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.
Find the area of a quadrant whose radius is 28 cm. Use $\pi = \dfrac{22}{7}$.
A circle has diameter 20 cm. Find the area of one quadrant. Use $\pi = 3.14$.
Find the perimeter of a quadrant whose radius is 7 cm. Use $\pi = \dfrac{22}{7}$.
The area of a quadrant is 78.5 cm². Find its radius. Use $\pi = 3.14$.
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.
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