Solving Ancient Problems in Ancient and Modern Ways: Group Response

 Ross, Jayden, Danielle 

Question: Problem 1.2.3 Volume of a pyramid frustum

Derive the formula for calculating the volume of a pyramid frustum by using the formula for calculating volume V of a pyramid, V = ⅓ B x h (B = base, h= height), and by dividing the frustum (base = a^2, top surface area = b^2, height = h) into a cuboid, four lateral rest prisms and four corner pyramids!

Ancient Solution:

Adding the volumes of each subdivision, we can derive the formula of the pyramid frustum as required.

We made a model of the pyramid frustum broken up into the shapes:

Modern Solutions:

We used modern methods to solve the problem in two ways:

Using the pyramid volume formula V = LWH/3, we can subtract the top of the pyramid from the larger pyramid (top of the pyramid + frustum).

To do this, we need the height of the top of the pyramid denoted by ‘x’ in the figure on the right. By using similar triangles we see that x/b = (x+h)/a. Rearranging this we get x = bh/(a-b).

Then the volume of the frustum is [a^2 * (h+ bh/(a-b))/3] - [b^2 * (bh/(a-b))/3]. Following algebraic simplification we come to the same result as the ancient method.

Solution 2: By integral

We can also solve this by integral, as the frustum has perfectly square cross sections that get cut-out parallel to the base. Therefore we would be summing those areas from 0 to h (the height of the pyramid. By centering the pyramid on the y-axis, we can find the dimension of each cross section by the distance of the y-axis to the slant curve of the pyramid. We will take that distance and multiply by 2 to get the entire side length, then square it to get the area of the cross section. Using the end-points of the frustum as (a/2,0) and (b/2,h), we can write this as x = [(b-a)/2h]y + a/2. Then A = (2x)^2 = ([(b-a)/h]y + a)^2. Taking the definite integral of this expression from 0 to h will yield the same formula as the ancient method.

Group Activity:

We connected this problem to a point in the grade 8 curriculum about learning 3d nets. For our group activity we decided to lean into this and do a trivia game about nets of 3d shapes. 

Homework:

We can (and have) extended this problem and solution method to finding other noteworthy formulae.

Show the total surface area of the square pyramid frustum is (2h(b+a)) * sqrt(1+[(b-a)/2h]) + a^2 + b^2.

A rectangular pyramid frustum has base L by W and upper face of k*L by k*W, where k is some scale factor. Show that the formula for the volume of this shape is LWH/3 * [k^2 + k + 1].

Show the volume of a triangular base frustum is [sqrt(3)/12]*h*[a^2 + ab + b^2]