How does area change when you double the radius?

Area scales with the square of the radius. Doubling r multiplies the area by 4. Tripling r multiplies it by 9. Formula: A = pi r squared. So a circle with r = 10 has 4 times the area of a circle with r = 5, even though the radius is only twice as large.

Area of a Circle Calculator

Q: How do I find the area of a circle from the circumference?

Start with circumference C = 2 pi r, so r = C / (2 pi). Then area = pi times r squared = pi times (C / (2 pi)) squared = C squared / (4 pi). For circumference 31.416, area = 31.416 squared / (4 pi) = 987.0 / 12.566 = 78.54 sq units.

Q: What is the area of a unit circle?

A unit circle has radius r = 1. Its area = pi times 1 squared = pi square units (approximately 3.14159). Its circumference = 2 pi (approximately 6.28318). The unit circle is fundamental in trigonometry for defining sine, cosine, and all six trigonometric functions.

Find the area, circumference, and diameter of any circle from its radius, diameter, or circumference.

⭕ What is the Area of a Circle?

The area of a circle is the total space enclosed within its boundary. It is one of the most fundamental quantities in geometry and is calculated using the formula A = pi times r squared, where r is the radius and pi is the mathematical constant approximately equal to 3.14159. The formula was derived by ancient Greek mathematicians, including Archimedes, who approximated pi by inscribing and circumscribing polygons around a circle.

The area of a circle appears in countless practical applications. In construction and architecture, circular areas determine the footprint of columns, pillars, round windows, and curved pools. In engineering, the cross-sectional area of a pipe or cable determines its flow capacity and resistance. In agriculture, irrigation systems covering a circular field require the area calculation to plan water usage. In everyday life, pizza size comparisons, paint coverage for circular walls, and fabric requirements for round tablecloths all use circle area formulas.

The key insight of the formula A = pi r squared is that area scales with the square of the radius. This means a circle with twice the radius has four times the area. This non-linear relationship is why large pizza sizes are significantly better value than small ones: a 14-inch pizza has about 3.4 times the area of an 8-inch pizza, not 1.75 times as the diameter ratio might suggest.

This calculator accepts three different input types, covering every common scenario. If you know the radius (distance from the center to the edge), use Radius mode. If you measured across the full circle, use Diameter mode (the calculator halves it automatically). If you measured the perimeter of the circle (for example, with a measuring tape around a cylindrical object), use Circumference mode and the calculator back-calculates the radius using r = C divided by (2 pi), then computes the area.

📐 Formula

From Radius (r):

A = πr²

A = Area of the circle (in square units)

r = Radius (distance from center to edge)

π = Pi ≈ 3.14159265

Example: r = 7 cm → A = π × 49 = 153.9380 cm²

From Diameter (d):

A = π(d/2)² = πd² / 4

d = Diameter = 2r (full width across the circle)

Example: d = 10 cm → r = 5 cm → A = π × 25 = 78.5398 cm²

From Circumference (C):

r = C / (2π) → A = C² / (4π)

C = Circumference = 2πr (total perimeter length of the circle)

Example: C = 31.416 cm → r = 5.000 cm → A = 78.540 cm²

📖 How to Use This Calculator

Steps

Select your input type - choose From Radius, From Diameter, or From Circumference depending on which measurement you know.

Enter your value - type any positive number. All measurements should use the same length unit (e.g. all in cm, or all in inches). The slider lets you explore quickly without typing.

Read the results - area, circumference, diameter, and radius all appear instantly to 4 decimal places. The formula note confirms the exact calculation performed.

Share your result - use Copy result to paste into a document, Copy link to bookmark exact inputs, or WhatsApp to send to a colleague.

💡 Example Calculations

Example 1 - Radius of 7 cm

A circular garden has a radius of 7 m. Find its area.

Radius r = 7 units

Area = π × 7² = 3.14159 × 49 = 153.9380 sq units

Circumference = 2 × π × 7 = 43.9823 units

Diameter = 2 × 7 = 14 units

Area = 153.9380 sq units | Circumference = 43.9823 units

Try this example →

Example 2 - Diameter of 10 cm

A circular pizza has a diameter of 10 inches. Find its area.

Diameter d = 10 units

Radius = 10 / 2 = 5 units

Area = π × 5² = π × 25 = 78.5398 sq units

Circumference = 2 × π × 5 = 31.4159 units

Area = 78.5398 sq units | Radius = 5 units

Try this example →

Example 3 - Circumference of 62.832 units

A circular fence has a total length of 62.832 m. Find the area it encloses.

Circumference C = 62.832 units

Radius = C / (2π) = 62.832 / 6.28318 = 10 units

Area = π × 10² = π × 100 = 314.1593 sq units

Diameter = 2 × 10 = 20 units

Area = 314.1593 sq units | Radius = 10 units

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Example 4 - Radius of 3.5 units

A circular drain cover has a radius of 3.5 cm. Find its area for cutting a replacement.

Radius r = 3.5 units

Area = π × 3.5² = π × 12.25 = 38.4845 sq units

Diameter = 2 × 3.5 = 7 units

Circumference = 2 × π × 3.5 = 21.9911 units

Area = 38.4845 sq units | Circumference = 21.9911 units

Try this example →

❓ Frequently Asked Questions

What is the formula for the area of a circle?+

The area of a circle is A = pi times r squared, where r is the radius. Pi is approximately 3.14159. For a circle with radius 5 cm, area = 3.14159 times 25 = 78.54 sq cm. If you know the diameter d, then r = d / 2, and area = pi times d squared / 4.

How do I find the area of a circle from the circumference?+

Use the formula A = C squared / (4 pi). For circumference C = 31.416, area = 987.0 / 12.566 = 78.54 sq units. Alternatively, find the radius first: r = C / (2 pi), then apply A = pi r squared.

What is pi and why is it used in circle calculations?+

Pi is the ratio of a circle's circumference to its diameter, approximately 3.14159265358979. It is irrational (its decimal expansion never repeats) and appears in every formula involving circles, spheres, and cylinders. All circles share this same circumference-to-diameter ratio, which is why pi is a universal constant.

How does the area change when the radius doubles?+

Area scales with the square of the radius. Doubling r multiplies area by 4. Tripling r multiplies area by 9. A circle with r = 10 has area 314.16, which is 4 times the area of a circle with r = 5 (area 78.54). This is why a 16-inch pizza has four times the area of an 8-inch pizza, not twice.

What is the area of a circle with diameter 12?+

Radius = 12 / 2 = 6. Area = pi times 6 squared = pi times 36 = 113.0973 sq units. Circumference = 2 times pi times 6 = 37.6991 units. This is a common textbook example: remember r = d / 2 always comes first when given the diameter.

How do I find the radius if I only know the area?+

Rearrange A = pi r squared to get r = square root of (A / pi). For area = 50 sq cm, r = sqrt(50 / 3.14159) = sqrt(15.9155) = 3.9894 cm. Then circumference = 2 pi r = 25.066 cm.

What is the area of a unit circle?+

A unit circle has radius r = 1. Area = pi times 1 = pi sq units (approximately 3.14159). Circumference = 2 pi (approximately 6.28318). The unit circle is fundamental in trigonometry as the reference for defining sine, cosine, and all six trigonometric functions.

What is pi r squared used for in real life?+

Circle area calculations are used in: engineering (pipe cross-sections for flow rate), construction (column footprints, manhole covers), agriculture (circular irrigation pivot area), manufacturing (circular gasket sizing), food service (pizza size comparisons), and astronomy (planetary cross-sectional area for shadow calculations).

What is the difference between area and circumference of a circle?+

Area measures the 2D space inside the circle in square units (A = pi r squared). Circumference is the length around the boundary of the circle in linear units (C = 2 pi r). For a circle with r = 5: area = 78.54 sq cm (space inside), circumference = 31.42 cm (fence length needed around it).

How accurate is this area of a circle calculator?+

This calculator uses JavaScript's Math.PI (15+ significant digits) for maximum precision. Results are displayed to 4 decimal places. For practical purposes like carpentry, engineering, or cooking, this is far more precise than required. For scientific or academic work, the results match textbook answers to at least 4 significant figures.

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📌 Quick Tips

💡The area of a circle grows with the square of the radius. Doubling the radius quadruples the area: A = pi times r squared.

💡To find the area from the diameter, use A = pi times (d/2) squared = pi times d squared divided by 4. This avoids the extra step of halving the diameter.

💡If you only know the circumference C, the area is A = C squared divided by (4 times pi). No need to find the radius first.

💡Pi (pi) is approximately 3.14159265. For most practical purposes, using 3.14 is sufficient. This calculator uses JavaScript's built-in Math.PI (15+ significant digits) for maximum precision.

💡The ratio of any circle's area to the square of its radius is always pi. This relationship holds for circles of any size, from atomic to astronomical.