Geometry Homework

Question

The sum of the measures of the interior angles of a convex regular polygon is 3,960 degrees. How many sides does the polygon have ? What is the measure of each interior angle?

Answer 1

A convex polygon is defined as a polygon with all its interior angles less than 180°. This means that all the vertices of the polygon will point outwards, away from the interior of the shape.

The sum of the interior angles in a polygon depends on the number of sides it has. The Polygon Sum Formula states that for any n−gon, the interior angles add up to (n−2)×180∘.

→n=8

(8−2)6×180°=1,080°

Once you know the number of sides you substitute. N = 8

As you can see there are 8 sides to this polygon.

IF YOU WANT TO FIND THE MEASURE FOR EACH ANGLE

Regular Polygon Interior Angle Formula: For any equiangular n−gon, the measure of each angle is (n−2)×180∘n.

In the picture below, if all eight angles are congruent then each angle is

(8−2)×180°

-----------------

N

6×180°

-----------

N

1080°

---------

N

Angle = 135°

Each angle is 135°

BUT IF YOU WANT TO KNOW THE NUMBER OF SIDES THEN

The interior angles of a polygon add up to 3960°∘. How many sides does it have?

Use the Polygon Sum Formula and solve for n.

(n−2) x 180°= 3960°

180°n − 360° = 3960°

180°n -360° + 360° = 3960° + 360°

180°n = 4320°

--------- --------

180° = 180°

n = 24 sides

WHAT IS MEASURE OF EACH ANGLE?

(n−2)×180°

-----------------

n

(24-2)x180°

------------------

24

22x180°

--------------

24

Angle = 165°

If you want to check your math?

MULTIPLY 24 X 165° = 3960°



Answer 2

The sum of the interior angles of a convex regular polygon is 180 times (n-2), where n is the number of sides. Since we already know the sum, we can write 3,960 = 180(n-2). If you solve for n in this equation, you will have the number of sides.

Since the number of interior angles will equal the number of sides, n also tells you the number of interior angles. If you take the sum of all the angles (3,960) and divide it by the number of angles (n) you will know what each angles measures.

I hope this helps! Let me know if you have any other questions.