Math · 34 · Geometry and Trigonometry

Circles

Notes

Full circle = $360^\circ$ = $2\pi$ radians — on the reference sheet.
Arcs and sectors are fractions of the circle: central angle over 360°.
Central angle = its arc. Inscribed angle = half its arc.
An angle inscribed in a semicircle (on a diameter) is 90°.
Tangent ⟂ radius at the point of contact — look for the right triangle.
A radius ⟂ to a chord bisects the chord.
Degrees to radians: $\times\tfrac{\pi}{180}$ . Radians to degrees: $\times\tfrac{180}{\pi}$ .

Area and circumference

A=\pi r^2\qquad C=2\pi r

On the reference sheet

Arc length

s=\dfrac{\theta}{360^\circ}\cdot 2\pi r

Not on the reference sheet

Sector area

A=\dfrac{\theta}{360^\circ}\cdot \pi r^2

Not on the reference sheet

In radians

s=r\theta\qquad A=\dfrac{1}{2}r^2\theta

θ in radians

Inscribed angle

\text{inscribed angle}=\dfrac{1}{2}\,\text{arc}

Conversion

\pi\text{ rad}=180^\circ

One proportion does it: angle / 360° = arc / circumference = sector / area.

10 official SAT questions on Circles, easiest first.