Each interior angle of a regular polygon is 162°. Find the sum of all the interior angles
Let's assume that the regular polygon has n sides.
The formula to find the sum of the interior angles of a polygon is given by:
Sum = (n-2) * 180°
Given that each interior angle of the regular polygon is 162°, we can set up the following equation:
162° = (n-2) * 180°
Simplifying the equation:
162° = 180°n - 360°
Rearranging the equation:
180°n = 162° + 360°
180°n = 522°
Dividing both sides by 180°:
n = 522° / 180°
n ≈ 2.9
Since the number of sides of a polygon must be a whole number, we round the value of n to the nearest whole number, which is 3.
Therefore, the regular polygon has 3 sides, which is an equilateral triangle.
Now, we can find the sum of the interior angles by using the formula:
Sum = (n-2) * 180°
Sum = (3-2) * 180°
Sum = 180°
So, the sum of all the interior angles of the regular polygon is 180°.
To find the sum of all the interior angles of a regular polygon, you can use the formula:
Sum of interior angles = (n - 2) * 180°
where n is the number of sides of the polygon.
In this case, we know that each interior angle of the regular polygon is 162°. Since all angles in a regular polygon are equal, we can set up an equation to find the number of sides of the polygon:
(n - 2) * 180° = n * 162°
Let's solve this equation step by step:
1. Distribute 180° on the left side:
180° * n - 360° = n * 162°
2. Move all the terms with n to one side of the equation:
180° * n - n * 162° = 360°
3. Factor out n on the left side:
n * (180° - 162°) = 360°
4. Simplify:
n * 18° = 360°
5. Divide both sides by 18° to solve for n:
n = 360° / 18°
n = 20
So, the regular polygon has 20 sides.
Now, plug the value of n into the sum of interior angles formula to find the sum:
Sum of interior angles = (n - 2) * 180°
Sum of interior angles = (20 - 2) * 180°
Sum of interior angles = 18 * 180°
Sum of interior angles = 3240°
Therefore, the sum of the interior angles of the regular polygon is 3240°.
To find the sum of all the interior angles of a regular polygon, we can use the formula:
Sum of interior angles = (n - 2) * 180°
In this formula, "n" represents the number of sides of the polygon. Since each interior angle is given as 162°, we can find the number of sides (n) by dividing 360° (the sum of the interior angles of a triangle) by 162°:
n = 360° / 162°
n ≈ 2.222
Since a polygon cannot have a non-integer number of sides, we can round the number of sides down to the nearest whole number, giving us:
n = 2
However, a polygon with only 2 sides is not possible, so we need to round up the value of n to the next whole number that is possible. In this case, that is 3 sides, which forms a triangle.
Now we can use the formula to find the sum of the interior angles:
Sum of interior angles = (n - 2) * 180°
Sum of interior angles = (3 - 2) * 180°
Sum of interior angles = 1 * 180°
Sum of interior angles = 180°
Therefore, the sum of all the interior angles of the regular polygon is 180°.