Question

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

Answer 1

`6` sides

Explanation:

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

`color(blue)(|bar(ul(color(white)(a/a)180^@(n-2)color(white)(a/a)|)))`

`ul("where")`:
`n=`number of sides

In your case, since the sum of the interior angles is `720^@`, then the formula must equal to `720^@`. Hence,

`720^@=180^@(n-2)`

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

`720^@/180^@=n-2`

`4=n-2`

`n=color(green)(|bar(ul(color(white)(a/a)color(black)6color(white)(a/a)|)))`

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

Answer 2

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 `color(blue)(t_n = a + (n - 1)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 + (n - 1)180`

`720 + 180 = 180n`

`900 = 180n`

`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!

Answer 3

`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 `rarr 1 Delta`
`4` sides `rarr 2 Delta s`
`5` sides `rarr 3 Delta s" "` and so on...

Each triangle has the sum of its angles as `180°`

Hence the formula: `"Sum int angles" = 180(n-2)`

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

`"number of " Delta s = = 720 div180 = 4 Delta s`

`"number of sides" = Delta s +2 = 4+2 = 6`