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.

 

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