Surface Area of Cylinders and Spheres

Many of my past and current Geometry students have had trouble memorizing the formulas for volume and surface area. Traditionally, teachers have asked students to memorize all of the formuals below.

Name Lateral Area Surface Area Volume
Prism L.A. = Ph S=L.A. + 2B
= Ph + 2B
V = Bh
Cylinder L.A. = Ph
= Ch
= 2πrh
S.A. = L.A. + 2B
= 2πr + 2πr²
V = Bh = πr²h
Regular/Square Pyramid L.A. = ½Pl S.A. = L.A. + B
= ½Pl + B
V = ⅓Bh
Cone L.A. = ½Pl
= ½Cl = ½(2πr)l
= πrl
S.A. = L.A. + B
= πrl + πr²
V = ⅓Bh
= ⅓πr²h
Sphere S.A. = 2πrh
= 2πr*2r
= 4πr²
V = 4/3πr³

This is a lot for anyone to remember. Our jobs as math teachers is to help students make connects between all of these formulas. The connections between them allow students to memorize very little and to generate these formulas based on their knowledge of how they are related. This also provides students with a much deeper understanding of the math.

Here are just a couple visual representations of some of these connections that I commonly share with my own students.

And in that vain, I created a Desmos graph that shows a visuallization of the connection between the latteral area of a cylinder and the surface area of a sphere. Click on the graph below to see the math behind it on

Triangle Inequality Theorem

Common Core State Standard: Know the triangle inequality theorem to determine the possible side lengths of a triangle.

Description: In this activity, students will discover and explore the triangle inequality theorem. It is similar to a popular activity where students are given different lengths of spaghetti and are asked to find relationships between the sides when trying to form triangles. It is a great way to still do this discovery lesson without needing the large amount of prep time needed to measure a bunch of side lengths.

Teacher Guide: Click Here

Desmos Activity: Click Here

Below is a sample of how students may try and create a triangle out of specific side lengths.

Volume of a Sphere

Common Core State Standard: Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

Description: In this activity, students will discover and explore the use of the formula for the volume of a sphere. Students will already need to know and how to use the formula for the volume of a cylinder.

Teacher Guide: Click Here

Desmos Activity: Click Here