Each exterior angle of a regular decagon has a measure of $(3 x+6)^{\circ}$. What is the value of $x$?
A. $x=8$
B. $x=10$
C. $x=13$
D. $x=18
Answer (1)
The problem states that each exterior angle of a regular decagon measures ( 3 x + 6 ) ∘ . The sum of exterior angles of any polygon is 36 0 ∘ . A regular decagon has 10 equal exterior angles, so each exterior angle measures 10 36 0 ∘ = 3 6 ∘ . Setting up the equation 3 x + 6 = 36 , we solve for x : 3 x = 30 , so x = 10 . The value of x is 10 .
Explanation
Find the measure of each exterior angle. We are given that each exterior angle of a regular decagon measures ( 3 x + 6 ) ∘ . We need to find the value of x . A decagon has 10 sides. The sum of the exterior angles of any polygon is 36 0 ∘ . Since it is a regular decagon, all exterior angles are equal. Therefore, each exterior angle measures 10 36 0 ∘ = 3 6 ∘ .
Set up the equation. Now we set up the equation 3 x + 6 = 36 and solve for x . Subtracting 6 from both sides, we get 3 x = 36 − 6 , which simplifies to 3 x = 30 .
Solve for x. Dividing both sides by 3, we find x = 3 30 = 10 . Therefore, the value of x is 10.
Examples
Understanding exterior angles is useful in architecture and design. For example, when designing a building with a regular polygonal shape, architects need to calculate the angles to ensure structural integrity and aesthetic appeal. Knowing that the sum of exterior angles is always 360 degrees helps in determining the angles at each vertex, ensuring that the structure fits together correctly and looks visually balanced.
