In this short video, we explore the relationship between cones, cylinders, and frusta (the plural form of frustum). The three shapes are connected by a general equation, and we explore the impact of changing the top radius for each of these shapes. TINKERCAD is the 3D modeling platform used for this activity.

The general equation for the volume of a frustum (volume V) with any cross-sectional area is defined as follows:

where, *height h*, and *areas A and B* are the independent variables.

In Wolfram Alpha, type this:`V = (h / 3) * (A + B + SQRT(A * B)`

(view)

Let us use a more specific cross-sectional area: the area of circle. Now, we have a conical frustum:

where, *height h, base radius R,* and *top radius r* are input variables. Also, the top radius must be a value between 0 and R; that is, *0 < r < R*.

In Wolfram Alpha, type this:`V = (pi / 3 ) * h * (r^2 + r*R + R^2)`

(view)

When r = 0, we have the formula for a cone:

In Wolfram Alpha, type this:`V = (pi / 3 ) * h * (0 + 0 + R^2)`

(view)

When r = R, we have the formula for a cylinder:

In Wolfram Alpha, type this:`V = (pi / 3 ) * h * (R*R + R*R + R*R)`

(view)

In this activity, we demonstrated that not every math problem has a numerical answer. There is no answer “at the back of the book” in the real world because design decisions require some careful thought and planning.

### Math Work is Work.

**One very important concept at MathForWork is the practice of working through ideas. **

TINKERCAD provides a great opportunity to work through different ideas related to geometry and trigonometry.

Mindless rote learning for standardized testing is not work. Mindful exploratory learning and computational thinking is meaningful work that leads to meaningful learning.

TINKERCAD has a simple interface that creates a wonderful environment for 3D modeling. This job skill requires a good understanding for how to choose and modify basic shapes for tasks in the workplace. Also, the 3D models can be exported for 3D printing in maker spaces, which allow the model to be inspected more closely for future reference.

TINKERCAD is a great entry-level software applications for learning 3D CAD modeling and design. It may feel like a toy, but TINKERCAD is * very useful*. The 3D CAD software used in basic design does not need to be too fancy, because the exported 3D models are standardized formats.

**CCSS.MATH.CONTENT.8.G.C.9: **Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

**MathForWork** delivers distributed learning systems, instruction support, and test preparation for all learners. Learn more at MATHFORWORK.COM. MathForWork is a Bitwise Thermodynamics project.