Geometry Lesson 3.1 - Interior Angles of a Polygon
It is important to know that each point of a polygon at which two sides
cross each other is called a vertex. At each vertex, there is an interior angle of the polygon. If the polygon has x sides, the sum, S, of the degree measures of these x interior sides is given by the formula S = (x - 2)(180).
For example, a triangle has 3 angles which add up to 180 degrees. A square has 4 angles which add up to 360 degrees. For every additional side, you have to add another 180 degrees.
Let's talk about a diagonal for a minute. What is a diagonal anyway? A diagonal is a line segment connecting two nonconsecutive vertices of the polygon. In the picture below, BD is a diagonal. As you can see, line segment BD divides quadrilateral ABCD into two triangles the sum whose angle measures is the same as the sum of all the angle measures of the polygon. So, the sum of the degree measures of the 4 angles of the quadrilateral is equal to 2 times 180 or 360.
Sample A:
The degree measures of the angles of quadrilateral ABCD are in the ratio 2:3:3:4. Find the degree measure of the biggest angle of quadrilateral ABCD.
How do we solve this? Since we know the sum of all four angles must be 360 degrees, we just need an expression adding the angles and equalling 360. Because they are in a ratio, they must have some common factor that we need to multiply each term by (call this x).
Steps:
(1) Add the terms 2x + 3x + 3x + 4x
(2) Equate the sum of the terms to 360
(3) Solve for x
(4) Determine the angle measures in degrees.
2x + 3x + 3x + 4x = 360
12x = 360
x = 360/12
x = 30
Even though we know x = 30 we aren't done yet. We multiply 30 times 4 to find the biggest angle. Since 30 times 4 = 120, the biggest angle is 120 degrees. Likewise, the other angles are 3*30=90, 3*3=90, and 2*30 = 60.
Sample B:
Find the sum of the degree measures of the angles of a hexagon. If the hexagon is regular, find the degree measure of each interior angle.
We can use the formula S = (x - 2)(180) to find sum of the degree measure of the hexagon.
A hexagon has 6 sides and x = number of sides in the formula.
Let x = 6 in the formula and simplify.
S = (6 - 2)(180)
S = 4(180)
S = 720
NOTE: A regular polygon is equiangular, so the degree measure of each angle can be found by dividing the sum of the angle by 6 (the number of interior angles of the regular polygon).
Degree measure of each angle = sum of all angles divided by the number of angles.
In other words, S/x. We know that S = 720 (sum of the measures of the angles) and x = 6 (the number of angles).
Then: 720/6 = 120
Sample C:
If the sum of the degree measures of the angles of a polygon is 3600, find the number of sides of the polygon.
Again, we can use the formula S = (x - 2)(180).
In this sample question, let S = 3600 and solve for x.
Look:
3600 = (x - 2)(180)
3600 = 180x - 360
3600 + 360 = 180x
3960 = 180x
3960/180 = x
22 = x
Exterior Angles of a Polygon
At each vertex of a polygon, an exterior angle may be formed by extending one side of the polygon so that the interior and exterior angles at that vertex are supplementary (add up to 180). In the picture below, angles a, b c and d are exterior and the sum of their degree measures is 360.
If a regular polygon has x sides, then the degree measure of each exterior angle is 360 divided by x or simply put 360/x.
Let's look at two sample questions.
Sample D:
Find the degree measure of each interior and exterior angle of a regular hexagon.
A hexagon has 6 sides.
Since x = 6 sides, the sum S can be found by using S = (x - 2)(180).
S = (10 -6)(180)
S = 4(180)
S = 720
Since there are 6 interior angles each with the same measure, the measure of each interior angle can be found by using S/x, where S = 720 and x = 6.
Then: 720/6 = 120.
We know that interior and exterior angles are supplementary (add up to 180) at each vertex and so the measure of each exterior angle is 180 - 120 = 60.
Sample B:
If the measure of each interior angle of a regular polygon is 150, find the number of sides of the polygon.
Because the measure of each interior angle is 150, the measure of an exterior angle drawn at any vertex in terms of this polygon is 180 - 150 = 30.
Measure of exterior angle = 30.
We equate 30 to 360/x and solve for x.
30 = 360/x
30x = 360
x = 360/30
x = 12 sides.
By the way, a geometric figure with 12 sides is called a dodecagon.
By Mr. Feliz
© 2007
