geometry help!!?

Question

"The length of the apothem in the regular pentagon shown below is 6 units. What is the area of the region bounded by the pentagon and the circumscribed circle?"

The way it was asked exactly by my geometry teacher.

I also have the drawing of a regular pentagon inscribed in a circle.

any help would be greatly appreciated!

wow, some of these answers are very good, but the thing is, i'm not in trigomnometry yet, and i don't know what you guys are talking about! i'm taking a geometry class, and our teacher wants everything to be explained the "long way"(geometrically), so that we can gain "knowlege"

Answer 1

The apothem is NOT the radius of the circle.

*I hope you have learned trigonometry by now.

First of all, the pentagon can be split into 5 triangles each with an angle of 72. Drop an altitude from one of these triangles from the 72 degree angle. You now have a 36-54-90 triangle, with the apothem adjacent to the 36 degree angle.

Using trig, half of one side of the pentagon is 6 tan 36, which is equal to 4.359. Thus, one side is 8.7185.

The perimeter of the pentagon is 5 * 8.7185, which is equal to 43.59.

The area of the pentagon is (apothem)(perimeter)/2.
Thus, the area of the pentagon is 130.777655.

The radius of the circle is the hypotenuse of the 36-54-90 triangle with the apothem adjacent to the 36 degree angle. The hypotenuse is 6 / cos 36, which is equal to 7.416.

The area of the circle is pi * r^2, which is equal to 172.797.

Subtract the area of the pentagon from the area of the circle...

172.797 - 130.777655 = 42.01969754

The answer is approximately 42.02.

Answer 2

If you draw a line from the center of the circle to each vertex of the pentagon you for 5 isosceles triangles. So the area of the pentagon is 5 times the area of one of these triangles.

The base of the triangle is a side of the pentagon and the two equal sides are = to r, the radius of the circle. Each base angle of the triangle is = to 54 degrees and the vertex angle is 72 degrees.The altitude of the triangle is the apothem and equals 6.

So 6/sin54 = r/sin 90 --> r = 6 / .809= 7.42

Thus 1/2 the triangle's base is sqrt (r^2-36)
So area of triangle is 6sqrt(r^2-36)
So area of pentagon is 30sqrt(r^2 -36)
Area of circle = pi r^2So the required area is
pi r^2 - 30sqrt(r^2-36)
=172.96 - 30(4.36)= 42.16 units ^2

Answer 3

the apothem is equal to 6 and that is the radius of the circle. Area is equal to pier^2, so the area is about 3.14*36 or rougly 113.04 units^2

since the circle is inscribed in the pentagon you only need find it's area

Answer 4

basically, the only thing that i can find in common with sacred geometry and alchemy is that they both deal with types of spirituality. Sacred geometry deals with the use of geometric patterns and shapes in the design and architecture of churches, temples, mosques or any place of spiritual gathering. These specific shapes mean certain things, depending on the religion or belief system. For example, the pentagram in Christianity refers to the five senses. In Catholicism, it can sometimes refer to the Devil, or the Evil One. Alchemy deals with the process of turning metals into gold, using chemicals and science. However, Alchemy is also a belief in finding the ultimate wisdom through chemistry. Therefore, alchemy is a spirituality all its own. All in all, the only thing that I can see in common between the two is that they both deal with a type, or types, of spirituality.

Answer 5

I believe you find the area of the pentagon and subtract it from the area of the circle.