Truncated Cone Formulas
Everything you ever wanted to know, and things you didn't even know you wanted to know about truncated cones all in one place. When it comes to truncated cones, aka conical frustums, the most commonly asked questions are
- How do you calculate the volume of a truncated cone?
- How do you calculate the surface area of a truncated cone?
- What are the dimensions of the two cones from which a conical frustum is derived?
- How do you develop a truncated cone from an annular sector? That is, to build a conical frustum of a certain size, what must the dimensions of the annular sector be?
Here is a handy geometric formula guide and reference to help you answer these questions and solve challenging problems involving truncated cones.
Volume of a Truncated Cone
For a truncated cone whose top and bottom radii are respectively A and B, and whose height is H, the the volume formula in terms of A, B, and H is
Volume = πH(A^2 + AB + B^2)/3
The derivation of this formula is discussed in more detail here. An example will help you see the formula in action. Suppose you have a plastic cup in the shape of conical frustum, and it has a bottom radius of 3cm, a top radius of 4.5 cm, and a height of 10 cm. The amount of liquid it can hold is the volume, which we can compute using the formula above.
π*10*(4.5^2 + 3*4.5 + 3^2)/3
= 447.677 cubic cm
= 0.447677 liters.
Surface Area of a Truncated Cone
Just as you can compute the volume by plugging A, B, and H into a formula, so can you compute the surface area of a truncated cone with a three-variable formula. There are two definitions of surface area depending on your purposes. One is the lateral surface area, that excludes the areas of the circular ends. The other is the total surface area, which includes both the areas of the top and bottom. Here are their formulas.
Lat. Surf. Area = π(A+B)*sqrt[ (B - A)^2 + H^2 ]
Tot. Surf. Area = π(A+B)*sqrt[ (B - A)^2 + H^2 ] + π(A^2 + B^2)
The derivation of these equations is discussed in more depth here. To continue with the example in the previous section, we can use the first formula to calculate the lateral surface area of a truncated cone whose height is 10 cm and whose radii are 3 cm and 4.5 cm.
π*7.5*sqrt[ 1.5^2 + 10^2 ]
= 238.255 square cm.
Dimensions of the Larger and Smaller Cones
A truncated cone can be developed in a couple of different ways. One method is to take a full cone and cut the top off parallel to the base. The tip you cut off is itself a smaller cone proportional to the larger cone you started with. This diagram explains the process visually.
If a given conical frustum has a height of H and radii of A and B at its circular ends, then what are the dimensions of the larger cone and smaller cone in terms of A, B, and H? You can answer that question with this set of formulas. For these equations, B is the radius of the larger circular face of the conical frustum and A is the radius of the smaller circular face.
Base Radius = B
Height = BH/(B - A)
Base Radius = A
Height = AH/(B - A)
Let's use the same example as before, a conical frustum with a height of 10 cm and radii of 4.5 cm and 3 cm. If we want to obtain this frustum by cutting a smaller cone off the top of a larger cone, the larger cone will have dimensions
Radius = 4.5 cm
Height = 4.5*10/(4.5 - 3) = 30 cm
and the smaller cone cut from the top will have dimensions
Radius = 3 cm
Height = 3*10/(4.5 - 3) = 20 cm
Dimensions of Annular Sector That Produces a Truncated Cone of Given Dimensions
A truncated cone can also be developed by taking a sector of an annulus and curling it so that the straight edges line up. This is similar to developing a cone from a circular sector. If you want to develop a truncated cone with measurements A, B, and H, you need to know three things about the annular sector: outer radius, inner radius, and angle.
Call the outer radius of the annular sector R, the inner radius r, and the angle θ. If you have the values of A, B, and H from the conical frustum (with B greater than A), then you can compute R, r, and θ with the following formulas.
θ in degrees = 360(B - A)/sqrt[ (B-A)^2 + H^2 ]
R = B*sqrt[(B-A)^2 + H^2]/(B - A)
r = A*sqrt[(B-A)^2 + H^2]/(B - A)
For example, let's use these formulas to determine the size of the annular sector needed to develop a conical frustum whose height is 10 cm and whose radii are 4.5 cm and 3 cm. We have H = 10, B = 4.5, and A = 3. This gives us
θ = 360(1.5)/sqrt[ 1.5^2 + 10^2 ] = 53.4 degrees
R = 4.5*sqrt[ 1.5^2 + 10^2 ] / 1.5 = 30.34 cm
r = 3*sqrt[ 1.5^2 + 10^2 ] / 1.5 = 20.22 cm