Conical Frustum / Truncated Cone: Geometry Problems with Solutions
Problems involving conical frustums (truncated cones) can be particularly tricky since they are not quite cones, and not quite cylinders, but a shape in between with characteristics of both. Whereas cones and cylinders have only two measurements to consider -- radius and height -- conical frustums have three: base radius, top radius, and height. These example problems in solid geometry can help you figure out volumes, dimensions, and surface areas of truncated cones.
Example 1: Conical Frustum Flattened into Sector of Annulus
If you cut the slanted side of a conical frustum and flatten it out, the 2-D shape you obtain is an annular sector. Given the dimensions of the conical frustum, can you determine the dimensions of the annular sector when you flatten it out? Given an annulus sector, can you determine the dimensions of the resulting conical frustum when you curl it into a three-dimensional shape? The answer to both of these questions is yes.
Example 1a: First, let's take an annular sector and see if we can figure out what size the truncated cone will be when we attach the straight sides together.
In the figure above, the annular sector has a total angle of 270°. That angle measure comes from 360° minus the 90° section cut away. Suppose its outer radius is 13 and its inner radius is 5, which means it has a thickness of 8 (since 13 - 5 = 8). The outer arc length is
2π*13*(270/360) = 39π/2
The inner arc length is
2π*5*(270/360) = 15π/2
When we curl it into a truncated cone, the base of the conical frustum will have a circumference of 39π/2, and the top of the frustum will have a circumference of 15π/2. We can then solve for the bottom and top radii:
bottom radius = (39π/2)/(2π) = 39/4 = 9.75
top radius = (15π/2)/(2π) = 15/4 = 3.75
Now we need to figure out the height of the frustum. If we look at it from the side, the profile is a trapezoid.
On the side of the trapezoid we can cut a right triangle, as shown in the diagram on the right. The hypotenuse of this triangle is 8, the difference in radii of the annulus. The bottom leg of this triangle has a length of 6, the difference in in radii of the truncated cone. The other leg of the triangle is the height of the frustum. Using the Pythagorean Theorem, we can figure the height:
8^2 = 6^2 + h^2
64 = 36 + h^2
28 = h^2
2*sqrt(7) = h
Example 1b: Now let's take a conical frustum of given dimensions and see what the shape of the annular sector is when we flatten it out.
Suppose a conical frustum has a bottom radius of 19, a top radius of 11, and a height of 15. We want to use this information to find the outer and inner radii of the annular sector and the angle of the sector.
The side length of the frustum can be found by solving for the hypotenuse of a right triangle seen on the profile, as in the previous example. One leg of the triangle is 15 and the other is 8 (the difference between 19 and 11). Therefore, the side length of the frustum is sqrt(15^2 + 8^2) = 17. This means when we flatten out the truncated cone into a sector of an annulus, the difference between the outer and inner radii of the annulus will be 17. I.e., the thickness of the annulus is 17.
The outer and inner arc lengths of the annular sector are 2π*19 = 38π and 2π*11 = 22π, which are the circumferences of the top and bottom of the conical frustum. With this preliminary work, we can now find the radii of the annulus sector and the angle.
The angle is (8/17)*360° = 169.4117°. In other words, you take the total angle of a circle and multiply it by the fraction whose numerator is the difference in radii of the conical frustum and whose denominator is the side length of the frustum.
The outer radius of the annulus is found by solving the equation
(169.4117/360)*2πR = 38π
which gives you R = 40.375. The inner radius is found by computing r = 40.375 - 17 = 23.375.
Example 2: Comparing a Conical Frustum to a Cylinder
A cylinder has a height of H and radius of 20. A truncated cone also has a height of H, with a top radius of 19 and a bottom radius of 21. Which figure has the greater volume.
One might think that they have the same volume since their heights are equal, and the average of the radii of the frustum equals the radius of the cone. But in fact they have unequal volume. Let's see which one is greater.
The volume of the cylinder is πH*20^2 = 400πH. The volume of the truncated cone is
πH(19^2 + 19*21 + 21^2)/3
As you can see, the conical frustum has the larger volume. Is this true in general? When a cylinder's radius is equal to the average radius of a conical frustum, and both objects have the same height, does the conical frustum always have the greater volume? It turns out the answer is yes.
Suppose a conical frustum has a height of H and radii A and B. Suppose a cylinder has a height of H and a radius of (A+B)/2. Their volumes are
πH(A^2 + AB + B^2)/3 and πH[(A+B)/2]^2
Factoring out the πH from both expressions gives us
(A^2 + AB + B^2)/3 and [(A+B)/2]^2
so we just need to compare these two quantities by applying the same operations to both sides and seeing if we can discover whey the expression on the left is always larger.
When A ≠ B, the expression on the left (A - B)^2 is always a positive number, that is, greater than 0. Thus (A^2 + AB + B^2)/3 is always greater than [(A+B)/2]^2 when A and B are different, and therefore the conical frustum has greater volume.