We can calculate interior angle of a polygon by using the formula given below.
Measure of Each Angle :
Here 'n' stands for number of sides.
Examples 1-4 : All the polygons are regular polygons. In each case, find the measure of each interior angle.
Example 1 :
Solution :
The sides of the polygon are AB, BC, CD, DE and EA.
So, the regular polygon above has 5 sides.
Sum of interior angles of polygon = (n - 2) x 180°
= (5 - 2) x 180°
= 3 x 180°
= 540°
Measure of each interior angle :
= 540°/5
= 108°
Example 2 :
Solution :
The sides of the polygon are AB, BC, CD, DE, EF, FG, GH and HA
So, the regular polygon above has 8 sides.
Sum of interior angles of a polygon = (n - 2) x 180°
= (8 - 2) x 180°
= 6 x 180°
= 1080°
Measure of each interior angle :
= 1080°/8
= 135°
Example 3 :
Solution :
The sides of the polygon are AB, BC, CD, DE, EF, FG, GH, HI, IJ, JK abd KA.
So, the regular polygon above has 8 sides.
Sum of interior angles of a polygon = (n - 2) x 180°
= (11 - 2) x 180°
= 9 x 180°
= 1620°
Measure of each interior angle :
= 1620°/11
= 147° 3'
Example 4 :
Solution :
The sides of the polygon are AB, BC, CD, DE, EF and FA.
So, the regular polygon above has 6 sides.
Sum of interior angles of a polygon = (n - 2) x 180°
= (6 - 2) x 180°
= 4 x 180°
= 720°
Measure of each interior angle :
= 720°/6
= 120°
Example 5 :
Find the value of x shown in the polygon shown below.
Solution :
The polygon above has 5 sides.
Sum of interior angles of the polygon = (5 - 2) x 180°
x + 115° + 90° + 90° + 160° = 3 x 180°
x + 455° = 540°
Subtract 455° from both sides.
x = 85°
Example 6 :
Find the value of x shown in the polygon shown below.
Solution :
The polygon above has 6 sides.
Sum of interior angles of the polygon = (6 - 2) x 180°
y + 130° + 90° + 170° + 75° + 110° = 4 x 180°
y + 575° = 720°
Subtract 575° from both sides.
y = 145°
Example 7 :
Find the value of x in the regular pentagon shown below.
Solution :
Sum of the interior angles of a regular pentagon :
= (5 - 2) x 180°
= 3 x 180°
= 540°
Measure of each interior angle of a regular polygon :
= 540°/5
= 108°
In a regular pentagon, all the interior angles are equal in measure.
m∠C = m∠D = 108°
In the figure above, ABCD is a quadrilateral.
Sum of the interior angles in a quadrilateral = 360°
m∠A + m∠B + m∠C + m∠D = 360°
38° + x + 108° + 108° = 360°
x + 254° = 308°
Subtract 254° from both sides.
x = 54°
Example 8 :
Find the measure of each interior angle of a regular hexagon.
Solution :
Sum of interior angles of an hexagon :
= (6 - 2) x 180°
= 4 x 180°
= 720°
Measure of each interior angle of a regular hexagon :
= 720°/6
= 120°
Example 9 :
If the sum of the interior angles of a polygon is 1260°, How many sides does the polygon have?
Solution :
Sum of interior angles of the polygon = 1260°
(n - 2) x 180° = 1260°
Divide both sides by 180°.
n - 2 = 7
Add 2 to both sides.
n = 9
So, the polygon has 9 sides.
Example 10 :
If the measure of each interior angle of a regular decagon is 2x°, find the value of x.
Solution :
Decagon is a 10-sided polygon.
Sum of interior angles of a decagon : (10 - 2) x 180°
= 8 x 180°
= 1440°
Since decagon is a 10-sided polygon, it will have ten interior angles
Given : The measure each interior angle of a regular decagon is 2x°.
10 x measure of each interior angle = 1440°
10(2x°) = 1440°
20x = 1440
Divide both sides by 20.
x = 72
Example 11 :
If the measure of each interior angle of a regular polygon is 150°, find the number of sides the polygon has.
Solution :
Let n be the number of sides the polygon has.
Sum of interior angles of a polygon : (n - 2) x 180°
n x 150° = (n - 2) x 180°
n x 150° = n x 180° - 2 x 180°
150n = 180n - 360
Subtract 180n from both sides.
-30n = -360
Divide both sides by -30.
n = 12
So, the polygon has 12 sides.
Example 12 :
If the measure of each interior angle of a regular dodecagon is (2x + 50)°, find the value of x.
Solution :
Dodecagon is a 12-dided polygon
Sum of interior angles of a dodecagon :
= (12 - 2) x 180°
= 10 x 180°
= 1800°
Measure of each interior angle of a regular dodecagon :
= 1800°/12
= 150°
Given : The measure of each interior angle of a regular dodecagon is (2x + 50)°.
Then,
(2x + 50)° = 150°
2x + 50 = 150
Subtract 50 from both sides.
2x = 100
Divide both sides by 2.
x = 50
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