# A conical tent is made by using a semi-circular piece of canvas of radius 8 feet.

A conical tent is made by using a semi-circular piece of canvas of radius 8 feet. Find the height of the tent and the number of cubic feet of air inside.

By manipulating I have found a way to get to the solutions provided by the textbook (that is $$h = 4\sqrt{3}$$ and $$V = \frac{64}{3}\sqrt{3}\pi$$) but I do not understand them. Here is what I have done so far

Let A be the area of the semi-circular piece of canvas of radius 8 so that

$$A = \frac{1}{2}\pi r^2 = \frac{1}{2}\pi 8^2 = 32\pi$$

Using the formula for the lateral area of a cone we have

$$\pi rs = A = 32\pi\implies s = \frac{32\pi}{8\pi} = 4$$

This is where I am stuck because I have a slant height smaller than the radius. However if I keep pushing forward I get

$$s^2 = h^2 + r^2 \implies h^2 = s^2 - r^2 = 4^2 - 8^2 \implies h = \sqrt{|-48|} = 4\sqrt{3}$$

I am quite close to the solution, but I cannot find a way to set up the problem correctly.

• The slant height is $8$, the radius of the piece of cloth. (Every point on the circumference of the base is at distance $8$ from the apex of the cone.) You've just got $r$ and $s$ reversed. Commented Oct 5, 2019 at 15:32

Some hints:

• If you take any cone, cut in a straight line from the top to the base and lay it flat, the resulting shape will be the sector of a circle. What is the relationship between the slant height of the cone and the radius of the circular sector?
• In the formula for the lateral area of a cone, $$r$$ is the radius of the circular base of the cone, which is not the same as the radius of the circular sector.
• However, the arc length of the sector must be the same as the circumference of the base of the cone. How does the length of a circular arc relate to the angle subtended at the center of the circle?
• Thank you I get it know, what I missed was the relationship between the arc length and the circumference of the base. It took me a long time to figure out but it was worth it. Commented Oct 5, 2019 at 16:04

Observe that the tent is a cone with side length $$s=8$$ and a circular base of radius $$r= 4$$. Thus, its height is

$$h=\sqrt{s^2-r^2} = \sqrt{48} = 4\sqrt 3$$

The corresponding volume is

$$v = \frac 13 \pi r^2h = \frac {64\pi}{3}\sqrt 3$$