POLYGON INTERIOR ANGLES SUM THEOREM

THE SUM OF THE INTERIOR ANGLE MEASURES OF AN n-SIDED CONVEX POLYGON IS

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(n - 2)*180

FOR EXAMPLE...

A pentagon has 5 sides so n = 5

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m A + m B + m C + m D + m E = (5 - 2)*180

= 3 * 180

= 540

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B

C

D

E

ANOTHER EXAMPLE...

An octagon has 8 sides so n = 8

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m A + m B + m C + m D + m E = (8 - 2)*180

= 6 * 180

= 1080

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B

C

D

E

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THEORY BEHIND THIS RULE

A diagonal of a polygon is a segment that connects any two non-consecutive vertices.

A

B

C

D

E

For example...

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There are two diagonals from vertex A.

A

B

C

D

E

THEORY BEHIND THIS RULE

This will make 3 triangles.

Each triangle's angles add up to 180^o.

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All the angles of this polygon (pentagon) must add up to

3 * 180 = 540^o.

A

B

C

D

E

THEORY BEHIND THIS RULE

1) Find the number of sides if the measure of an interior angle is 177^o in a regular polygon.

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MORE EXAMPLES

1) Find the number of sides if the measure of an interior angle is 177^o in a regular polygon. SOLUTION

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Remember that the sum of the angles is (n-2)*180.

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The sum of the angles would be 177n

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Therefore 177n = (n-2)*180

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177n = 180n - 360

360 = 3n

120 = n (This is the number of sides)

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MORE EXAMPLES

2) Find the measure of each interior angle if a regular polygon has 15 sides.

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MORE EXAMPLES

2) Find the measure of each interior angle if a regular polygon has 15 sides. SOLUTION

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Remember that the sum of the angles is (n-2)*180.

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The sum of the angles would be: (each interior angle) * n

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Therefore (interior angle)15 = (15-2)*180

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(interior angle)15 = 13 * 180

(interior angle)15 = 2340

interior angle = 156

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MORE EXAMPLES