}$.

Turf is the area:

$$\frac{\pi r^2}{4} = \frac{\frac{22}{7}\times (3.5)^2}{4} = \frac{\frac{22}{7}\times 12.25}{4} = \frac{38.5}{4} = 9.625 \text{ m}^2.$$

Edging along the curved arc only is the arc length:

$$\frac{\pi r}{2} = \frac{\frac{22}{7}\times 3.5}{2} = \frac{11}{2} = 5.5 \text{ m}.$$

Final answer: 9.625 m² of turf and 5.5 m of curved edging.

Where Do Quadrants Show Up?

Quadrants matter because a 90° corner is one of the most common shapes in the built world, and the quarter-circle is how a curve fits into that corner. A learner often treats the formula as exam-only, yet it shows up the moment a straight edge meets a rounded one.

• Architecture uses quarter-round windows and arches, where the glass area is a quadrant.

• Plumbing and engineering rely on 90° pipe bends, whose swept region is a quadrant.

• Landscaping fits quarter-circle beds and lawns into square corners, sizing turf and edging separately.

• Design and manufacturing round square corners with a quarter-circle fillet to spread stress.

The formula's real usefulness is that it separates the two questions a corner curve raises, how much surface it covers (area) and how much edge it has (perimeter), which are answered by two different formulas. A helpful reference on the general shape is the circular sector.

What Are the Most Common Mistakes With Area of a Quadrant?

Two mistakes cause most errors, and both come from mixing up related circle formulas.

Mistake 1: Dividing by 2 instead of 4

Where it slips in: problems that mention "quarter" but the student pictures "half."

Don't do this: write $\dfrac{\pi r^2}{2}$. The first-instinct error is reaching for the semicircle formula because both start with "part of a circle."

The correct way: a quadrant is one of four equal parts, so divide by 4: $\dfrac{\pi r^2}{4}$.

Mistake 2: Forgetting the two radii in the perimeter

Where it slips in: perimeter questions where the arc feels like the whole boundary.

Don't do this: report only the arc $\dfrac{\pi r}{2}$ as the perimeter.

The correct way: the boundary of a quadrant is two straight radii plus one arc, so the perimeter is $2r + \dfrac{\pi r}{2}$.

Conclusion

• The area of a quadrant is one-fourth of a circle: $\dfrac{\pi r^2}{4}$.

• In terms of diameter it is $\dfrac{\pi d^2}{16}$.

• The formula comes from the 90°-of-360° fraction of the whole circle.

• The perimeter is $2r + \dfrac{\pi r}{2}$, the two radii plus the arc.

• A quadrant is the 90° sector, and four quadrants rebuild the full circle.

To practise quadrant and sector problems with a teacher, explore Bhanzu's geometry tutor or a middle school math tutor, or browse math classes online.

A Practical Next Step

Test your understanding with these problems: find the area of a quadrant of radius 21 cm, then find the perimeter of the same quadrant, and check that four of these areas rebuild the full circle $\pi(21)^2$. Next, given a quadrant of area $38.5$ cm², find its arc length. If the divisor trips you, return to the full-circle derivation above. Want a live Bhanzu trainer to work through circle mensuration with you? Book a free demo class.

Practice Questions on Area of a Quadrant

Try these on your own, then check the answers below.

1. Find the area of a quadrant whose radius is 28 cm. Use $\pi = \dfrac{22}{7}$.

2. A circle has diameter 20 cm. Find the area of one quadrant. Use $\pi = 3.14$.

3. Find the perimeter of a quadrant whose radius is 7 cm. Use $\pi = \dfrac{22}{7}$.

4. The area of a quadrant is 78.5 cm². Find its radius. Use $\pi = 3.14$.

5. Four quadrants of radius 5 cm are joined to rebuild a full circle. Find the total area. Use $\pi = 3.14$.

Answers: 1. 616 cm² 2. 78.5 cm² 3. 25 cm 4. 10 cm 5. 78.5 cm²

Read More

• Segment of a circle — the region between a chord and its arc.

• Parts of a circle — radius, diameter, chord, arc, and sector defined.

• Diameter of a circle — the measurement behind the $\pi d^2/16$ form.

• Perimeter of a circle formula — the circumference the quadrant arc is a quarter of.

• Chord of a circle — the straight cut that defines segments and sectors.