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Solving for Missing Measures

Pick the right formula based on WHERE the angle or point is (center, on circle, inside, or outside), then plug in what you know and solve.

CENTER∠ = arcON CIRCLE∠ = ½ arcINSIDE∠ = ½(a+b)OUTSIDE∠ = ½(a−b)

The four cases: where the vertex is tells you which formula to use.

How It Works

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Step 1: Where is the angle?

Look at where the vertex (point) of the angle is. At the center? On the circle? Inside? Outside? This tells you which formula to use.

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Step 2: Pick the formula

Center → angle = arc. On circle → angle = ½ arc. Inside → angle = ½(arc + arc). Outside → angle = ½(big arc − small arc). Memorize these four!

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Step 3: Fill in what you know

Write down the formula. Put in the numbers you know. Use x for what you don't know. Then solve the equation step by step.

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Step 4: Check your work

Make sure arcs in the same circle add up to 360°. Make sure your answer makes sense (angles can't be negative, arcs can't be more than 360°).

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For segment lengths

Intersecting chords: a × b = c × d. Two secants from outside: whole₁ × outside₁ = whole₂ × outside₂. Tangent-secant: tangent² = whole × outside.

Key Formulas

Quick Reference: Angle at Center
angle = arc
Central angle equals intercepted arc.
Quick Reference: Angle on Circle
angle = ½ × arc
Inscribed angle equals half its intercepted arc.
Quick Reference: Angle Inside
angle = ½ × (arc₁ + arc₂)
Two chords crossing: half the sum of arcs.
Quick Reference: Angle Outside
angle = ½ × (big arc − small arc)
Two secants from outside: half the difference of arcs.
Intersecting Chords
a × b = c × d
Products of segments are equal.
Secant-Secant
whole₁ × outside₁ = whole₂ × outside₂
Products of whole and external segments are equal.