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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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Noah G Share
Jun 14, 2016

Answer:

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!

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Jun 12, 2016

Answer:

#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.

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Jun 2, 2017

Answer:

#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#

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