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Inscribed and Circumscribed Polygons

A free Geometry lesson from the “Measurement, Circles, and 3D Solids” unit, with a worked example and practice problems including step-by-step solutions.

A polygon inscribed in a circle has all its vertices on the circle. A polygon circumscribed around a circle has all its sides tangent to the circle. For an inscribed quadrilateral, opposite angles are supplementary (sum to 180 degrees). Every regular polygon fits perfectly inside its circumscribed circle, with the circle's radius equal to the distance from the center to any vertex.

What you'll learn

Why it matters: Architectural rose windows, gear teeth design, surveying landmark layouts, and any 'tiling' that fits a polygon snugly inside or around a circular boundary use these relationships.

Worked example

Problem. Quadrilateral ABCD is inscribed in a circle. Angle A = 70 degrees. Find angle C.

  1. Opposite angles of an inscribed quadrilateral sum to 180.
  2. A + C = 180, so C = 180 - 70 = 110.

Answer: 110

Practice problems

1. A polygon with all vertices on a circle is:

Choices: Inscribed in the circle · Circumscribed around the circle

Show solution
  1. Warm-up: First identify exactly what the question is asking: A polygon with all vertices on a circle is:
  2. For circle problems, connect the formula or theorem to the given radius, diameter, chord, arc, or center information.
  3. 'Inscribed' means inside, with vertices on the circle.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Inscribed in the circle

2. A polygon with all sides tangent to a circle is:

Choices: Inscribed in the circle · Circumscribed around the circle

Show solution
  1. Warm-up: First identify exactly what the question is asking: A polygon with all sides tangent to a circle is:
  2. For circle problems, connect the formula or theorem to the given radius, diameter, chord, arc, or center information.
  3. 'Circumscribed' means around, with sides touching the circle once each.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Circumscribed around the circle

3. Inscribed quadrilateral ABCD with A = 60 degrees. Find C in degrees.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Inscribed quadrilateral ABCD with A = 60 degrees. Find C in degrees.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. A + C = 180, so C = 120.
  4. Check the result by substituting or estimating: the response should match 120 and make sense in the original problem.

Answer: 120

4. Inscribed quadrilateral with B = 95 degrees. Find D in degrees.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Inscribed quadrilateral with B = 95 degrees. Find D in degrees.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. B + D = 180, so D = 85.
  4. Check the result by substituting or estimating: the response should match 85 and make sense in the original problem.

Answer: 85

5. Inscribed quadrilateral with A = 45 and B = 100. Find C in degrees.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Inscribed quadrilateral with A = 45 and B = 100. Find C in degrees.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. A + C = 180, so C = 135.
  4. Check the result by substituting or estimating: the response should match 135 and make sense in the original problem.

Answer: 135

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