Circle Calculator

Circle Calculator: What You Can Compute

A circle calculator is one of the most practical geometry tools because circles show up in real-world measurements all the time: wheels, pipes, round tables, coins, lenses, circular gardens, curved architectural features, and technical drawings. In school math, circles connect algebra, geometry, and trigonometry — and in everyday work they show up when you need to measure a round object, estimate material, or convert between radius, diameter, and circumference.

This calculator is designed as an “all-in-one” circle tool. You can start from whichever value you already know (radius, diameter, circumference, or area) and calculate the rest. Then, if your problem involves a slice of a circle, you can move to the arc/sector tab to compute arc length, sector area, chord length, and segment area. If you’re working with a ring-shaped region (like a washer, donut, gasket, or circular frame), use the annulus tab. And if you’re solving coordinate geometry problems, the circle equation tab formats the standard and general equations from the center and radius.

Key Circle Terms: Radius, Diameter, Circumference, and Area

The radius (r) is the distance from the circle’s center to any point on its boundary. The diameter (d) is the distance across the circle through the center, and it is always twice the radius: d = 2r. The circumference (C) is the distance around the circle, and the area (A) is the amount of surface enclosed by the circle. Together, these values form the foundation of most circle calculations.

Why π Matters (and How This Tool Uses It)

The constant π (pi) represents the ratio of a circle’s circumference to its diameter. No matter how large or small a circle is, C/d is always π. Since π is irrational, it cannot be written exactly as a fraction or terminating decimal. In most calculators and engineering contexts, π is approximated using a high-precision value.

This tool lets you choose a π mode: the standard JavaScript value (Math.PI), a fixed decimal approximation, or the common fraction 22/7 (useful for quick estimates). For most work, Math.PI is the best option. 22/7 can slightly overestimate circumference and area, especially for precise tasks.

Solving a Circle Backwards: Finding Radius from Area or Circumference

Sometimes you don’t start with the radius — you start with the result of a measurement or a requirement. For example, you might know the area of a circular garden plot and want the radius, or you might know the circumference of a pipe and want its diameter. These are “inverse” circle problems, and they’re very common in design, purchasing, and planning.

Once you compute radius, it becomes easy to compute everything else. That’s why the calculator’s “Solve for Radius” mode shows you not only r, but also the implied diameter, circumference, and area so you can confirm the values match your expectation.

Arc Length and Sector Area: Circle “Slices” in Degrees

A sector is a slice of a circle bounded by two radii and an arc. A central angle (θ) describes how big that slice is. If θ is measured in degrees, then θ/360 represents the fraction of the full circle. This makes arc length and sector area straightforward:

These formulas show up in projects like determining the length of a curved track, estimating material along a curved edge, computing the area of a pie-shaped region, or sizing a rotating component with a limited swing angle. If you double the radius, arc length doubles and sector area quadruples — a helpful intuition when scaling designs.

Chord Length and Segment Area: Straight vs Curved Boundaries

A chord is the straight line connecting two points on a circle. If you know the radius and the central angle θ between those two radii, chord length is:

A circular segment is the region between the chord and its corresponding arc. For the minor segment (when θ ≤ 180°), segment area can be computed by subtracting the triangle area from the sector area. The triangle area formed by two radii and the chord is (1/2)r²·sin(θ) (with θ in radians inside the sine function, but calculators handle that conversion).

The calculator handles both minor and major segments. Major segment area is simply the full circle area minus the minor segment area. Segment calculations are useful in machining, reservoir volume estimation (partial fill shapes), lens geometry, and any design where a curved “cap” area matters.

Annulus (Ring) Area: Washers, Donuts, and Circular Frames

An annulus is the region between two concentric circles (same center). You’ll see it in gaskets, washers, rings, circular cutouts, and round frames. The annulus area is:

This works because you subtract the inner circle area from the outer circle area. The tool also shows the outer and inner circumferences, which can help if you’re calculating edging material, seals, or perimeter contact.

Circle Equation in Coordinate Geometry

In coordinate geometry, a circle can be described by its center (h, k) and radius r. The standard equation is:

Expanding the squares produces the general form: x² + y² + Dx + Ey + F = 0. Some textbooks prefer general form for graphing and algebraic manipulation, while standard form makes the center and radius obvious. This calculator prints both forms so you can use whichever your assignment, CAD tool, or analysis step requires.

Common Mistakes to Avoid

• Mixing radius and diameter: always check whether your measurement is to the center (radius) or across the full circle (diameter).
• Forgetting squared units for area: area is measured in unit² (cm², m², in²), not in plain units.
• Angle confusion: arc/sector formulas here use degrees; the calculator converts internally for trig functions.
• Invalid ring inputs: inner radius must be smaller than outer radius for an annulus.
• Over-rounding: keep more decimals for intermediate work, then round at the end for final reporting.