The sum of the interior angles of a convex polygon is three times the sum of its exterior … — JAMB UTME

The sum of the interior angles of a convex polygon is three times the sum of its exterior angles. Calculate the number of sides of the polygon.

Cevap: 8

The polygon has 8 sides.

The sum of the exterior angles of any convex polygon is always

360^\circ

. Since the sum of the interior angles is three times this value, the interior angle sum is

3 \times 360^\circ = 1080^\circ

. Setting

(n - 2) \times 180^\circ = 1080^\circ

gives

n - 2 = 6

, which yields

n = 8

sides.

Determine the exterior angle sum

The sum of exterior angles for any convex polygon is

360^\circ

The exterior angles of any convex polygon sum to a full turn (

360^\circ

Compute the sum of the interior angles

Sum of interior angles =

3 \times 360^\circ = 1080^\circ

The question states that the interior angle sum is three times the exterior angle sum.

Apply the interior angle sum formula

(n - 2) \times 180^\circ = 1080^\circ

The interior angle sum of an

n

-sided convex polygon is given by

(n - 2) \times 180^\circ

Solve for

n

n - 2 = \frac{1080^\circ}{180^\circ} = 6 \implies n = 8

Dividing

1080^\circ

180^\circ

gives

6

, and adding

2

gives

n = 8

Relationship between the sum of interior and exterior angles of a convex polygon

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