Circle Calculator - Radius, Area, Circumference

The Circle Calculator converts between radius, diameter, circumference and area from any one known measurement. Add a central angle in degrees or radians to obtain arc length and sector area. Answers can appear as decimals or as exact multiples of pi when exam form is preferred.

A circle in the plane is fixed by a single length once pi ties the four standard quantities together. The page redraws a labelled figure with the radius marked to the entered value, and every formula on the page is followed by substituted numbers for the reference radius 5 case.

Calculate a circle from any measurement

A circle is determined by any one of its standard length measurements because radius, diameter, circumference and area are linked by fixed ratios involving pi. Enter whichever field matches what is known and leave the others blank. The engine recovers the radius first, then fills the remaining values with the same precision used internally until display rounding.

Live recalculation updates the diagram and the result panel as soon as a valid single input is present. Units on circumference and radius stay consistent; area is reported in square units for the chosen length unit. The case card names which input seeded the solution so the path is visible.

Calculate the area of a circle

Area measures the region inside the circle boundary in square units of the radius. Students usually start from the radius, but the calculator can also back-solve radius from a known area when that is the given quantity on a worksheet. The relation is quadratic in radius, so one positive root is returned.

A = pi * r^2

For radius r = 5:

A = pi * 5^2 = 25 * pi approx 78.54

Calculate the circumference

Circumference is the perimeter distance around the circle, equal to pi times the diameter or two pi times the radius. It shares the same pi factor as area but grows linearly with radius rather than quadratically. Starting from circumference alone is common on lab sheets that measure distance with a string.

C = 2 * pi * r = pi * d

For r = 5:

C = 2 * pi * 5 = 10 * pi approx 31.42

Convert between radius and diameter

Diameter is the longest chord through the center and equals twice the radius in every circle. Radius is half the diameter. Because both describe the same scale, entering either one is enough to unlock area and circumference without a second measurement. The diagram labels both when both are displayed.

d = 2 * r
r = d / 2

When r = 5, diameter d = 10. When d = 10, radius r = 5 and the area and circumference values match the radius 5 fixture above.

Calculate arc length and sector area

An arc is a portion of the circumference cut off by a central angle at the center. A sector is the pie-slice region bounded by two radii and that arc. Both quantities scale with the fraction of the full turn, whether the angle is entered in degrees or radians. The calculator accepts either unit and converts internally.

For central angle theta in degrees:

arc length L = (theta / 360) * 2 * pi * r
sector area = (theta / 360) * pi * r^2

With r = 5 and theta = 90 degrees, L = (1/4) * 10 * pi = 2.5 * pi and sector area = (1/4) * 25 * pi = 6.25 * pi.

Solve a circle with radius 5

Take r = 5 as the classroom reference. Diameter follows immediately from doubling the radius. Circumference and area reuse the same r in the standard formulas, and the decimal approximations below use pi approx 3.14159 for display only while exact forms stay in terms of pi.

1. Diameter

d = 2 * 5 = 10

2. Circumference

C = 2 * pi * 5 = 10 * pi approx 31.42

3. Area

A = pi * 5^2 = 25 * pi approx 78.54

Give answers in terms of pi

Exam questions often ask for exact multiples of pi instead of early rounding. Keeping C = 10 * pi and A = 25 * pi avoids introducing decimal error before later steps multiply or add results. The toggle on the result panel switches between decimal approximations and pi form without changing the underlying computation.

Exact form is especially useful when arc length and sector area must be combined with other pi multiples on the same page. Students can add coefficients while pi remains factored out. Decimal mode still shows approx 31.42 and approx 78.54 for the radius 5 case when a numeric estimate is enough.

Frequently asked questions

Summary

Circle Calculator returns the measures named in the sections above, with substituted arithmetic for every formula and a labelled figure drawn to the entered dimensions. Worked numbers on this page match the geometry reference fixtures.