A regular octagon is inscribed in a circle of radius 8 cm. Find its perimeter (in cm).
Answer (1)
To find the perimeter of a regular octagon inscribed in a circle, we need to understand a few geometric relationships.
Understanding Inscribed Octagon : A regular octagon means all the sides and angles are equal. When it is inscribed in a circle, the vertices of the octagon touch the circle.
Determine the Side Length :
The circle is called the circumcircle of the octagon, and the radius given is 8 cm.
We can use the formula for the side length s of a regular octagon inscribed in a circle: s = 2 r sin ( n π ) where r is the radius of the circle, and n is the number of sides of the polygon, which is 8 for an octagon.
Calculating Side Length :
We substitute r = 8 and n = 8 into the formula: s = 2 × 8 × sin ( 8 π )
Evaluate the sine function (use a calculator): sin ( 8 π ) ≈ 0.3827 .
Thus, s ≈ 2 × 8 × 0.3827 = 6.1232 cm.
Calculate the Perimeter :
Since it is a regular octagon, all sides are equal, so the perimeter P is: P = 8 × s = 8 × 6.1232 = 48.9856 cm.
Therefore, the perimeter of the regular octagon is approximately 48.99 cm.
