Circle Formulas

Equation of a circle

In an x−y coordinate system, the circle with center (a,b) and radius r is the set of all points (x,y) such that:

$$(x-a)^2 + (y-b)^2 =r^2$$

Circle centered at the origin:

$$x^2 + y^2 = r^2$$

Parametric equations

\begin{aligned} x &= a + r\,\cos t \\ y&= b + r\,\sin t \end{aligned}

where t is a parametric variable.

In polar coordinates the equation of a circle is:

$$r^2 – 2\cdot r \cdot r_0\cdot cos(\Theta – \phi ) + r_0^2 = a^2$$

Area of a circle

$$A = r^2\pi$$

Circumference of a circle

$$C = \pi \cdot d = 2\cdot \pi \cdot r$$

Theorems:

Chord theorem

The chord theorem states that if two chords, CD and EF, intersect at G, then:

$$CD \cdot DG = EG \cdot FG$$

Tangent-secant theorem

If a tangent from an external point D meets the circle at C and a secant from the external point D meets the circle at G and E respectively, then

$$DC^2 = DG \cdot DE$$

Secant – secant theorem

If two secants, DG and DE, also cut the circle at H and F respectively, then:

$$DH \cdot DG = DF \cdot DE$$

Tangent chord property

The angle between a tangent and chord is equal to the subtended angle on the opposite side of the chord.