The diagram shows part of a regular polygon. The interior angle and the exterior angle at a vertex are marked. The size of the interior angle is 7 times the size of the exterior angle. Work out the number of sides of the polygon.

Question

UpStudy AI · Accepted Answer

Let the exterior angle be $$ x $$ degrees. Then the interior angle is given as $$ 7x $$ degrees. Since the interior and exterior angles at a vertex add up to $$ 180^\circ $$, we have

$$
7x + x = 180.
$$

Combining like terms,

$$
8x = 180.
$$

Solving for $$ x $$:

$$
x = \frac{180}{8} = 22.5.
$$

For a regular polygon, the exterior angle is also given by

$$
x = \frac{360}{n},
$$

where $$ n $$ is the number of sides. Substituting the value of $$ x $$:

$$
22.5 = \frac{360}{n}.
$$

Solving for $$ n $$:

$$
n = \frac{360}{22.5} = 16.
$$

Thus, the polygon has $$ 16 $$ sides.
The polygon has 16 sides.

The diagram shows part of a regular polygon. The interior angle and the exterior angle at a vertex are marked. The size of the interior angle is 7 times the size of the exterior angle. Work out the number of sides of the polygon.

Upstudy AI Solution

Answer

Solution

Mind Expander

Related Questions

Latest Geometry Questions