a. The measure of four interior angles of a regular pentagon are 95,
120, 80 and 135. Find the measure of the fifth interior angle.
b. Find the area of an equilateral triangle with an apothem 3.5 meters
long and each side 12 meters long.
c. Find the area of a regular octagon with an apothem 13.3 feet long
and each side 11 feet long.
d. Find the area of a regular hexagon with an apothem 7.8 millimeters
long and each side 9 millimeters long.
e. Suppose a chord in a circle is 80 centimeters long and it is 30
centimeters from the center of the circle. Find the measure of a
radius of the circle

a.

The exterior acute angles of a pentagon (or any other convex polygon)
must add up to 360 degrees, so each must on average be 360/5 = 72
degrees.

But then each interior angle must on average be 180-72 = 108 degrees,
which means that the sum of the interior angles must be 5*108 = 540
degrees.

Knowing that four of the angles add up to 95+120+80+135 = 430, we
conclude that the fifth angle is 540-430 = 110.

The final answer is: 110 degrees.


b.

If the perpendicular distance from the center of a triangle to any of
its sides is 3.5 and the length of a side is 12, then the total area
of the triangle may be divided into three smaller triangles of height
3.5 and base 12.

The area of each of these smaller triangles is 3.5*12/2 = 21, so the
three together make a total area of 3*21 = 63.

The final answer is: 63 square meters.


c.

In the case of this octagon, we may divide its area into eight
triangles of height 13.3 and base 11. The area of each triangle is
13.3*11/2 = 73.15, and the eight together make a total area of 8*73.15
= 585.2.

The final answer is: 585.2 square feet.


d.

The hexagon is composed of six triangles of height 7.8 and base 9.
Each triangle has area 7.8*9/2 = 35.1, so the six together make an
area of 6*35.1 = 210.6.

The final answer is: 210.6 square millimeters.


e.

The chord and the two radii that intersect its endpoints form an
isosceles triangle of height 30 and base 80.

Let us bisect the isosceles triangle along its height into a pair of
right triangles. In each of these right triangles, the two shorter
sides are of length 30 and 80/2 = 40. Observe that the hypotenuse is a
radius of the circle.

The length of the hypotenuse is calculated from the Pythagorean theorem:

    c^2 = a^2 + b^2
    c^2 = 30*30 + 40*40
        = 900 + 1600
        = 2500
      c = sqrt(2500)
        = 50.

The final answer is: 50 centimeters.


If you find my answer inaccurate or incomplete, please let me know so
that I have a chance to meet your needs before you assign a rating.

Regards,

leapinglizard

Hi,

A nifty formula to know to what the interior angles must add up to is:
total degrees = 180(n-2)
Where n is the number of sides.
Which means as leapinglizard has shown, that the interior angles of a
pentagon add up to 540 degrees.

Thanks.
endo