# You need to construct a regular polygon. When you draw two sides, the interior angle created between them is 120°. What will be the sum, in degrees, of the measures of the interior angles of this polygon when it is completed?

Dec 30, 2015

$\text{Sum of interior angles} = {720}^{\circ}$

#### Explanation:

Since the shape in question is a regular polygon, we know that all of the interior angles must be the same: ${120}^{\circ}$. We must now consult the following formula:

$\text{interior angle "+" exterior angle" = 180^@" " rArr" " "exterior angle" = 180^@ - "interior angle}$

Thus, the size of one of this shape's exterior angles will be:

$\text{exterior angle} = {180}^{\circ} - 120 = {60}^{\circ}$

The sum of the exterior angles of a regular polygon always total to ${360}^{\circ}$. This can be expressed mathematically, and then adapted:

$\text{size of exterior angle "+" number of exterior angles"=360^@" " rArr" " "number of exterior angles" = 360^@/"size of exterior angle}$

We can now calculate the number of exterior angles the shape has and, since an exterior angle is formed by the extension of one side of the shape, this will also equate to the number of sides the shape has:

$\text{number of exterior angles" = "number of sides} = {360}^{\circ} / {60}^{\circ} = 6$

Our calculation would suggest that our shape is a regular hexagon, which does indeed have interior angles of ${120}^{\circ}$.

We can now find the total of all of our interior angles: we can use this equation to do so:

$\text{sum of interior angles} = 180 \left(n - 2\right)$

Where $n =$ the number of sides the shape has. Therefore:

$\text{sum of interior angles} = 180 \left(6 - 2\right) = 180 \times 4 = {720}^{\circ}$

To verify, dividing our answer by $6$ should result in the size of one of the shape's interior angles, as dictated by the question:

$\frac{720}{6} = {120}^{\circ} \textcolor{g r e e n}{\text{ TRUE}}$

Observe that both of our equations to find the total of the shape's interior angles require us to know the number of sides the shape has. This was a piece of information with which we were not provided: this is why we had to work this out using exterior angles.