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KB20448558: Creating a frustum in TINKERCAD

Summary

This short video demonstrates how to transform a cone into a frustum in TINKERCAD. This article also discusses how the formula for the volume of a cone relates to the formula for the volume of a conical frustum.

Video

More information

We have seen how transform a cone into a cylinder in TINKERCAD. Let us now look at how a cone shape relates to a conical frustum.

We begin with the formula for calculating the volume of a conical frustum:

\[V_{frustum} = {\pi h \over 3}{(R^2 + Rr + r^2)}\]

The height h and the base radius R will stay constant for this solution.

The only property of the shape that we change is the top radius r.

This formula can be represented as a function with one variable (the top radius):

\[f(r) = {\pi h \over 3}{(R^2 + Rr + r^2)}\]

With this function, we can set the top radius to be any value between zero and the value of the base radius. Any value of the top radius r greater than zero and less than the base radius transforms the cone into a conical frustum:

\[f(r) = {\pi h \over 3}{(R^2 + Rr + r^2)}\] \[0 < r < R\]

Let us look at how the volume of a conical frustum relates to a geometric cone. Again, the height and the base radius will be treated as constants for this exercise.

We will set the top radius of the function equal to zero:

\[f(0) = {\pi h \over 3}{(R^2 + 0 \cdot R + 0^2)}\]

The zero coefficients eliminate the last two terms within the parantheses:

\[f(0) = {\pi h \over 3}{(R^2 + \not{0} + \not{0})}\]

We are left with:

\[f(0) = {\pi h \over 3}{(R^2)}\]

Re-arranged for clarity, we have:

\[f(0) = {\pi R^2 h \over 3}\]

In summary, when the top radius is equal to zero (r = 0), we have the formula for the volume of a cone.

\[V = {\pi R^2 h \over 3}\]

QUESTION

What happens when we set the top radius equal to base radius (r = R)? Show your work.




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