The exterior and interoir angles of a regular polygon are in the ratio 2:7 how many sides does it have
do this one just like the other one.
2x+7x=180
x=20
n*140 = (n-2)*180
n = 9
To find the number of sides of a regular polygon given the ratio of its exterior and interior angles, we can use the formula:
Sum of the exterior angles = 360 degrees
Sum of the interior angles = (n-2) * 180 degrees (where n is the number of sides of the polygon)
Let's assume that an exterior angle measures 2x degrees and an interior angle measures 7x degrees.
Since the exterior angles of any polygon add up to 360 degrees, we can write the equation:
2x * n = 360
Simplifying the equation we get:
2x = 360 / n
Using the formula for the sum of interior angles, we can also write the equation:
7x * n = (n - 2) * 180
Simplifying the equation we get:
7x = 180 - 360 / n
Now we have two equations:
2x = 360 / n
7x = 180 - 360 / n
To solve the equations simultaneously, we'll find the value of x first:
From the first equation: 2x = 360 / n
x = (360 / n) / 2
x = 180 / n
Now replacing x in the second equation with its value:
7 * (180 / n) = 180 - 360 / n
Multiply both sides by n:
7 * 180 = 180n - 360
1260 = 180n - 360
Add 360 to both sides:
1620 = 180n
Divide both sides by 180:
9 = n
Therefore, the regular polygon has 9 sides.
To find the number of sides in a regular polygon given the ratio of exterior and interior angles, we can use the formula:
Interior angle = (180 * (n - 2)) / n
Exterior angle = 360 / n
Let's assume the ratio of exterior angle to interior angle is 2:7. This means the exterior angle is 2x and the interior angle is 7x (where x is a common factor).
Using the formula for the interior angle, we can equate it with 7x:
(180 * (n - 2)) / n = 7x
To simplify this equation, we can cross multiply:
180(n - 2) = 7xn
Expanding the left side:
180n - 360 = 7xn
Now, let's equate the exterior angle (2x) with the expression 360/n:
2x = 360/n
To eliminate the fraction, we can multiply both sides by n:
2nx = 360
Now, we have a system of equations:
180n - 360 = 7xn
2nx = 360
To solve this system, we can substitute the second equation into the first equation:
180n - 360 = 7(360/n)
Multiply both sides by n to eliminate the fraction:
180n^2 - 360n = 2520
Rearrange the equation:
180n^2 - 360n - 2520 = 0
Now, we have a quadratic equation. Solving this equation will give us the value(s) of n, which represents the number of sides of the regular polygon.