Circle and Disk Formulas

Circumference, area, arcs, sectors and segments, chords, tangents and secants, and the equation of a circle — the formulas for circles, from the first lessons on π to the geometry of tangent and secant segments.

16 formulas · Grades 6–9

★ Favourites
All 16 formulas on this page
  1. Circumference (via diameter)
  2. Circumference (via radius)
  3. Area of a circle
  4. Relationship between diameter and radius
  5. Area of a circle via diameter
  6. Property of two tangent segments from a point
  7. Equation of a circle centred at the origin
  8. Product of segments of two secants
  9. A secant and a tangent from one point
  10. Chord length via the central angle
  11. Chord length via the inscribed angle
  12. Length of a circular arc
  13. Area of a circular sector
  14. Radius of a circle from its equation
  15. Area of a circular segment
  16. Area of an annulus (ring)

Circumference (via diameter)

Grade 6
C=π⋅dC = \pi \cdot d

Calculating the circle's boundary length from its diameter.

d
d is the diameter — it passes through the centre from edge to edge.
Legend:
  • d — Diameter (d)

Circumference (via radius)

Grade 6
C=2πrC = 2 \pi r

Calculating the circle's boundary length from its radius.

r
r is the radius — from the centre to any point on the circle.
Legend:
  • r — Radius (r)

Area of a circle

Grade 6
S=πr2S = \pi r^2

Area of the region enclosed by a circle of radius r.

rS
S is the area of the whole shaded part of the circle.
Legend:
  • r — Radius (r)

Relationship between diameter and radius

Grade 6
d=2rd = 2r

The diameter of a circle equals twice its radius.

rrd
The diameter consists of two radii.
Legend:
  • r — Radius (r)

Area of a circle via diameter

Grade 6
S=πD24S = \frac{\pi D^2}{4}

Area of a circle expressed through its diameter.

DS
The same circle area, but expressed via the diameter.
Legend:
  • D — Diameter (D)

Property of two tangent segments from a point

Grade 8
AB=ACAB = AC

Two tangent segments to a circle drawn from the same point are equal.

ABC
Two tangent segments from point A are equal to each other.
Legend:
  • AB — Tangent segment AB

Equation of a circle centred at the origin

Grade 9
r2=x2+y2r^2 = x^2 + y^2

Radius of a circle centred at the origin passing through the point (x, y).

r(x, y)
The circle's centre coincides with the origin.
Legend:
  • x — Coordinate x
  • y — Coordinate y

Product of segments of two secants

Grade 9
AQ⋅BQ=CQ⋅DQAQ \cdot BQ = CQ \cdot DQ

If two secants are drawn from a point outside a circle, the products of their segments are equal.

QABCD
The products of the segments of both secants from point Q are equal.
Legend:
  • AQ — Segment AQ
  • BQ — Segment BQ
  • CQ — Segment CQ

A secant and a tangent from one point

Grade 9
AQ⋅BQ=CQ2AQ \cdot BQ = CQ^2

The square of the tangent segment equals the product of the secant's segments, from the same point.

QABC
QC is the tangent, QA and QB are the secant's segments.
Legend:
  • AQ — Segment AQ
  • BQ — Segment BQ

Chord length via the central angle

Grade 9
AB=2rsin⁡α2AB = 2r \sin \frac{\alpha}{2}

Length of a chord from the radius and the central angle.

αAB
α is the central angle subtending the chord AB.
Legend:
  • r — Radius (r)
  • alpha — Angle α (in degrees)

Chord length via the inscribed angle

Grade 9
AB=2rsin⁡αAB = 2r \sin \alpha

Length of a chord from the radius and the inscribed angle.

αAB
α is the inscribed angle subtending the chord AB.
Legend:
  • r — Radius (r)
  • alpha — Angle α (in degrees)

Length of a circular arc

Grade 9
l=πrα180∘l = \frac{\pi r \alpha}{180^\circ}

Calculates the length of an arc with central angle α.

αL
L is the length of the highlighted arc, α is its central angle.
Legend:
  • r — Radius (r)
  • alpha — Angle α (in degrees)

Area of a circular sector

Grade 9
S=πr2α360∘S = \frac{\pi r^2 \alpha}{360^\circ}

Area of a sector with central angle α.

αS
A sector is the part of a circle between two radii.
Legend:
  • r — Radius (r)
  • alpha — Angle α (in degrees)

Radius of a circle from its equation

Grade 9
r=(x−a)2+(y−b)2r = \sqrt{(x-a)^2+(y-b)^2}

Radius of a circle centred at (a, b) passing through the point (x, y).

r(a, b)(x, y)
The circle's centre is at (a, b), the point (x, y) lies on it.
Legend:
  • x0 — Coordinate x0
  • y0 — Coordinate y0
  • x — Coordinate x
  • y — Coordinate y

Area of a circular segment

Grade 9
S=R22(θ−sin⁡θ)S = \frac{R^2}{2} (\theta - \sin \theta)

Area of a circular segment (the part of a circle cut off by a chord); the angle θ is given in radians.

Sθ
S is the segment cut off by the chord, θ is the central angle.
Legend:
  • R — Radius (R)
  • theta — Angle θ (in radians)

Area of an annulus (ring)

Grade 9
S=π(R2−r2)S = \pi (R^2 - r^2)

Area of the ring between two concentric circles of radii R and r.

Rr
The ring is the region between circles of radii R and r.
Legend:
  • R — Outer radius (R)
  • r — Inner radius (r)

Related topics