A circle is inscribed in a square with an area of 25 cm². Determine the area of a regular octagon inscribed in this circle.

univerkov | August 31, 2021 | education

Knowing the area of the square ABCD, we calculate the length of its side.

AB ^ 2 = Savsd.

AB = √Savsd = √25 = 5 cm.

Since a circle is inscribed in the square, its diameter is equal to the side of the square.

D = AB = 5 cm, then R = D / 2 = 5/2 = 2.5 cm.

The diagonals of an octagon inscribed in a circle divide it into eight equal triangles, the two sides of each of which are the radii of the circle.

In the triangle KOM, OK = OM = R = 2.5 cm.The central angle KOM = 360/8 = 450.

Then Skom = OK * OM * Sin45 / 2 = 2.5 * 2.5 * √2 / 4 = 6.25 * √2 / 4.

Then S8 = 8 * Scom = 8 * 6.25 * √2 / 4 = 12.5 * √2 cm2.

Answer: The area of the octagon is 12.5 * √2 cm2.

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