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
- Distinguish polygons inscribed IN a circle (all vertices on the circle) from those circumscribed AROUND a circle (all sides tangent to the circle)
- Use the inscribed-quadrilateral theorem (opposite angles sum to 180)
- Apply the relationship between a regular polygon and its circumscribed circle
Worked example
Problem. Quadrilateral ABCD is inscribed in a circle. Angle A = 70 degrees. Find angle C.
- Opposite angles of an inscribed quadrilateral sum to 180.
- A + C = 180, so C = 180 - 70 = 110.
- Connect the result back to Inscribed and Circumscribed Polygons so the geometric relationship is explicit.
- Check the result against the original diagram, units, or proof statement.
Answer: 110
Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (multiple choice and typed response). Counts include repeated prompts and variants.
Practice problems
1. A polygon with all vertices on a circle is:
Choices: Inscribed in the circle · Circumscribed around the circle
Show solution
- 'Inscribed' means inside, with vertices on the circle.
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
- 'Circumscribed' means around, with sides touching the circle once each.
Answer: Circumscribed around the circle
3. Inscribed quadrilateral ABCD with A = 60 degrees. Find C in degrees.
Show solution
- A + C = 180, so C = 120.
Answer: 120
4. Inscribed quadrilateral with B = 95 degrees. Find D in degrees.
Show solution
- B + D = 180, so D = 85.
Answer: 85
5. Inscribed quadrilateral with A = 45 and B = 100. Find C in degrees.
Show solution
- A + C = 180, so C = 135.
Answer: 135
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