Area of a Quarter Circle: Formula & Examples

This calculator finds the area of a quarter circle when you enter its radius (or diameter). Type the radius in your chosen unit, and it returns the area in square units, along with the full circle area for comparison. The formula is simple: area = (π × r²) / 4, where r is the radius.

Formula

The area of a quarter circle is exactly one-fourth the area of a full circle. If you know the radius r, the formula is:

A = (π r²) / 4

• A – area of the quarter circle (same square units as r²)
• r – radius (any unit: cm, m, in, ft, etc.)
• π – pi, approximately 3.14159

If you only know the diameter d, use r = d/2 first, then the same formula. Many people mistakenly use the diameter directly in the formula, which gives an area four times too large.

Worked Example

Using the default values from the calculator above (r = 10 cm):

1. Square the radius: 10² = 100 cm²
2. Multiply by π: 100 × 3.14159 ≈ 314.159 cm²
3. Divide by 4: 314.159 / 4 = 78.54 cm²

The quarter circle area is 78.54 cm², and the full circle area is 314.16 cm². The calculator rounds to two decimals, matching this result.

Why Divide by 4?

A quarter circle is a 90° sector. A full circle is 360°, so a quarter is 360° / 4 = 90°. Since the area of a sector is proportional to its central angle, the area of any sector is (angle/360) × π r². For a quarter, angle/360 = 1/4, so you get (π r²)/4. This works for any angle: a 60° sector would be (60/360) × π r² = (π r²)/6.

When to Use This Formula

Quarter circles appear in many real‑life situations:

• Designing a rounded corner on a table, countertop, or garden bed
• Calculating the area of a semicircular or quarter‑circular window
• Finding the area of a quadrant in a coordinate plane (e.g., the region x ≥ 0, y ≥ 0 inside a circle)
• Estimating material needed for a quarter‑round trim or a curved patio

In geometry problems, the quarter circle often appears as part of a composite shape. You can subtract its area from a rectangle or add it to another shape to find a total area.

Common Mistakes and How to Avoid Them

• Using the diameter instead of the radius: The formula uses r². If you have the diameter, halve it first. For example, a quarter circle with diameter 10 cm has radius 5 cm, area = (π × 25)/4 ≈ 19.63 cm², not (π × 100)/4 ≈ 78.54 cm².
• Forgetting to square the radius: r² means r × r, not r × 2. A radius of 10 gives 100, not 20.
• Mixing units: If the radius is in meters, the area is in square meters. Convert all measurements to the same unit before calculating.
• Rounding too early: Keep π as a constant (3.14159 or the calculator’s value) until the final step to avoid rounding errors.

How to Read the Result

The result is always in square units. If you entered centimeters, the area is in square centimeters (cm²). For inches, it’s square inches (in²). This is because you multiply two lengths together. To convert to a different square unit, use the appropriate conversion factor: 1 m² = 10,000 cm², 1 ft² = 144 in², etc. The calculator gives the area directly, so you don’t need to do any extra steps.

Limitations of the Formula

The formula assumes a perfect mathematical circle and a perfectly straight 90° angle. In real objects, a “quarter circle” might be slightly irregular, especially if it’s a cut piece of material. In such cases, the formula gives a close approximation but not an exact measurement. Also, if you’re working with a three‑dimensional object (like a quarter of a cylinder), you’d need a volume formula, not this area formula.

Frequently Asked Questions

How do I find the area of a quarter circle if I only know the diameter?

Divide the diameter by 2 to get the radius, then use A = (π r²)/4. For example, a diameter of 20 cm gives a radius of 10 cm, so the area is (π × 100)/4 ≈ 78.54 cm².

What is the perimeter of a quarter circle?

The perimeter includes the curved arc plus the two straight radii. The arc length is (π r)/2, and the two radii add 2r. So the total perimeter is (π r)/2 + 2r = r(π/2 + 2). For r = 10 cm, that’s about 35.7 cm.

Can I use the formula for a semicircle too?

Yes, but you’d divide by 2 instead of 4: A = (π r²)/2. A quarter circle is a 90° sector, while a semicircle is a 180° sector.

Why is the area of a quarter circle not just (π r²)/4? Is there any special case?

No special case – it’s always (π r²)/4 for a perfect quarter circle. The only variation is if you’re dealing with an ellipse or another shape, but for a true circle, the formula is fixed.

How do I calculate the area if the quarter circle is part of a larger shape?

Break the shape into known parts. Find the area of the rectangle or triangle that contains the quarter circle, then add or subtract the quarter circle’s area as needed. For example, a square with a quarter circle cut out has area = side² − (π r²)/4.

What if I only know the arc length?

If you know the arc length L, then L = (π r)/2, so r = 2L/π. Then plug that r into the area formula. For instance, if the arc length is 15.7 cm, r ≈ 10 cm, and the area is about 78.5 cm².

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in r | Radius | cm | 10 | 0
out area_q | Quarter circle area | cm² | (pi * r^2) / 4 | 2
out area_full | Full circle area | cm² | pi * r^2 | 2