Unlocking the Secrets of Inscribed Polygons

An inscribed angle has its vertex on the circle, and it measures exactly half the arc it intercepts. That half relationship is why every angle inscribed in a semicircle is a right angle.

• Inscribed Polygon: A polygon is said to be inscribed in a circle if all its vertices lie on the circle. The circle is then called the circumscribed circle of the polygon.

• Any side of the inscribed polygon is also a chord of the circle.
• The angle subtended by a chord at the center is double the angle subtended by it at any point on the remaining part of the circle. This becomes particularly interesting for inscribed polygons.

• The sum of the opposite angles of any quadrilateral inscribed in a circle is (180^circ).
• In an inscribed triangle, the length of its sides can determine the circle’s radius using certain relationships, such as the circumradius formula.

Worked examples: inscribed polygons

Practice questions on inscribed polygons

1. A regular hexagon is inscribed in a circle of radius (4 text{ cm}). What is the length of one side of the hexagon?
2. An equilateral triangle is inscribed in a circle. If one side of the triangle is (9 text{ cm}), can you determine the circle’s radius?

1. Each side of the hexagon is equal to the chord of the circle subtended by (60^circ). Using the formula for chord length: ( c approx 2 times 4 times sin(30^circ) = 4 text{ cm}).
2. For an equilateral triangle inscribed in a circle with a side length of (9 text{ cm} ), by using the Pythagorean theorem on the right triangle formed with the radius and half the triangle’s side, the circle’s radius is determined to be (3sqrt{3} text{ cm} ).

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