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# If the sum of interior angle measures of a polygon is 720°, how many sides does the polygon have?

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#### Explanation

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Noah G Share
Jun 14, 2016

An alternative approach.

#### Explanation:

Consider a triangle, a polygon with three sides. The sum of the interior angle measures is 180˚.

Consider any quadrilateral, a polygon with four sides. The sum of the interior angles measures 360˚. We can therefore deduce that for each polygon with an additional side has 180˚ more than the previous figure.

This forms an arithmetic series. Note: An arithmetic series is a sequence of numbers where a common difference is added or subtracted from previous terms to give the next terms. For example, 2, -1, -4 forms an arithmetic series, with a common difference of 3.

The general term of an arithmetic series is given by $\textcolor{b l u e}{{t}_{n} = a + \left(n - 1\right) d}$.

We know ${t}_{n}$, which is 720˚, and $a$, which is 0˚ ( a figure with one line would have an angle measure of 0˚), and $d$ is $180$.

$720 = 0 + \left(n - 1\right) 180$

$720 + 180 = 180 n$

$900 = 180 n$

$5 = n$

Since the figure with angles measuring 0˚ is 1 lines, then the figure with interior angles of 720˚ has $1 + 5 = 6$ sides.

Practice exercises:

1. The interior angles of a polygon add up to 3960˚. How many sides does this polygon have?

Hopefully this helps, and good luck!

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9
Jun 12, 2016

$6$ sides

#### Explanation:

Recall that the formula for the sum of the interior angles in a regular polygon is:

$\textcolor{b l u e}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} {180}^{\circ} \left(n - 2\right) \textcolor{w h i t e}{\frac{a}{a}} |}}}$

$\underline{\text{where}}$:
$n =$number of sides

In your case, since the sum of the interior angles is ${720}^{\circ}$, then the formula must equal to ${720}^{\circ}$. Hence,

${720}^{\circ} = {180}^{\circ} \left(n - 2\right)$

Since you are looking for $n$, the number of sides the polygon has, you must solve for $n$. Thus,

${720}^{\circ} / {180}^{\circ} = n - 2$

$4 = n - 2$

$n = \textcolor{g r e e n}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{6} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

Since $n = 6$, then the polygon has $6$ sides.

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#### Explanation

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Jun 2, 2017

$6$sides

#### Explanation:

You are probably aware of the fact that there is a formula for calculating the sum of the interior angles of a polygon.

Any convex polygon can be divided into triangles by drawing all the possible diagonals from ONE vertex to all the others.

If you do this for a number of shapes and count the number of triangles, you will find that the number of triangles is always $2$ less than the number of sides:

$3$ sides $\rightarrow 1 \Delta$
$4$ sides $\rightarrow 2 \Delta s$
$5$ sides $\rightarrow 3 \Delta s \text{ }$ and so on...

Each triangle has the sum of its angles as 180°

Hence the formula: $\text{Sum int angles} = 180 \left(n - 2\right)$

So to find the number of sides, it will help to find the number of triangles first, then we can just add $2$

$\text{number of } \Delta s = = 720 \div 180 = 4 \Delta s$

$\text{number of sides} = \Delta s + 2 = 4 + 2 = 6$

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