Find each interior angles the following regular polygons in degrees and grades
(a) Triangle
(b) Quadrilateral (c) Pentagon (d) Hexagon
(e) Heptagon
(1) Octagon (g) Nonagon
(h) Decagon
polygons in degrees and grades​

(a) 120° or 133.33 grades.

(b)135° or 150 grades.

(c) 144° or 160 grades.

(d) 150° or 166.67 grades.

(e) 154.29° or 171.43 grades.

(f) 157.5° or 175 grades.

(g) 160° or 177.78 grades.

(h) 162° or 180 grades.

A regular polygon is a polygon that has all sides and all angles equal.

Each interior angle, k (measured in degrees), of a regular polygon is given by;

k = s ÷ n            -----------(k)

Where;

s = sum of the interior angle of the polygon.

n = number of sides of the polygon

To get s, we use

s = (n - 2) x 180         [This is the formula to calculate the sum of the interior angles of a polygon]

Substituting s this into equation (i) gives

k = (n - 2) x 180 ÷ n

k = 180(n - 2) ÷ n             -------------(ii)

(a) Triangle.

A triangle has n = 3 sides, therefore, each interior angle of a triangle is found by substituting n = 3 into equation (ii)

k = 180(3 - 1) ÷ 3

k = 180(2) ÷ 3

k = 180 x 2 ÷ 3

k = 360 ÷ 3

k = 120°

Convert to grade.

Remember that;

90° = 100 grades

∴ 120° = [tex]\frac{120 * 100}{90}[/tex] grades

⇒ 120° = 133.33 grades.

Therefore, each interior angles of a regular triangle is 120° or 133.33 grades.

(b) Quadrilateral.

A quadrilateral has n = 4 sides, therefore, each interior angle of a quadrilateral is found by substituting n = 4 into equation (ii)

k = 180(4 - 1) ÷ 4

k = 180(3) ÷ 4

k = 180 x 3 ÷ 4

k = 540 ÷ 4

k = 135°

Convert to grade.

Remember that;

90° = 100 grades

∴ 135° = [tex]\frac{135 * 100}{90}[/tex] grades

⇒ 135° = 150 grades.

Therefore, each interior angles of a regular quadrilateral is 135° or 150 grades.

(c) Pentagon

A pentagon has n = 5 sides, therefore, each interior angle of a pentagon is found by substituting n = 5 into equation (ii)

k = 180(5 - 1) ÷ 5

k = 180(4) ÷ 5

k = 180 x 4 ÷ 5

k = 720 ÷ 5

k = 144°

Convert to grade.

Remember that;

90° = 100 grades

∴ 144° = [tex]\frac{144 * 100}{90}[/tex] grades

⇒ 144° = 160 grades.

Therefore, each interior angles of a regular pentagon is 144° or 160 grades.

(d) Hexagon

A hexagon has n = 6 sides, therefore, each interior angle of a hexagon is found by substituting n = 6 into equation (ii)

k = 180(6 - 1) ÷ 6

k = 180(5) ÷ 6

k = 180 x 5 ÷ 6

k = 900 ÷ 6

k = 150°

Convert to grade.

Remember that;

90° = 100 grades

∴ 150° = [tex]\frac{150 * 100}{90}[/tex] grades

⇒ 150° = 166.67 grades.

Therefore, each interior angles of a regular hexagon is 150° or 166.67 grades.

(e) Heptagon

A heptagon has n = 7 sides, therefore, each interior angle of a heptagon is found by substituting n = 7 into equation (ii)

k = 180(7 - 1) ÷ 7

k = 180(6) ÷ 7

k = 180 x 6 ÷ 7

k = 1080 ÷ 7

k = 154.29°

Convert to grade.

Remember that;

90° = 100 grades

∴ 154.29° = [tex]\frac{154.29 * 100}{90}[/tex] grades

⇒ 154.29° = 171.43 grades.

Therefore, each interior angles of a regular heptagon is 154.29° or 171.43 grades.

(f) Octagon

An octagon has n = 8  sides, therefore, each interior angle of a octagon is found by substituting n = 8 into equation (ii)

k = 180(8 - 1) ÷ 8

k = 180(7) ÷ 8

k = 180 x 7 ÷ 8

k = 1260 ÷ 8

k = 157.5°

Convert to grade.

Remember that;

90° = 100 grades

∴ 157.5° = [tex]\frac{157.5 * 100}{90}[/tex] grades

⇒ 157.5° = 175 grades.

Therefore, each interior angles of a regular octagon is 157.5° or 175 grades.

(g) Nonagon

A nonagon has n = 9 sides, therefore, each interior angle of a nonagon is found by substituting n = 9 into equation (ii)

k = 180(9 - 1) ÷ 9

k = 180(8) ÷ 9

k = 180 x 8 ÷ 9

k = 1440 ÷ 9

k = 160°

Convert to grade.

Remember that;

90° = 100 grades

∴ 160° = [tex]\frac{160 * 100}{90}[/tex] grades

⇒ 160° = 177.78 grades.

Therefore, each interior angles of a regular nonagon is 160° or 177.78 grades.

(h) Decagon

A decagon has n = 10 sides, therefore, each interior angle of a decagon is found by substituting n = 10 into equation (ii)

k = 180(10 - 1) ÷ 10

k = 180(9) ÷ 10

k = 180 x 9 ÷ 10

k = 1620 ÷ 10

k = 162°

Convert to grade.

Remember that;

90° = 100 grades

∴ 162° = [tex]\frac{162 * 100}{90}[/tex] grades

⇒ 162° = 180 grades.

Therefore, each interior angles of a regular decagon is 162° or 180 grades.

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