Answer:
11a)
A regular hexagon (6 sides) is given. Because it is regular, every interior angle has the same value.
We know that the sum of the interior angles of a triangle is 180°. Using this information, we can break this hexagon up into 4 triangles:
Given each triangle's interior angles sum to 180°, and we have 4 triangles, the total sum of the interior angles of the entire hexagon is
180*4 = 720
The sum of the interior angles of the hexagon is 720°.
11b)
The same idea can be applied:
This time a regular decagon (10 sides) is given. This shape can be broken up into 8 triangles (This value will always be the number of sides - 2).
We can now multiply to find the total sum of the interior angles.
180*8 = 1440
The sum of the interior angles of the decagon is 1440°.
(The formula to solve for interior angle sum of regular shapes is 180 * (number of sides - 2)
11c)
To find the measure of one interior angle of a regular octagon (8 sides), we must take the total sum of the interior angles and divide that by 8 (to find the value of 1 angle).
First, find the interior sum value using interior angle sum formula:
180 * (8-2) = 180 * 6 = 1080°
Now we can divide this by 8 to find the sum of one interior angle:
1080/8 = 135°
The value of one interior angle of a regular octagon is 135°.
(The formula to solve for one interior angle of a regular shape is
[180 * (number of sides - 2)] / number of sides
11d)
The sum of the exterior angles of any polygon is 360°.
An easy way to demonstrate this idea is with an equilateral triangle (every interior angle is 60°). If the interior angle is 60°, the exterior angle is 120° (supplemental theorem).
A triangle has 3 angles: 120 * 3 = 360°. The sum of exterior angles is 360°.
For a heptagon (7 sides), or any other polygon, the same result will be found.
(In order to algebraically solve this however, you would find the value of one interior angle using the formula above, subtract that value from 180 to find the value of one exterior angle, and then multiply the value of one exterior angle by 7 for a heptagon).
11e)
Given the sum of exterior angles is 360°, we can simply divide 360 by the number of sides to find the value of one exterior angle.
360 / 7 = 51.42857...
The measure of one exterior angle of the heptagon is about 51.4°.