Every circle has the same handful of named parts — and almost every circle formula is just a relationship between two of them. Once you can label the radius, diameter, chord, arc, sector, segment, tangent, and secant on a figure, the rest of circle geometry follows naturally. This guide walks each part one-by-one with the formula that depends on it.
The center is the defining point of a circle — every point on the circle is exactly the radius distance away from it. The radius (plural: radii) is the most-used measurement in circle formulas because it’s the simplest. Anything else you can compute (area, circumference, diameter, sector, chord length) ultimately reduces to a formula involving r.
Formulas that use the radius: Area A = πr², Circumference C = 2πr, Diameter d = 2r, Equation of a circle (x − h)² + (y − k)² = r².
The diameter is the longest chord in any circle — a straight line through the center, ending on the circle on both sides. Its length is always exactly twice the radius: d = 2r. If you only know the diameter, you can still compute everything: r = d/2, A = πd²/4, C = πd.
A common student trap: confusing diameter with radius in the area formula. If you mistakenly plug d into A = πr², you’ll get an answer 4× too large. Always halve first if the figure labels the diameter.
A chord is any segment whose endpoints lie on the circle. The diameter is the special chord that happens to pass through the center; every other chord is shorter than the diameter.
Chord length formula: c = 2r × sin(θ/2), where θ is the central angle subtending the chord (the angle between the two radii drawn to the chord’s endpoints).
Example: In a circle of radius 10, a chord subtended by a 60° central angle has length c = 2 × 10 × sin(30°) = 20 × 0.5 = 10. (When θ = 60°, the chord equals the radius — that’s the equilateral-triangle case.)
An arc is a piece of the circumference. There are two kinds:
Arc length formula: L = r × θ (radians), or L = (θ°/360) × 2πr (degrees).
Example: In a circle of radius 6, a 90° arc has length (90/360) × 2π × 6 = (1/4) × 12π = 3π ≈ 9.42.
A sector is the pie-slice region between two radii — bounded by the radii on two sides and an arc on the curved side. Think pizza slice.
Sector area formula: A_s = ½ × r² × θ (radians), or A_s = (θ°/360) × πr² (degrees).
Example: A 45° sector in a circle of radius 8 has area (45/360) × π × 64 = (1/8) × 64π = 8π ≈ 25.13.
A segment is easily confused with a sector but it’s a different region. Imagine drawing a single chord across a circle — the chord divides the circle into two regions, each bounded by the chord and an arc. Each region is a segment. (A sector, in contrast, is bounded by two radii plus an arc.)
Segment area formula: A_seg = ½ × r² × (θ − sin θ), with θ in radians.
Mnemonic: a sector is what you’d cut with two straight knife strokes from the center; a segment is what you’d cut with one straight stroke across.
A tangent is a line that just barely touches the circle — meeting it at exactly one point (the “point of tangency”) without crossing into the interior. The key property:
A tangent line is always perpendicular to the radius drawn to the point of tangency.
This is the foundation of dozens of geometry-proof problems and shows up in calculus when you find the tangent line to a curve. If a problem mentions “the tangent at point P”, instantly draw the radius OP — the angle there is 90°.
A secant is a line that cuts the circle at exactly two points. Visualize it as a chord whose endpoints have been extended into a full line on both sides.
The relationship that examiners love: if two secants are drawn from an external point P, the product of the two segments (external × full) is the same for both secants. This is the Power of a Point theorem.
The circumference is the total distance around the circle — its perimeter. In school textbooks the word “perimeter” is used; in geometry the technical word is “circumference”. Both refer to the same length:
C = 2π × r = π × d
That ratio C/d = π (≈ 3.14159) is identical for every circle. It’s the most famous constant in mathematics.
Picture a circle with center O. Inside:
Re-drawing this figure from memory is the single best study exercise for circle vocabulary. Once the labels become automatic, all the formulas are just relationships between these parts.
How many parts of a circle are there? The canonical school-geometry list has 9–10 named parts: center, radius, diameter, chord, arc (minor / major), sector, segment, tangent, secant, circumference. Higher-level treatments add the inscribed angle, central angle, and the equation of a circle.
Is the radius half the diameter? Yes. r = d/2, or equivalently d = 2r. This is the most-used identity in circle geometry — it lets you swap between the two whenever a formula uses the one you don’t know.
What’s the difference between a chord and a secant? A chord is a segment (finite, both endpoints on the circle). A secant is the line obtained by extending that chord infinitely in both directions.
Why is a diameter also called a chord? A chord is any segment with both endpoints on the circle. The diameter happens to be the chord that also passes through the center — making it the longest possible chord, but still a chord.
For all the formulas in one place, see the Circle Formula reference page. To compute area, circumference, radius or diameter instantly from any one input, use the Circle Geometry Calculator. For a problem that involves an inscribed shape or proof, the AI Math Solver can walk you through it from a photo.