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Geometry

Circle Basics

Lesson

A circle is the set of all points at a fixed distance from a center. That fixed distance is the radius.

Key parts

  • Radius (rr): center to edge.
  • Diameter (d=2rd = 2r): across, through the center.
  • Chord: segment between two points on the circle. The diameter is the longest chord.
  • Tangent: a line touching the circle at exactly one point. Perpendicular to the radius at that point.
  • Arc: a piece of the circle.
  • Central angle: vertex at the center; its measure equals the arc it intercepts.

Perpendicular bisector of chord

A line from the center perpendicular to a chord bisects the chord. Useful for solving with the Pythagorean theorem on half-chord triangles.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

Diameter if radius = 5\text{Diameter if radius = 5}

Problem 2

Radius if diameter = 16\text{Radius if diameter = 16}

Problem 3

Arc measure intercepted by a 60∘ central angle\text{Arc measure intercepted by a } 60^\circ \text{ central angle}

Problem 4

Sum of all arcs around a full circle\text{Sum of all arcs around a full circle}

Practice

Standard problems matching the lesson.

Problem 5

Diameter if r=12\text{Diameter if } r = 12

Problem 6

Radius if d=30\text{Radius if } d = 30

Problem 7

Arc for 90∘ central angle\text{Arc for } 90^\circ \text{ central angle}

Problem 8

Central angle for 120∘ arc\text{Central angle for } 120^\circ \text{ arc}

Problem 9

Semicircle arc measure\text{Semicircle arc measure}

Problem 10

Quarter-circle central angle\text{Quarter-circle central angle}

Problem 11

Two arcs partition circle 2:3. Larger arc measure?\text{Two arcs partition circle 2:3. Larger arc measure?}

Problem 12

Longest chord of circle with r=9\text{Longest chord of circle with } r = 9

Problem 13

Diameter if r=10\text{Diameter if } r = 10

Problem 14

Arc that’s 1/6 of the circle: measure?\text{Arc that's 1/6 of the circle: measure?}

Problem 15

Two perpendicular diameters. Each arc measure?\text{Two perpendicular diameters. Each arc measure?}

Problem 16

Arc that’s half a semicircle: measure?\text{Arc that's half a semicircle: measure?}

Problem 17

Central angle for arc 45∘\text{Central angle for arc } 45^\circ

Problem 18

Central angle for arc that is 1/8 of circle\text{Central angle for arc that is 1/8 of circle}

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 19

Chord length 8 at distance d from center; r=5. Find d.\text{Chord length 8 at distance } d \text{ from center; } r=5. \text{ Find } d.

Problem 20

Diameter 26, chord 10. Distance from center?\text{Diameter 26, chord 10. Distance from center?}

Problem 21

Chord 6, distance from center 4. Radius?\text{Chord 6, distance from center 4. Radius?}

Problem 22

Chord 8 at distance 3 from center. Radius?\text{Chord 8 at distance 3 from center. Radius?}

Problem 23

Central angle 270∘ as fraction of circle (decimal)\text{Central angle } 270^\circ \text{ as fraction of circle (decimal)}

Problem 24

Concentric circles radii 3 and 7. Width of the ring?\text{Concentric circles radii 3 and 7. Width of the ring?}

Problem 25

Half of diameter 14\text{Half of diameter 14}

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