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Circle Geometry Study Guide

Key concepts, formulas, diagrams, and worked examples for all five topics.

Quick Formula Reference

Angle at Centerangle = arc
Angle on Circleangle = ½ × arc
Angle Inside Circleangle = ½(arc₁ + arc₂)
Angle Outside Circleangle = ½(big arc − small arc)
Intersecting Chordsa × b = c × d
Two TangentsPA = PB
1

Tangents and Points of Tangency

A tangent is a line that touches a circle at exactly one point.

Key Idea: A tangent line is always perpendicular (makes a 90° angle) to the radius at the point where it touches the circle.
OTPradiustangent90°

A tangent line touches the circle at one point and makes a 90° angle with the radius.

Concepts

  1. What is a tangent?: A tangent is a straight line that just barely touches the outside of a circle. It touches at exactly ONE point. Think of it like a ball sitting on the floor — the floor is the tangent line.
  2. The 90° rule: Draw a line from the center of the circle to where the tangent touches. That line (the radius) makes a perfect 90° angle with the tangent. Always. This is the most important rule.
  3. Two tangents from the same point: If you draw two tangent lines from the same point outside the circle, those two lines are always the same length. This is super useful for solving problems.
  4. Tangent-radius triangles: Since the radius meets the tangent at 90°, you often get right triangles. You can use the Pythagorean theorem (a² + b² = c²) to find missing lengths.

Formulas

Tangent-Radius Rule
tangent ⊥ radius (90° angle)
The tangent line is perpendicular to the radius at the point of tangency.
Two-Tangent Rule
If PA and PB are tangents from point P, then PA = PB
Two tangent segments from the same external point are equal in length.
Tangent-Secant Angle
angle = ½ × (far arc − near arc)
When a tangent and a secant meet outside the circle, the angle equals half the difference of the intercepted arcs.

Worked Example

Q: A tangent and a radius meet at point T on a circle. The radius is 5 cm. A line from the center to an external point P is 13 cm. How long is the tangent from P to T?
  1. The tangent and radius form a 90° angle at T.
  2. So we have a right triangle: radius = 5, hypotenuse = 13.
  3. Use Pythagorean theorem: tangent² + 5² = 13²
  4. tangent² + 25 = 169
  5. tangent² = 144
  6. tangent = 12 cm
Answer: 12 cm
2

Secants and Intersecting Chords

Secants are lines that cut through a circle at two points. Chords are segments inside the circle.

Key Idea: When two chords cross inside a circle, or two secants meet outside, there are special rules to find missing lengths and angles.
Pabcda × b = c × d

When two chords cross inside a circle, multiply the pieces: a × b = c × d

Concepts

  1. What is a chord?: A chord is a line segment with both endpoints on the circle. The diameter is the longest possible chord — it goes through the center.
  2. What is a secant?: A secant is a line that passes through a circle, hitting it at TWO points. Think of it as a chord that keeps going past the circle.
  3. Chords crossing inside: When two chords cross inside a circle, multiply the pieces: (piece 1) × (piece 2) = (piece 3) × (piece 4). This always works!
  4. Secants from outside: When two secants come from the same point outside the circle, use: (whole length 1) × (outside part 1) = (whole length 2) × (outside part 2).
  5. Angle from intersecting chords: The angle where two chords cross = half the sum of the two arcs they cut off. Formula: angle = ½(arc1 + arc2).

Formulas

Intersecting Chords (lengths)
a × b = c × d
When two chords intersect, the products of their segments are equal.
Intersecting Chords (angle)
angle = ½ × (arc₁ + arc₂)
The angle formed equals half the sum of the two intercepted arcs.
Two Secants from Outside
(whole₁) × (outside₁) = (whole₂) × (outside₂)
For two secants from the same external point, the products of whole and external segments are equal.
Secant-Tangent from Outside
(whole secant) × (outside part) = tangent²
When a secant and tangent come from the same point, the secant product equals the tangent squared.

Worked Example

Q: Two chords cross inside a circle. One chord is split into pieces of 4 and 6. The other chord has one piece of 3. Find the missing piece.
  1. Use the intersecting chords rule: a × b = c × d
  2. 4 × 6 = 3 × d
  3. 24 = 3d
  4. d = 8
Answer: 8
3

Arcs and Angles

Arcs are curved pieces of the circle. Angles can be at the center, on the circle, or outside.

Key Idea: A central angle equals its arc. An inscribed angle equals HALF its arc. This is the key rule for most circle angle problems.
OABCarc BCangle Aangle A = ½ × arc BC

An inscribed angle (vertex on circle) equals HALF the intercepted arc.

Concepts

  1. What is an arc?: An arc is a piece of the circle's edge. A minor arc is the small piece (less than half). A major arc is the big piece (more than half). The whole circle = 360°.
  2. Central angle: A central angle has its vertex (point) at the CENTER of the circle. The central angle equals the arc it cuts off. Simple: angle = arc.
  3. Inscribed angle (THE BIG RULE): An inscribed angle has its vertex ON the circle. It equals HALF the arc it intercepts. If the arc is 80°, the inscribed angle is 40°. This is the rule you'll use the most.
  4. Angles in a semicircle: If an inscribed angle intercepts a semicircle (half the circle = 180° arc), the angle is 90°. Half of 180° = 90°. So any angle in a semicircle is a right angle.
  5. Angles outside the circle: When two secants (or a secant and tangent) meet OUTSIDE the circle: angle = ½ × (big arc − small arc). Notice it's the DIFFERENCE, not the sum.

Formulas

Central Angle
central angle = intercepted arc
A central angle is equal to the arc it intercepts.
Inscribed Angle
inscribed angle = ½ × intercepted arc
An inscribed angle is half the arc it intercepts.
Angle Inside Circle
angle = ½ × (arc₁ + arc₂)
An angle formed inside the circle (by two chords) equals half the sum of intercepted arcs.
Angle Outside Circle
angle = ½ × (big arc − small arc)
An angle formed outside the circle equals half the difference of intercepted arcs.

Worked Example

Q: An inscribed angle intercepts an arc of 120°. What is the inscribed angle?
  1. Use the inscribed angle rule: angle = ½ × arc
  2. angle = ½ × 120°
  3. angle = 60°
Answer: 60°
4

Inscribed Quadrilaterals

A quadrilateral inscribed in a circle has all four corners on the circle.

Key Idea: Opposite angles in an inscribed quadrilateral always add up to 180°. Always.
ABCD75°110°105°70°A + C = 180° and B + D = 180°

Opposite angles in an inscribed quadrilateral always add up to 180°.

Concepts

  1. What does inscribed mean?: Inscribed means the shape is inside the circle with all its corners (vertices) touching the circle. Think of it as the shape fitting perfectly inside.
  2. The 180° rule: For any quadrilateral inscribed in a circle: opposite angles add up to 180°. If angle A = 70°, then the angle across from it = 110°. Because 70 + 110 = 180.
  3. Using this to solve problems: If you know one angle, you can find the opposite angle right away. Just subtract from 180°. If angle B = 95°, angle D = 180° - 95° = 85°.
  4. Connecting to arcs: Each inscribed angle = half its intercepted arc. So the arcs intercepted by opposite angles must add up to 360° (the whole circle). This helps when you know arc measures.

Formulas

Opposite Angles Rule
angle A + angle C = 180°
Opposite angles in an inscribed quadrilateral are supplementary (add to 180°).
Other Pair Too
angle B + angle D = 180°
Both pairs of opposite angles add to 180°, not just one pair.
Arc Connection
Each angle = ½ × (its intercepted arc)
Each angle in the inscribed quadrilateral is an inscribed angle, so it equals half its intercepted arc.

Worked Example

Q: In an inscribed quadrilateral ABCD, angle A = 75° and angle B = 110°. Find angles C and D.
  1. Opposite angles add to 180°.
  2. Angle C is opposite angle A: C = 180° - 75° = 105°
  3. Angle D is opposite angle B: D = 180° - 110° = 70°
  4. Check: 75° + 110° + 105° + 70° = 360° ✓
Answer: Angle C = 105°, Angle D = 70°
5

Solving for Missing Measures

Putting it all together to find missing angles, arcs, and segment lengths.

Key Idea: 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.

Concepts

  1. 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.
  2. 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!
  3. 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.
  4. 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°).
  5. For segment lengths: Intersecting chords: a × b = c × d. Two secants from outside: whole₁ × outside₁ = whole₂ × outside₂. Tangent-secant: tangent² = whole × outside.

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.

Worked Example

Q: Two chords intersect inside a circle. The angle formed is 65°. One intercepted arc is 80°. Find the other intercepted arc.
  1. Use the inside-angle formula: angle = ½(arc₁ + arc₂)
  2. 65 = ½(80 + arc₂)
  3. 130 = 80 + arc₂
  4. arc₂ = 50°
Answer: 50°

Circle Geometry Tutor — Study Guide