Question

Marcus says that the sum of the exterior angles of a decagon is greater than that of a heptagon because a decagon has more sides. Liam says that the sum of the exterior angles for both polygons is the same. Is either of them correct? Explain your reasoning. (Please use at least two complete sentences.)

Answer 1

Liam is correct - The sum of the exterior angles of a polygon will always be 360.

Answer 2

Liam is correct, as the sum of the exterior angles for both polygons (heptagon and decagon) is the same that is 360°.

What is exterior angle?

"An exterior angle is the angle between a side of a polygon and an extended adjacent side."

What is decagon?

"A decagon is a polygon having ten sides, ten interior angles and ten vertices."

What is heptagon?

"A heptagon can be defined as a polygon having seven sides, seven interior angles and seven vertices."

First consider a decagon,

The interior angle of decagon is 144° , so the corresponding exterior angle is 36°.

In decagon there are ten interior angles. So, there are 10 exterior angles.

So, sum of the exterior angles of a decagon is, 10 × 36 = 360°

Now, consider a heptagon.

The interior angle of heptagon is 5π/7 radians which is approximately 128.57°.

The corresponding exterior angle of heptagon is 2π/7 radians ≈ 51.43°

In heptagon, there are seven interior angles. So, there must be corresponding 7 exterior angles.

So, sum of the exterior angles of a heptagon is, 2π/7 × 7 = 2π

that is, 360°.

Hence, the sum of the exterior angles for both polygons is the same.

This means, Liam is correct.

Learn more about exterior angles here:

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