A regular polygon has 16 sides. If one of its angles measures $(5h-29)^{\circ}$, what is the value of $h$?
A. $h=56.5$
B. $h=37.3$
C. $h=33.6$
D. $h=9
Answer (1)
Calculate the measure of each interior angle of a regular 16-sided polygon using the formula: n ( n − 2 ) × 180 , where n = 16 . The result is 157. 5 ∘ .
Set the expression for the interior angle, ( 5 h − 29 ) ∘ , equal to the calculated measure: 5 h − 29 = 157.5 .
Solve the resulting equation for h : 5 h = 186.5 , which gives h = 5 186.5 .
Find the value of h : h = 37.3 .
Explanation
Problem Analysis We are given a regular polygon with 16 sides, and one of its interior angles measures ( 5 h − 29 ) ∘ . Our goal is to find the value of h . To do this, we first need to calculate the measure of each interior angle of a regular 16-sided polygon.
Calculate the measure of each interior angle The formula for the measure of each interior angle of a regular n -sided polygon is given by: n ( n − 2 ) × 18 0 ∘ In our case, n = 16 , so we have: 16 ( 16 − 2 ) × 18 0 ∘ = 16 14 × 18 0 ∘ = 16 252 0 ∘ = 157. 5 ∘ Thus, each interior angle of the regular 16-sided polygon measures 157. 5 ∘ .
Set up the equation Now we set the given expression for the interior angle, ( 5 h − 29 ) ∘ , equal to the calculated measure of the interior angle, 157. 5 ∘ :
5 h − 29 = 157.5
Solve for h Next, we solve the equation for h :
5 h = 157.5 + 29 5 h = 186.5 h = 5 186.5 h = 37.3
Final Answer Therefore, the value of h is 37.3 .
Examples
Understanding the angles in regular polygons is crucial in many fields, such as architecture and engineering. For instance, when designing a building with a regular polygonal base, architects need to calculate the exact angles to ensure structural integrity and aesthetic appeal. Knowing how to find the measure of interior angles and solve for unknowns helps in creating precise and stable designs.
