Circle calculator, area, arc and equation, with steps.

A circle calculator finds the radius, diameter, circumference and area of a circle from any one of them. This one also finds the arc length, sector area, chord and segment from a central angle, writes the equation of a circle in standard and general form (and reads it back), and finds the circle that passes through three points. Each answer shows the formula with your numbers and a drawing.

How to use the circle calculator

1. Choose what to find: Radius, area, circumference, Arc and sector, Ring (annulus), Equation of a circle, From general form or Through 3 points.
2. For the first one, pick what you know in I know the, type the Value, choose the Value of pi your class uses, and add a unit if you want it on the answers. A value such as 10π or 25pi is read as a multiple of π.
3. Read the big cards. When the radius is exact, the circumference and the area also appear in terms of π, like 10π.
4. For arc and sector, say what you know: the central angle, the arc length, the chord length or the sector area. The other three are worked out from it.
5. Open Steps for the working. Copy result copies a short summary, and Copy link keeps your numbers in the link.

The formulas

Worked example: from the radius

For r = 5: the diameter is 10, the circumference is 2 × π × 5 = 10π = 31.4159 and the area is π × 5² = 25π = 78.5398. With π replaced by 3.14 the circumference is 31.4 and the area is 78.5, which is what many textbooks expect.

Worked example: from the circumference or the area

If the circumference is 10, the radius is 10 ÷ 2π = 1.5915 and the area is π × 1.5915² = 7.9577. If the area is 50, the radius is √(50 ÷ π) = 3.9894 and the diameter is 7.9788. Going backwards from area always needs the square root, which is the step people forget.

Typing a value with pi

Many textbook questions give the answer in terms of π. If the circumference is 10π, type 10π (or 10pi) with Circumference selected. The radius is 10π ÷ 2π = 5, and the area is 25π. If the area is 36π, the radius is √36 = 6, the diameter is 12 and the circumference is 12π. When the number is not a perfect square, such as an area of 2π, the radius is exactly √2 and the page shows that next to the decimal 1.4142.

Which value of pi to use

For a radius of 7, the full π gives a circumference of 43.9823 and an area of 153.9380. With 22/7 the circumference is exactly 44 and the area is exactly 154, which is why that value shows up in school questions with a radius of 7. With 3.14 you get 43.96 and 153.86. Use whichever your teacher or textbook asks for, and the working shows the number used.

Semicircle and quarter circle

For r = 5 the full area is 78.5398. A semicircle is half of it, 39.2699, and a quarter circle is 19.635. The perimeter of a semicircle is not half of the circumference. It is the curved part, π × 5 = 15.708, plus the straight edge, the diameter 10, which makes 25.708. Forgetting that straight edge is the usual slip.

Area of a ring (annulus)

A ring is a circle with a smaller circle cut out of the middle. Its area is π(R² − r²). With an outer radius of 8 and an inner radius of 5, that is π × (64 − 25) = 39π = 122.5221. The width of the ring is R − r = 3. The page also shows both circumferences, 50.2655 and 31.4159, and the area of each circle on its own.

Arc, sector, chord and segment

Take r = 5 and a central angle of 60°, which is π/3 = 1.0472 radians. The arc is 5 × 1.0472 = 5.2360. The sector is ½ × 25 × 1.0472 = 13.0900. The chord is 2 × 5 × sin 30° = 5, which is the same as the radius, because a 60° sector is an equilateral triangle. The segment between the chord and the arc is 13.0900 minus the triangle (10.8253), so 2.2647, and its height is 5 × (1 − cos 30°) = 0.6699.

Arc and sector from the length, chord or area

You do not always start with the angle. With r = 10 and an arc length of 12, the angle is s ÷ r = 1.2 radians, which is 68.7549°. The sector area is ½ × 10 × 12 = 60, the chord is 2 × 10 × sin 0.6 = 11.2928 and the segment is 13.398. If you know the chord instead, θ = 2 × arcsin(chord ÷ 2r): a chord of 10 in a circle of radius 10 gives 60°. The chord cannot be longer than the diameter, and the angle you get is the shorter of the two arcs. From a sector area of 50 with r = 10, θ = 2 × 50 ÷ 100 = 1 radian, or 57.2958°.

The equation of a circle

A circle with centre (3, −2) and radius 4 is (x − 3)² + (y + 2)² = 16. Expand it and you get the general form x² + y² − 6x + 4y − 3 = 0, because 9 + 4 − 16 = −3. To go back, complete the square: from x² + y² − 4x + 6y − 12 = 0, add 4 and 9 to both sides to get (x − 2)² + (y + 3)² = 25, so the centre is (2, −3) and the radius is 5. If the right side comes out zero or negative, there is no circle, and the page says so.

A circle through three points

Three points that are not in a line give exactly one circle. For (0, 0), (4, 0) and (0, 2) the centre is (2, 1) and r² = 5, so r = √5 ≈ 2.2361 and the equation is (x − 2)² + (y − 1)² = 5. This is also the circumcircle of the triangle those points make.

Common problems

• Mixing up radius and diameter. The area uses the radius. A pizza of 12 inches across has r = 6, not 12.
• Units of area. Area is in square units. If r is in cm, the area is in cm². Pick the unit and the page adds it to every answer.
• Degrees against radians. The formula s = rθ only works with θ in radians. The calculator converts for you.

For more shapes use the area and volume calculator. For distances between points, see the midpoint and distance calculator, and for circle angles the unit circle chart.