The instructions say, "The measure of an exterior angle of a regular polygon is given. Find the measure of an interior angle. Then find the number of sides. For another problem just like this, they gave the number 18 and I subtracted 18 from 180 and solved it like this, 162n=(n-2)180. However in this case they gave the variable x. How do I find what the number is instead of x so I can solve my problem?
In the actual case they had 18°
So in the general case they now have x°
So let's do exactly the same thing
So if we have n sides and the exterior angle is x°
then the interior angle is 180-x
So (180-x)n = (180)(n-2)
180x - nx = 180n + 360
solving for n:
180x - 360 = 180n + nx
180x - 360 = n(180-x)
n = (180x - 360)/(180-x) ---> number of sides
solving for x:
x(180 - n) = 180n + 360
x = (180n + 360)/(180 - n)
To find the number of sides and the measure of an interior angle of a regular polygon given the measure of an exterior angle, you can follow these steps:
Step 1: Understand the relationship between exterior and interior angles of a regular polygon.
In a regular polygon, the exterior angle and the interior angle that are next to each other form a linear pair. A linear pair of angles adds up to 180 degrees, so we have the equation:
(exterior angle) + (interior angle) = 180 degrees
Step 2: Set up an equation using the given information.
Let's say the measure of the exterior angle is x degrees. Since we know that the sum of the exterior and interior angle is 180 degrees, we can write the equation as:
x + (interior angle) = 180 degrees
Step 3: Solve the equation to find the value of the interior angle.
Rearrange the equation to isolate the interior angle:
(interior angle) = 180 degrees - x
Step 4: Substitute the value of the interior angle into the formula for the sum of interior angles in a polygon.
The formula for the sum of the interior angles in a polygon is given by:
sum of interior angles = (number of sides - 2) * 180 degrees
Substitute the value of the interior angle into the formula:
(number of sides - 2) * 180 degrees = 180 degrees - x
Step 5: Solve for the number of sides.
Simplify the equation:
(number of sides - 2) = (180 degrees - x) / 180 degrees
To find the value of the number of sides, add 2 to both sides:
number of sides = (180 degrees - x) / 180 degrees + 2
Therefore, by substituting the given value of x into the formula, you will find the number of sides of the polygon.
For example, if x is given as 18, you would substitute x = 18 into the formula:
number of sides = (180 degrees - 18) / 180 degrees + 2
number of sides = 162 / 180 + 2
number of sides = 0.9 + 2
number of sides = 2.9 (approximately)
Since the number of sides must be a whole number for a regular polygon, you would round up to the nearest whole number:
number of sides ≈ 3
So, for this particular example with x = 18, the regular polygon would have 3 sides.
To solve the problem with a variable x instead of a specific value, you would keep x as a variable in the final equation until you have more information or until you can simplify it further.