If the ratio of the interior angle to the exterior angle is 5:1 for a regular polygon, find
a. the size of each exterior angle
b. the number of sides of the polygon
c. the sum of the interior angles
d. Name the polygon
Oops
n=12
exterior: 30
interior: 150
n=12
10(180) = 1800
dodecagon
The ratio of interior and exterior angle of regular polygon is is 5:1, find the number of diagonals of the regular polygon.
interior: (n-2)180/n
exterior: 360/n
(n-2)180/n ÷ 360/n = (n-2)/2 = 5
n = 8
a:
exterior: 45
interior: 135
b: n=8
c: 6(180) = 1080
d: octagon
To find the answers to these questions, let's break down each step:
a. To find the size of each exterior angle, we can use the fact that the ratio of the interior angle to the exterior angle is 5:1. Since a regular polygon has equal interior angles, the sum of the interior angles is always (n - 2) * 180 degrees, where n is the number of sides of the polygon.
Let's set up the equation:
Interior angle / Exterior angle = 5 / 1
Since the sum of the interior angles of a regular polygon is (n - 2) * 180 degrees, and the exterior and interior angles form a linear pair (they add up to 180 degrees), we can write the equation:
((n - 2) * 180) / (180 - x) = 5 / 1
Simplifying the equation:
(n - 2) / (180 - x) = 5
Cross-multiplying and solving for x (exterior angle):
5(180 - x) = n - 2
900 - 5x = n - 2
5x = 902 - n
x = (902 - n) / 5
Thus, we have found the equation to calculate the size of each exterior angle.
b. To find the number of sides of the polygon, we need to substitute the value of x (exterior angle) into the equation for a regular polygon:
360 / x = n
By substituting the equation for x, we get:
360 / ((902 - n) / 5) = n
360 * (5 / (902 - n)) = n
1800 / (902 - n) = n
Solving for n in this equation will give us the number of sides of the polygon.
c. The sum of the interior angles can be derived by using the formula for a regular polygon:
Sum of Interior Angles = (n - 2) * 180 degrees
By substituting the value of n, we can find the sum of the interior angles.
d. To name the polygon, we can use the number of sides.
For example, a polygon with 3 sides is called a triangle, 4 sides is a quadrilateral, 5 sides is a pentagon, 6 sides is a hexagon, and so on.
Using the value of n, we can determine the name of the polygon.
By following these steps, we can find the answers to the questions.