Math Question?

Question

The interior angle of a regular polygon is 3 times its exterior angle. FInd the number of sides of the polygon.

Please explain it to me! Thanks!

It's not a square. The INTERIOR angle of the square is NOT 3 times the EXTERIOR. It's the other way round.

Answer 1

The polygon is an octogon (8 sides).

To find the measure of each exterior angle of a regular polygon, you just divide 360 degrees by the number of sides. So the measure of each exterior angle of an equilateral triangle is 120 degrees, the measure of each exterior angle of a regular quadrilateral (a square) is 90 degrees, and the measure of each exterior angle of a regular pentagon is 72 degrees.

Therefore we can use this information to find the measure of each interior angle of a regular polygon! If we look at the relationship between one interior angle of a polygon and the exterior angle at that vertex, we see that together they make a straight angle, 180 degrees. Therefore, if we know the measure of an interior angle, we can find the measure of its adjacent exterior angle by subtracting from 180 degrees. For example, in an equilateral triangle, the exterior angle is 360 divided by 3 angles = 120 degrees. So each interior angle is 180 - 120 = 60 degrees.

Answer 2

Here is one way to solve this problem.

A regular polygon's sides are all the same size, as are all of its angles. The exterior angles of any polygon add up to 360 degrees. Since a regular polygon's interior angles will all be the same, so will its exterior angles. Once you find the value of one exterior angle, you know that is the value for all the other exterior angles. If you divide 360 by that value, you will be able to find how many angles there are. Once you know how many angles there are, you know that's the number of sides of the polygon.

Since the exterior angle is what you want to find, we can call that value X. Since we also don't know what the interior angle is, we can call it Y. The problem tells you that X is 3 times Y, or X=3*Y.

The exterior angle is what you get if you extend a side of the polygon, and measure the angle it forms with an adjacent side. Now, that means when you extend a side, it forms two angles with another side of the polygon.

One angle (the one that's inside the polygon) is the interior angle, and the one on the outside is the exterior angle. Notice that added together, they form a straight line (that straight line is just the extended side). You know that the angle a straight line makes is 180 degrees.

X (the interior angle) + Y (the exterior angle) = 180
Now you have 2 unknown variables, but just one equation. This would be a problem, but you can express X in terms of Y using the previously formed equation:
X = 3*Y
Plug that into the other equation, and you get:
3 * Y + Y =180
One equation, one variable. That's what you want so you can solve. The rest is simple arithmetic:
4 * Y = 180
Y = 45 degrees

How many 45 degree angles add up to 360 degrees?
360/45 = 8

The polygon has 8 angles, thus it also has 8 sides and is an octogon.