Inscribed and Circumscribed Circles Dr. Jason Gershman

Circumscribed Circles To the right are examples of circles circumscribed about a square, rectangle, parallelogram, and quadrilateral. These figures have many interesting properties which can be used to solve problems.

Square in a circle Given the inscribed square has area 25 – Find the area of the circle. – Find the area in pink which is outside the square but inside the circle. – Suppose that you are playing a game whereby if you hit the area in pink, you win the game. Find the probability that you win the game.

Rectangle in circle Given the inscribed rectangle has length 6 and width 8 – Find the area of the circle. – Find the area in pink which is outside the square but inside the circle. – Find the ratio of the area in pink which is east of the rectangle but inside the circle to the area in pink which is north of the rectangle but inside the circle.

Equilateral Triangle in a Circle Given an equilateral triangle with side length 12 with a circle circumscribed about the triangle. Find the area and circumference of the circle.

Quadrilateral in a Circle Using a variation on Heron’s Formula, find the area of a quadrilateral of side lengths 4, 5, 7, and 10 with a circle circumscribed about it. Hint: First find the semiperimeter.

Inscribed Circles To the right are examples of circles inscribed in a square, equilateral triangle, regular hexagon, and a star. These figures have many interesting properties which can be used to solve problems.

Circle in a Square For a square of side length 4. Find the area of the circle inscribed in that square. Find the ratio of the area of the circle inscribed in that square to the area of the circle circumscribed about that same square of side length 4.

Circle in Equilateral Triangle For an equilateral triangle of side length 6. Find the area of the circle inscribed in that triangle. Find the ratio of the area of the circle inscribed in that triangle to the area of the circle circumscribed about that same triangle of side length 6.

Circle in Hexagon For a circle of radius 2 inscribed in a regular hexagon, find the area of the hexagon. Find the ratio of the area of the regular hexagon with circle radius 2 inscribed in it to the area of a square with circle of radius 2 inscribed in it. What do you expect to happen to the area of an n-gon with circle of radius 2 inscribed in it as the value of n goes to infinity.

Circle in a Star Suppose the radius of the circle at the right is 8 inches. Find the area of the star.