Circles

What the exam asks

• Measurement: circumference, area, arc length and sector area, with central angles given in degrees or radians.
• Angle and segment theorems: central and inscribed angles, tangent lines (perpendicular to a radius), chords and the perpendicular from the center.
• Circles in the $xy$-plane: read the center and radius from $(x - h)^2 + (y - k)^2 = r^2$, complete the square, write an equation from given information, and decide whether a point lies inside, on or outside a circle.
• Line-circle intersections: tangency and the number of intersection points, often with a parameter $k$.

The reference sheet gives $A = \pi r^2$ and $C = 2\pi r$, and states that a circle has $360^\circ$ of arc, or radians. You must know everything else in this lesson yourself.

Core ideas

Arcs and sectors

A central angle cuts off the same fraction of the circumference (the arc) and of the area (the sector) as it does of the full angle.

The fraction method is the safest: arc over circumference, sector over area, and central angle over $360^\circ$ (or over $2\pi$) are all the same fraction.

Angle theorems

• A central angle (vertex at the center) has the same measure as its intercepted arc.
• An inscribed angle (vertex on the circle) is half its intercepted arc, so it is half the central angle that intercepts the same arc.
• Inscribed angles that intercept the same arc are equal.
• An angle inscribed in a semicircle (its sides meet the ends of a diameter) is $90^\circ$.
• Two radii form an isosceles triangle, so the base angles of triangle $AOB$ are equal.

Tangents and chords

• A tangent line is perpendicular to the radius at the point of tangency. That creates a right triangle, so the Pythagorean theorem applies.
• The two tangent segments from one external point are equal in length. The angle between them and the central angle between the two radii sum to $180^\circ$.
• A segment from the center perpendicular to a chord bisects the chord. For a chord of length $c$ at distance $d$ from the center, $r^2 = d^2 + \left(\frac{c}{2}\right)^2$.

Circle equations

The signs flip: $(x + 2)^2$ means $h = -2$. If the equation is expanded, like $x^2 + y^2 + ax + by = c$, in each variable by adding and to sides.

Point position: compute $(x - h)^2 + (y - k)^2$ for the point and compare it with $r^2$. Less means inside, equal means on the circle, greater means outside.

Line and circle: substitute the line into the circle’s equation to get a quadratic. Its discriminant gives the number of intersection points: positive means two, zero means one (tangent), negative means none. Geometrically, a tangent line is exactly $r$ from the center.

Worked examples

Exam technique

• Radius first. Convert every given (diameter, circumference, area or equation) into $r$ before doing anything else.
• Draw the radius to the point of tangency and mark the right angle. Most tangent problems become Pythagorean problems.
• Think in fractions of the circle. The arc, the sector and the central angle are all the same fraction of their totals.
• Desmos is a circle machine. Type the equation exactly as given, even if it’s expanded; click points on the graph to read the center and radius; add a line with a slider to find tangency; shade $(x - h)^2 + (y - k)^2 < r^2$ to test points.

Common mistakes

Quick recap

• $C = 2\pi r$ and $A = \pi r^2$; always find $r$ first.
• Arc length $$; sector area .