The polygon below is a regular pentagon. Calculate the size of the angle

X

Y

Z

x = 108

y = 36

z = 72

=============================================================

Explanation:

Check out the diagram below. I have added letters of x, y and z in places to help find the values of y and z. Note the triangle on top is isosceles (since a regular polygon has all sides equal; therefore the triangle on top has the top diagonal sides equal).

Before we find either y or z, let's find x.

For any regular polygon, the interior angles are all the same measure. They sum to 180(n-2). In this case, n = 5, so the angles sum to 180(5-2) = 540. Each individual interior angle is 540/n = 540/5 = 108 degrees

x = 108

Another way to find this interior angle is to first find the exterior angle. For any convex polygon (regular or not), the exterior angles always add to 360. When we talk about regular polygons, each individual exterior angle is 360/n. So in this case, we have 360/5 = 72 as one exterior angle. The adjacent interior angle is therefore x = 180-72 = 108. So there are two ways to find the measure of an interior angle.

--------

Referring to the diagram, specifically the isosceles triangle on top, we can see that it has angles of x, y and y. They add to x+y+y = x+2y. Set this equal to 180, plug in x = 108 and solve for y

x+2y = 180

108+2y = 180

2y = 180-108

2y = 72

y = 72/2

y = 36

--------

The bottom most triangle is a congruent copy of the triangle on top. We have another isosceles triangle with the same side lengths as before. This triangle also has x, y and y as mentioned above.

Notice the adjacent angles of y and z in the bottom left corner. They must add to 108 as this was the measure of the interior angle of a regular pentagon. So,

y+z = x

36+z = 108

z = 108-36

z = 72

step-by-step explanation:

34.5 cm

step-by-step explanation:

generally for a circular segment of angle, θ° and radius r

arc length

= (θ/360) x circumference

= (θ/360) x 2πr

in our case, θ = 60°, π=3.142 and r = 33 cm

substitute into above formula

arc length

= (60/360) x 2(3.142)( 33) = 34.5 cm