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Find an interior angle of a regular polygon or a missing angle

Split a polygon into triangles

An $n$ -sided polygon can be split into $n - 2$ triangles, so its interior angles sum to $(n - 2) 18 0^{\circ}$ . In a regular polygon, divide that sum by $n$ . For a missing angle, subtract the known angles from the sum.

Problem

(1) What is each interior angle of a regular pentagon? (2) A hexagon has interior angles $11 2^{\circ}$ , $x^{\circ}$ , $12 0^{\circ}$ , $10 0^{\circ}$ , $12 8^{\circ}$ and $13 3^{\circ}$ . Find $x$ .

Answer

(1)

10 8^{\circ}

. (2)

x = 12 7^{\circ}

The pentagon has equal angles; the hexagon in the second example need not be regular.

Step-by-step solution

1. Regular pentagon

A pentagon has $5 - 2 = 3$ triangles' worth of interior angle, so the sum is $3 \cdot 18 0^{\circ} = 54 0^{\circ}$ . Its five angles are equal:

54 0^{\circ} /5 = 10 8^{\circ} .

2. Hexagon with one unknown angle

A hexagon's interior angles sum to $(6 - 2) 18 0^{\circ} = 72 0^{\circ}$ . The five known angles total

112 + 120 + 100 + 128 + 133 = 593.

Therefore $x = 720 - 593 = 12 7^{\circ}$ . The six angles now add to $72 0^{\circ}$ , as required.

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