Problems 1-4 : Find the measure of each interior angle in the regular polygon given.
Problem 1 :
Problem 2 :
Problem 3 :
Problem 4 :
Problem 5 :
What is the value of a in the polygon shown below?
Problem 6 :
Find the value of k in the polygon shown below.
Problem 7 :
Find the value of y in the polygon shown below.
Problem 8 :
What is the measure of each interior angle of a regular heptagon?
Problem 9 :
Sum of the interior angles of a polygon is 1800°. Find the number of sides the polygon has.
Problem 10 :
In a dodecagon, if the measure of each interior angle is 3x°, What is the value of x?
Problem 11 :
The average of all interior angles of a polygon is 156°. Find the number of sides the polygon has.
Problem 12 :
In a regular nonagon, the measure of each interior angle is (3x - 40)°. What is the value of x?
1. Answer :
Count the number of sides in the given regular polygon.
The given regular polygon has 5 sides.
Formula to find the sum of interior angles of a polygon :
= (n - 2) x 180°
Substitute n = 5.
= (5 - 2) x 180°
= 3 x 180°
= 540°
Measure of each interior angle :
= 540°/5
= 108°
2. Answer :
Count the number of sides in the given regular polygon.
The given regular polygon has 8 sides.
Formula to find the sum of interior angles of a polygon :
= (n - 2) x 180°
Substitute n = 8.
= (8 - 2) x 180°
= 6 x 180°
= 1080°
Measure of each interior angle :
= 1080°/8
= 135°
3. Answer :
Count the number of sides in the given regular polygon.
The given regular polygon has 8 sides.
Formula to find the sum of interior angles of a polygon :
= (n - 2) x 180°
Substitute n = 11.
= (11 - 2) x 180°
= 9 x 180°
= 1620°
Measure of each interior angle :
= 1620°/11
= 147° 3'
4. Answer :
Count the number of sides in the given regular polygon.
The given regular polygon has 8 sides.
Formula to find the sum of interior angles of a polygon :
= (n - 2) x 180°
Substitute n = 6.
= (6 - 2) x 180°
= 4 x 180°
= 720°
Measure of each interior angle :
= 720°/6
= 120°
5. Answer :
The given polygon has 5 sides.
Sum of interior angles of the polygon = (5 - 2) x 180°
a + 120° + 90° + 90° + 155° = 3 x 180°
a + 455° = 540°
Subtract 455° from both sides.
x = 85°
6. Answer :
The given polygon has 6 sides.
Sum of interior angles of the polygon = (6 - 2) x 180°
k + 125° + 90° + 175° + 70° + 115° = 4 x 180°
k + 575° = 720°
Subtract 575° from both sides.
k = 145°
7. Answer :
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°
40° + x + 108° + 108° = 360°
x + 256° = 308°
Subtract 256° from both sides.
x = 52°
8. Answer :
What is the measure of each interior angle of a regular heptagon?
Sum of interior angles of a heptagon :
= (7 - 2) x 180°
= 5 x 180°
= 900°
Measure of each interior angle of a regular hexagon :
= 900°/7
≈ 128.57°
9. Answer :
Sum of the interior angles of a polygon is 1800°. Find the number of sides the polygon has.
Sum of interior angles of the polygon = 1800°
(n - 2) x 180° = 1800°
Divide both sides by 180°.
n - 2 = 10
Add 2 to both sides.
n = 12
Therefore, the polygon has 12 sides.
10. Answer :
Decagon is a 12-sided polygon.
Sum of interior angles of a dodecagon : (12 - 2) x 180°
= 10 x 180°
= 1800°
Since dodecagon is a 12-sided polygon, it will have twelve interior angles
Given : The measure each interior angle of a regular dodecagon is 3x°.
12 x measure of each interior angle = 1800°
12(3x°) = 1800°
36x = 1800
Divide both sides by 36.
x = 50
11. Answer :
Let n be the number of sides the polygon has.
Given : The average of all interior angles of a polygon is 156°.
(Sum of all interior angles of the polygon)/n = 156°
Sum of interior angles of the polygon = n x 156°
(n - 2) x 180° = n x 156°
n x 180° - 2 x 180° = n x 156°
180n - 360 = 156n
Subtract 156n from both sides.
24n - 360 = 0
Add 360 to both sides.
24n = 360
Divide both sides by 24.
n = 15
Therefore, the polygon has 15 sides.
12. Answer :
Dodecagon is a 9-dided polygon
Sum of interior angles of a nonagon :
= (9 - 2) x 180°
= 7 x 180°
= 1260°
Measure of each interior angle of a regular nonagon :
= 1260°/9
= 140°
Given : The measure of each interior angle of a regular nonagon is (3x - 40)°.
(3x - 40)° = 140°
3x - 40 = 140
Add 40 to both sides.
3x = 180
Divide both sides by 3.
x = 60
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