Circle Area Calculator
Calculate the area of a circle from its radius, diameter or circumference.
About
Circle Area Calculator
The area of a circle is A = πr². Enter a radius above and the calculator returns the area with its working, the exact multiple of π, and the same figure converted into square feet, acres and hectares.
What the formula means
A circle is every point sitting the same distance from a centre, and that distance is the radius. It is the only measurement a circle has. The diameter is 2r, the circumference is 2πr, and the area is πr².
The radius is squared, so area responds sharply to size. Double the radius and the area quadruples. Treble it and the area goes up ninefold. Two pizzas of 12 inches do not equal one of 24 inches: the large one carries four times the area of each small one, so it beats both together.
Where pi r squared comes from
Cut a circle into many thin wedges and lay them alternately point-up and point-down. They form something close to a rectangle. Its height is the radius, and its length is half the circumference, πr, because half the wedges point each way. Area is height times length, giving r × πr = πr². The more wedges you cut, the closer the shape gets to a true rectangle, and the argument becomes exact in the limit.
A worked example
Take a radius of 30 units.
- Square the radius: 30² = 900
- Multiply by π: 900π
- As a decimal: 900 × 3.14159265 = 2827.4334 square units
The exact form, 900π, is often what coursework asks for, and it makes clear that the decimal is a rounding. The calculator prints the exact multiple whenever the answer lands on one.
Starting from diameter or circumference
Round objects are measured across, not from the centre, so most real problems start with a diameter. Halve it first. A circle 20 cm across has a radius of 10 cm and an area of 100π = 314.16 cm².
You can also work straight from the diameter with A = πd²/4, which avoids the halving step and is the form used in engineering for pipe bores. A pipe of 100 mm bore has a cross-section of π × 10,000 / 4 = 7854 mm².
Circumference is the practical measurement for anything too large to reach across, such as a tree trunk or a tank. Divide by 2π to recover the radius, or use A = C²/4π directly. A trunk measuring 1.5 m around has a cross-sectional area of 2.25/12.566 = 0.179 m².
The pizza arithmetic
Pizza is sold by diameter and eaten by area, which is why the pricing rarely matches the value. A 12 inch pizza has a radius of 6 inches and an area of 36π, about 113.1 square inches. A 16 inch has a radius of 8 and an area of 64π, about 201.06 square inches.
The 16 inch carries 1.778 times the food of the 12 inch, though it sounds only a third larger. If the larger costs less than 78% more, it is the better buy. The same reasoning applies to cake tins, round tables and any circular thing priced by its width.
Rings and sectors
An annulus is the area between two concentric circles, the shape of a washer, a running track lane or a pipe wall seen end on. Its area is π(R² − r²), the outer circle minus the inner one.
A sector is a wedge, a fraction of the whole circle set by its angle. Its area is (θ/360) × πr² in degrees. A quarter circle is 90/360, so a quarter of the area. A segment, the region cut off by a straight chord, needs the sector area minus the triangle formed by the two radii and the chord.
Units and conversions
| Radius in | Area in | Useful conversion |
|---|---|---|
| metres | m² | 10,000 m² = 1 hectare |
| feet | ft² | 43,560 ft² = 1 acre |
| centimetres | cm² | 10,000 cm² = 1 m² |
| inches | in² | 144 in² = 1 ft² |
Area conversions square the linear factor. A metre is roughly 3.28 feet, so a square metre is 3.28² = 10.76 square feet, not 3.28. Forgetting to square the factor is the most frequent conversion error here.
Where it gets used
Flow and pressure. The volume a pipe can carry depends on its cross-sectional area, which is why doubling a pipe's diameter quadruples its capacity at the same velocity. Pipe sizing starts with πr².
Irrigation and land. A centre-pivot irrigation system waters a circle. A pivot with a 400 m arm covers π × 160,000 = 502,655 m², which is 50.3 hectares.
Materials. The load a column or a bolt can carry depends on its cross-sectional area, so stress calculations begin here.
Everyday buying. Rugs, tables, tins and paddling pools are all sold by diameter and used by area.
Circles in construction and land
Circular slabs, patios and fire pits are priced by area, and the material comes from area multiplied by depth. A patio 4 m across needs π × 4 = 12.57 m² of paving, and at 100 mm of sub-base that is 1.26 m³ of hardcore before compaction.
Round tanks and silos are sized the same way. Capacity is the base area multiplied by the working height, and the base area is where the calculation starts. A grain silo 6 m across holds π × 9 = 28.27 m² per metre of depth, so 5 m of grain is 141 m³.
Why circles appear so often
Of all shapes with a given perimeter, the circle encloses the most area. This is the isoperimetric property, and it is the reason so many natural and engineered forms are round.
A circle of 40 m circumference encloses 127.3 m². A square of the same 40 m perimeter encloses only 100 m². A 19 × 1 rectangle with that perimeter encloses 19 m². If you have a fixed length of fencing and want the most land, curve it.
The same property runs the other way for containment. A circle needs the least edge for a given area, so a round pipe uses the least metal for a given flow, a round tank the least steel for a given volume, and a round cell the least membrane for a given interior.
Approximating pi, and how close is close enough
| Value used for π | Error | On a 1 hectare circle |
|---|---|---|
| 3 | 4.5% low | 450 m² short |
| 3.14 | 0.05% low | 5 m² short |
| 22/7 | 0.04% high | 4 m² over |
| 3.14159265 | negligible | under 1 cm² |
For a homework answer 3.14 is fine. For anything you are ordering material against, use the full value, which the calculator above does.
Measuring a circle you cannot reach across
Tree trunks, tanks, columns and pipes are all easier to measure around than across. Wrap a tape, read the circumference, and divide by π for the diameter or by 2π for the radius.
Foresters do this so routinely that they use a tape marked in diameter rather than length, so the division is already done. A trunk reading 60 cm on such a tape has a true circumference of 188.5 cm.
Common mistakes
Using diameter as radius. This quadruples the answer. It is the single most common error, and the reason engineers often prefer the πd²/4 form.
Confusing area with circumference. Circumference is 2πr, the distance round the edge. Area is πr², the space inside. Fencing a circular field needs the first; seeding it needs the second.
Not squaring the conversion factor. Converting square metres to square feet needs 10.76, not 3.28.
Rounding π early. Using 3.14 introduces about 0.05% of error, which is invisible on homework and matters across a hectare.
Common questions
Frequently asked questions
A = pi r squared, where r is the radius. A circle of radius 30 has an area of 900 pi, which is 2827.4334 square units.
Halve the diameter to get the radius, or use A = pi d squared over 4 directly. A circle 20 cm across has an area of 314.16 square centimetres.
Divide the circumference by 2 pi to get the radius, or use A = C squared over 4 pi. A trunk measuring 1.5 m around has a cross-section of 0.179 square metres.
It quadruples, because the radius is squared. This is why one 16 inch pizza carries 1.778 times the food of a 12 inch one despite sounding only a third bigger.
Circumference is the distance around the edge, 2 pi r. Area is the space inside, pi r squared. Fencing a round field uses circumference; seeding it uses area.
Subtract the inner circle from the outer one: pi times R squared minus r squared, where R is the outer radius.
For a sector, multiply the full area by the angle over 360. For a segment cut off by a chord, take the sector area and subtract the triangle formed by the two radii.
Area conversions square the linear factor. One metre is 3.28 feet, so one square metre is 3.28 squared, which is 10.76 square feet.