If the sum of the measures of the interior angles of a polygon is 5400 how many sides does it have

Think of it backwards. To get the total measure of the interior angles of a polygon, you divide the polygon into triangles, where a base of each triangle makes up a side of the polygon, and every triangle shares a vertex towards the center of the polygon. We know that the sum of the interior angles in triangles is 180 degrees. Two of the 3 interior angles of each triangle make up the interior angles of the entire polygon. If you add up the third angle of each triangle, you will get 360 degrees. So, the sum of the interior angles of a polygon is equal to the number of sides the polygon has times 180 degrees minus 360 degrees. Written as an expression, you get:
180n-360, where n is the number of sides a polygon has.
Set this equal to 5400 to get the
5400=180n-360
180n=5760
n=32

So the polygon has 32 sides.

Formula

Sum of all interior angles = (n - 2) x 180,  
n being the number of sides


Find n

(n - 2) x 180 = 5400


Apply distributive property to remove bracket

180n - 360 = 5400


Add 360 on both sides

180n = 5760


Divide by 180 on both sides

n = 32


The polygon has 32 sides