The sum of the interior angles of a regular polygon equals 1260 degrees. How many sides does a polygon have?
For a regular polygon with nnn sides,
\text{Sum of interior angles} = (n-2)\times 180\degreeSum of interior angles=(n−2)×180°start text, S, u, m, space, o, f, space, i, n, t, e, r, i, o, r, space, a, n, g, l, e, s, end text, equals, left parenthesis, n, minus, 2, right parenthesis, times, 180, degree
[How do we get this?]
For a regular polygon with nnn sides,
\text{interior angle} = \dfrac{\text{Sum of interior angles}}{\text{number of sides}}interior angle=
number of sides
Sum of interior angles
start text, i, n, t, e, r, i, o, r, space, a, n, g, l, e, end text, equals, start fraction, start text, S, u, m, space, o, f, space, i, n, t, e, r, i, o, r, space, a, n, g, l, e, s, end text, divided by, start text, n, u, m, b, e, r, space, o, f, space, s, i, d, e, s, end text, end fraction
We can find the number of sides, nnn, and use it to find the measure of each interior angle.
Hint #22 / 4
Finding nnn
\begin{aligned} \text{Sum of interior angles}&=(n-2)\times180\degree\\\\ 1260\degree&=(n-2)\times180\degree\\\\ n&=9 \end{aligned}
Sum of interior angles
1260°
n
=(n−2)×180°
=(n−2)×180°
=9
So, the given polygon is 999 sided or a nonagon.
Hint #33 / 4
\begin{aligned} &\phantom{=}\text{Each interior angle}\\\\ &=\dfrac{\text{Sum of interior angles}}{n}\\\\ &=\dfrac{1260\degree}{9}\\\\ &=140\degree \end{aligned}
=Each interior angl9
1260°
=140°
The measure of each interior angle is 140, degree.
1260 = 180*7
So, a 9-gon fits the bill
Why did you multiply by 7?
actually, you divide by 180. A n-sided polygon's interior angles sum to (n-2)*180. So, here we have
(n-2)*180 = 1260
n-2 = 1260/180
n-2 = 7
n = 9
To determine how many sides a regular polygon has given the sum of its interior angles, we can use the formula:
Sum of interior angles = (n - 2) * 180 degrees,
where n represents the number of sides in the polygon.
In this case, we have the equation:
1260 degrees = (n - 2) * 180 degrees.
To solve for n, we need to isolate it on one side of the equation. We can do this by dividing both sides of the equation by 180:
1260 degrees / 180 degrees = n - 2.
Simplifying the left side gives us:
7 = n - 2.
Now, let's solve for n by adding 2 to both sides of the equation:
7 + 2 = n.
Therefore, the number of sides in the polygon is:
n = 9.
Hence, the polygon has 9 sides.