How to find the volume of a frustum?

Question

The frustum has: radii of the top circle base is 9. the radii of the bottom circle is 12. The slant height is 6. 

I know you draw the frustum into a cone by extending it. Then you do the volume of the cone which is 1/3 bh. I can't seem to figure out the height though. So I can't figure how to get the volume. I was planning to subtract the volume of the small cone (the part that was extended) from the overall cone...and I was going to do Pythagorean theorem to find the height. But it doesn't make sense to me. 

if I am remembering correctly there is a theorem that says something about the slant height of the frustum to be half of the slant height of the whole cone. I did Pythagorean theorem  but the numbers just totally canceled. This is the source of my problem. Illustrating: (12^2+h^2=12^2) <---------that was what I tried to do to find the height of the cone. Anyways please help me find the volume of the frustum.

Thank you so much for any help provided

Answer 1

For this question you need the formula for the Volume for a conical frustum which is

V= pi(H)/3(R²+Rr+r²)

where H is the height

R is the bottom radius

r is the top radius

 

In this question you were given all except the standing height. Since you were given the slanting height you can derive it from that 

So you have to set it up using a right angle triangle and remember the application of similar triangles in the link below

https://www.dropbox.com/s/q67lmcs53ilo503/20130512_153545.jpg

Then you can solve for the remaining length of upper slant

https://www.dropbox.com/s/h74ztb4pl52h2zv/20130512_153552.jpg

After doing that, you can solve for the Height of the full cone using Pythagoras theorem.

Since you know the ratio of the two triangles, you can use that to determine the  

height of the frustum, the lower height.

After doing that you just Plug the missing height into the equation with the other quantities given

If you have any question, do not hesitate to ask

Please leave a vote if that helped :) 

Answer 2

You're on the right track

If you extend the height and slant height, we have a "missing" cone.  Let's call the missing cone's slant height x.

So now we can think of a proportion: smaller missing cone to bigger overall cone (actual + missing)

small radius/big radius = small slant height/big slant height

9/12 = x/(x+6)

Then we cross multiply: 9(x+6) = 12x

9x+54 = 12x

54 = 3x

x = 18

So the slant height of the "missing" cone is 18 and the slant height of the overall cone (actual + missing) is 18+6 = 24

Now we can use pythagorean theorem to find the height of the "missing" cone:

h² + 9² = 18²

h² + 81 = 324

h² = 324 - 81 = 243

h = √243 = 9√3

We can repeat the process with the larger cone (actual + missing).  We'll call the height of the larger cone big H.

H² + 12² = 24²

H² + 144 = 576

H² = 432

H = 12√3

So now we can find the volume of each and subtract:

Volume of "missing" cone = 1/3 π r²h = 1/3 π (9)²(9√3) = 1/3 π (81)(9√3) = 243 π √3

Volume of (actual + missing) cone = 1/3 π R²H = 1/3 π (12)²(12√3) = 576 π √3

Finally, we just subtract: (actual+missing) - (missing) = 576 π √3 - 243 π √3 = 333 π √3

Volume of frustrum = 333 π √3

 

Answer 3

The formula for the volume of a conic or pyramidal frustrum with bases a, b, and height h is

V = (a + b + √(ab))h/3.

In your case, a = π*9² = 81π and b = π*12² = 144π.

For the height we can derive it from the slant height. Imagine cutting the frustrum with a plane through the axis of rotation. You get an isoceles trapezoid. Dropping heights from the top vertices of the trapezoid cuts off two congruent right triangles. One leg's length is 12 - 9 = 3, and the hypotenuse 6 (same as slant height of the frustrum).

The other leg is the vertical height h of the frustrum, which is 3√3 using the pythagorean theorem.

Now we can use the above formula.

V = (81π + 144π + √(81π*144π))*3√3 / 3 = 333π√3

Answer 4

Extend the slant side to get double cones. By similarity, the upper cone has a height of 18. So,

the volume of frustrum

= the volume of the whole cone - the volume of the upper cone

= (1/3) pi (12^2 * 24 - 9^2 * 18)

= 666 pi

---------

Attn: To get the height of the upper cone, you can count mentally: for every 12-9 = 3 units change horizontally, the vertical change is 6. Therefore, for every 9 units change horizonally, the vertical change is 3*6 = 18 units.