A regular polygon has each interior angle equal to 120°. How many sides does it have?
a. 6
b. 5
c. 8
d. 10
Answer (2)
The problem states that a regular polygon has each interior angle equal to 12 0 ∘ .
The formula for each interior angle of a regular n -sided polygon is n ( n − 2 ) × 18 0 ∘ .
Set up the equation n ( n − 2 ) × 180 = 120 and solve for n .
The solution to the equation is n = 6 , which means the polygon has 6 sides.
6
Explanation
Problem Analysis We are given that the polygon is regular and each interior angle is 12 0 ∘ . We need to find the number of sides of the polygon.
Setting up the equation The formula for the measure of each interior angle of a regular n -sided polygon is given by: n ( n − 2 ) 18 0 ∘ We set this equal to 12 0 ∘ and solve for n :
n ( n − 2 ) 18 0 ∘ = 12 0 ∘
Solving for n Multiplying both sides by n , we get: ( n − 2 ) × 180 = 120 n Expanding the left side, we have: 180 n − 360 = 120 n Subtracting 120 n from both sides, we get: 60 n − 360 = 0 Adding 360 to both sides, we have: 60 n = 360 Dividing both sides by 60, we find: n = 60 360 = 6
Final Answer Therefore, the regular polygon has 6 sides. This means the polygon is a hexagon.
Examples
Understanding the properties of polygons, like the relationship between interior angles and the number of sides, is useful in various fields. For example, architects use these principles when designing buildings with specific aesthetic or structural requirements. Knowing that a hexagon has interior angles of 120 degrees helps in creating tessellated patterns or structures with hexagonal symmetry, ensuring that the design is both visually appealing and geometrically sound.
The regular polygon has 6 sides, confirming that it is a hexagon. Each interior angle measures 120°. This is derived from the formula relating the number of sides to the interior angle of the polygon.
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