Circle Calculator

How to use this calculator

1. Pick what you actually haveAny one of the four measures determines the circle completely. Choosing the right one avoids converting by hand first.
2. Keep units consistentThe calculator is unit-agnostic — enter centimetres and you get square centimetres. Mixing units is the most common source of a wrong answer.
3. Add an angle for a sectorA sector is a fraction of the circle, so 90° gives a quarter. The chord and segment are given too, which is what pipe and tank problems usually need.
4. Read the fill percentageThe last rows show how much of the enclosing square the circle occupies — useful for material yield when cutting circles from sheet.

Everything follows from the radius

A circle has one degree of freedom. Fix any single measure and every other is determined, which is why the calculator works backwards to the radius first whatever you enter.

From the diameter, halve it. From the circumference, divide by 2π. From the area, divide by π and take the square root. Then the rest follow directly.

The relationship worth internalising is that area scales with the <em>square</em> of the radius while circumference scales linearly. Double the radius and the circumference doubles, but the area quadruples.

That has real consequences. A 12-inch pizza has 2.25 times the area of an 8-inch one, not 1.5 times — which is why the larger size is almost always better value despite looking only modestly bigger. A pipe of twice the diameter carries roughly four times the flow at the same velocity, which is why small increases in bore make such a difference to plumbing.

It also means a circle is remarkably efficient at enclosing area. Of all shapes with a given perimeter, the circle encloses the most — a fact that explains why bubbles are round, why pressure vessels are cylindrical, and why a circular pen uses least fencing for a given field.

The circle fills about 78.5% of its enclosing square, and that figure is π/4 exactly. It is worth knowing because it tells you the material yield when cutting discs from sheet: rather more than a fifth of the sheet becomes offcut no matter how carefully you nest them.

A sector is simply a fraction of the whole, so both arc length and sector area are that fraction of the circumference and area. A segment — the region between a chord and the arc — is the sector minus the triangle formed by the two radii, which is why it needs the sine term.

Worked example: a radius of 5

Diameter: 2 × 5 = 10.

Circumference: 2π × 5 = 31.4159.

Area: π × 5² = π × 25 = 78.5398 square units.

Now a 90° sector, which is a quarter of the circle.

Arc length: 31.4159 × 90 ÷ 360 = 7.854.

Sector area: 78.5398 × 90 ÷ 360 = 19.635.

Chord: the straight line joining the two ends of the arc. It is 2r·sin(θ/2) = 2 × 5 × sin 45° = 7.0711. Note this is the diagonal of a square of side 5, as it must be for a right-angled sector.

Segment: the area between the chord and the arc, which is the sector less the triangle. That gives ½r²(θ − sin θ) = 0.5 × 25 × (1.5708 − 1) = 7.1350.

So of the 19.635 sector, 12.5 is the triangle and 7.135 is the segment beyond the chord.

Now check the scaling claim. Double the radius to 10: circumference becomes 62.832, exactly double. Area becomes 314.159, exactly four times. The square relationship holds precisely.

And the enclosing square: a circle of radius 5 sits inside a 10 by 10 square of area 100. The circle occupies 78.54 of that, which is 78.54% — π/4, as it always is regardless of size.

Common circles

The radius-2 row is a curiosity worth noticing: circumference and area are numerically equal, both 4π. That happens only at radius 2, because C = 2πr and A = πr² are equal when 2r = r², so r = 2. It is a coincidence of units rather than anything geometric — the two quantities have different dimensions and the equality vanishes if you change the unit.

Where circle geometry actually matters

Circles turn up constantly in practical work, usually in the form of a volume or a flow rate rather than a plain area.

A cylinder's volume is the circular area times the length, so pipe capacity, tank volume and cylinder displacement all start with πr². A pipe of 50 mm bore holds 1.96 litres per metre; one of 100 mm holds 7.85 litres per metre — four times, not twice.

Flow rate follows the same square relationship at a given velocity, which is why undersized pipework throttles a system so severely and why plumbers care about bore rather than nominal size.

Cutting discs from sheet material is a yield problem governed by the π/4 figure. Even perfect hexagonal nesting leaves around 9% waste; a simple grid layout leaves 21.5%. That is worth knowing before ordering material.

Circular areas for landscaping — a round patio, a fire pit base, a circular lawn — need the area to order aggregate or turf, and the circumference to order edging. The two scale differently, so a modest increase in size adds much more material than edging.

Sectors appear in pie charts, where the angle for a category is its share of the total times 360°, and in any pivoting or hinged arrangement where something sweeps through an arc.

Segments matter for partially filled horizontal tanks. Working out how much liquid is in a cylindrical tank lying on its side from a dipstick reading is a segment area problem, and it is not linear — the same rise in level corresponds to very different volumes near the bottom, middle and top.

For the ratio between diameter and area, one shortcut is worth remembering: area is roughly 0.785 times the diameter squared. That avoids halving to get the radius and is quicker mentally when working from a stated pipe or duct size.

Assumptions and limitations

• A perfect circle — real objects have tolerances, and pipe bore differs from nominal size
• Unit-agnostic: output units follow whatever you enter
• Area is in square units, so a radius in cm gives cm²
• Angles are in degrees, converted internally to radians
• Does not handle ellipses, which need both semi-axes
• Does not calculate cylinder or sphere volumes directly — multiply the area by length yourself
• Segment area assumes a minor segment for angles below 180°

Questions people ask about this in United Kingdom