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Geometry Unit 7: Circles

Circle theorems relate angles to the arcs they cut off. The coordinate equation of a circle comes from the distance formula.

Covers: Arcs and central angles, inscribed angles, chords and tangents, arc length, circle equations

Study guide

Circles have their own vocabulary. A radius goes from the center to the circle, a diameter passes through the center (twice the radius), a chord connects two points on the circle, a secant line crosses the circle twice, and a tangent line touches it at exactly one point.

A central angle has its vertex at the center, and its measure equals the measure of its intercepted arc. Arcs are minor (under 180°), major (over 180°) or semicircles. An inscribed angle has its vertex on the circle, and it measures half its intercepted arc. So inscribed angles that intercept the same arc are equal, and an angle inscribed in a semicircle is always 90°. Opposite angles of a quadrilateral inscribed in a circle are supplementary.

A tangent line is perpendicular to the radius at the point of tangency, and two tangent segments from the same outside point are equal. Chords equidistant from the center are equal, and a radius perpendicular to a chord bisects it.

Arc length is (θ/360°)·2πr, the fraction of the circumference. In radians, arc length is s = rθ.

The equation of a circle with center (h, k) and radius r is (x − h)² + (y − k)² = r². To find the center and radius from an expanded equation, complete the square for x and y.

When two chords intersect inside a circle, each angle formed equals half the sum of the intercepted arcs. When two secants, two tangents or a secant and a tangent meet outside the circle, the angle equals half the difference of the intercepted arcs (far arc minus near arc). For two chords that intersect inside a circle, the products of their segment lengths are equal (a·b = c·d). For two secants from an outside point, (outside part)·(whole secant) is the same for both. For a tangent and a secant from the same point, tangent² = (outside part)·(whole secant).

An angle formed by a tangent and a chord at the point of tangency measures half its intercepted arc, the same as an inscribed angle.

Key ideas

• A central angle equals its intercepted arc.
• An inscribed angle is half its intercepted arc.
• A tangent is perpendicular to the radius at the point of tangency.
• Arc length is a fraction of the circumference.
• Equation: (x − h)² + (y − k)² = r².

Formulas and rules

Common mistakes

• Forgetting to halve the arc for inscribed angles.
• Sign errors in the center of a circle equation.

Words to know

Inscribed angle theorem

An inscribed angle is half its intercepted arc.

Tangent line to a circle

Perpendicular (90°) to the radius at the point of tangency.

Arc length formula

(θ/360)·2πr

Equation of a circle

(x − h)² + (y − k)² = r²

Angle inscribed in a semicircle

Always a right angle (90°).

Radius / diameter

Center to circle; across through the center (d = 2r).

Chord

A segment joining two points on a circle.

Secant

A line that crosses a circle twice.

Central angle

Vertex at the center; equals its intercepted arc.

Minor / major arc

Less than 180°; greater than 180°.

Inscribed quadrilateral

Opposite angles are supplementary.

Tangent segments

Two tangents from the same outside point are equal.

Sample practice questions

39 more questions like these, from level 1 to 5, are in the practice set.