Re: Need help Solid Geometry

Quote:

Originally Posted by

**keosan22** We had this as a seatwork..couldn't answer it. My classmates answered it but I am doubtful about their answers I'll explain later

1. Given right circular cylinder circumscribed about the regular hexagonal pyramid with base a regular hexagon of side S= 6m, height of cylinder 10m

compute:

a. area of the base of the cylinder

...

basically it says we have a hexagonal pyramid (pyramid with a hexagon for it's base, my classmates just used S for radius r...and I'm very sure that's wrong...unless someone explains to me how did S = r,)

- a regular hexagon consists of 6 equilateral triangles and therefore the radius of the circumscribed circle of the hexagon has the same length as the side of the hexagon.
- The area of a regular hexagon is determined by: $\displaystyle a_{6-gon} = 6 \cdot \overbrace{ \frac12 \cdot s \cdot \underbrace{\frac12 \cdot s \cdot \sqrt{3}}}_{\text{height of triangle}}^{\text{area of one triangle}}$

Quote:

I know everything else to solve this but...I need to know:

a. how to solve for the area of a hexagon? (or for any other polygon for future reference since our instructor said we'd be facing more problems like this)

...

If you have a regular n-gon then this shape consists of n isosceles triangles with the radius of the circumscribed circle as legs and the side of the n-gon as base. The half of such an isosceles triangle is a right triangle. Assuming that n and r are known then

$\displaystyle s = 2 \cdot r \cdot \sin\left(\frac{2 \pi}{2n} \right)$

The area of one triangle is consequently: $\displaystyle a_\Delta = \frac12 \cdot 2 \cdot r \cdot \sin\left(\frac{2 \pi}{2n} \right) \cdot r \cdot \cos\left(\frac{2 \pi}{2n} \right) = r^2 \cdot \sin\left(\frac{\pi}{n} \right)\cdot \cos\left(\frac{\pi}{n} \right)$

Then the area of a regular n-gon is:

$\displaystyle a_{n-gon} = n \cdot a_\Delta$

Re: Need help Solid Geometry

in your first question,

for me, i think the best way to solve for the area of any polygon is to draw or make triangles on them,and make sure that you are very familiar with the sine law or cosine law so that you could easily solve for the sides of the triangle,the area of the triangle multiply iby the number of triangles you had drawn is equal to the area of the polygon.

in "how S=r?"

simply because we know that in one revolution there is 360 degrees,so knowing that hexagon has 6 sides, the angle in every triangle would be 360/6 = 60,since it is a regular hexagon, the two remaining angles in a single triangle would be (180-60)/2 = 60.

therefore the triangle is said to be an equilateral triangle which has equal sides,

therefore S=r.

I hope I helped somehow.