A regular polygon has an interior angle measuring 144°.
a. How many sides does the polygon have?
b. What is the sum of the interior angles of the polygon?
c. What is the measure of an exterior angle of this polygon?
The correct answer and explanation is :
To solve these problems, we will first break down each part systematically.
a. How many sides does the polygon have?
For a regular polygon, the formula for the measure of an interior angle is:
$$
\text{Interior Angle} = \frac{(n-2) \times 180^\circ}{n}
$$
Where $n$ is the number of sides of the polygon.
Given that the interior angle is 144°, we can substitute this into the formula and solve for $n$:
$$
144 = \frac{(n-2) \times 180}{n}
$$
Multiply both sides by $n$ to eliminate the denominator:
$$
144n = (n-2) \times 180
$$
Distribute the 180:
$$
144n = 180n – 360
$$
Now, subtract 144n from both sides:
$$
0 = 36n – 360
$$
Add 360 to both sides:
$$
360 = 36n
$$
Now, divide by 36:
$$
n = 10
$$
So, the polygon has 10 sides.
b. What is the sum of the interior angles of the polygon?
The sum of the interior angles of any polygon is given by the formula:
$$
\text{Sum of Interior Angles} = (n-2) \times 180^\circ
$$
Substitute $n = 10$ into the formula:
$$
\text{Sum of Interior Angles} = (10-2) \times 180 = 8 \times 180 = 1440^\circ
$$
So, the sum of the interior angles of the polygon is 1440°.
c. What is the measure of an exterior angle of this polygon?
The measure of an exterior angle of a regular polygon can be found using the formula:
$$
\text{Exterior Angle} = \frac{360^\circ}{n}
$$
Substitute $n = 10$:
$$
\text{Exterior Angle} = \frac{360^\circ}{10} = 36^\circ
$$
So, the measure of an exterior angle of the polygon is 36°.
Summary:
- The polygon has 10 sides.
- The sum of the interior angles is 1440°.
- The measure of an exterior angle is 36°.
This problem highlights the relationship between the interior and exterior angles of a polygon and the formulas used to calculate them based on the number of sides.