# Surface Area of a Sphere

This article presents an intuitive approach to understanding how to derive the equation for the surface area of a sphere. It is inspired by the concept of limit from Calculus (particularly multivariate integration).

## For Teachers and Parents

### What is this Hint System

The hint system is designed to allow students to engage with the problems with little to no support (Discovery/Inquiry-Based Math). The hints gradually give more and more information until we end up with a full explanation (Traditional Lecture). Since Math is not a ‘spectator sport,’ it is hoped that repeated use of this hint system in multiple lessons will encourage students to use fewer and fewer hints and learn to solve problems creatively for themselves. This is, of course, only one possible arrangement of these hints. An instructor will be better able to adapt the presentation of the hints to the progress a specific student has made. This is beneficial to the student until they learn to use the system and self-regulate. If the student needs hints, they can choose to set aside their particular approach temporarily while filling in the gaps in the order the text presents them. Later they can analyze how the approach they took compares to that of the text.

### Isn't this Basically Integral Calculus?

Yes! I am using this lesson as a Trojan Horse. Exposing students to these ideas early so they have horizon knowledge about what is coming down the track later is important. It gives their subconscious extra years to digest the ideas before they become everyday companions.

### Why no Rigor?

Why no Rigor? Sorry, the treatment is not rigorous. To my Pure Math friends, sometimes rigor can get in the way of clarity when first encountering a concept. I promise a rigorous treatment of calculus, straight ‘outta’ Real Analysis, will come in future years.

## Introduction

### Finding the Surface Area of a Sphere: Your First Encounter

To start, I'd like to invite you to try to come up with a solution on your own. Take a few minutes and think about how you have found equations for the volumes and surface areas of other 3-dimensional shapes (cylinders, cones, pyramids, etc.). Also, consider the equations and ways you found those equations for relevant 2-dimensional shapes (like the circle). Can you think of any way to reuse or extend one of those techniques? Can you create your own method? Please don't work too hard. If you find yourself getting frustrated, take a break or move on to the next section. There are some nice pointers to get you started. Please stop often and reassess if you have new ideas about how to solve the problem. The fewer hints you use, the stronger your creative and problem-solving skills will become. Remember there are lots of ways to solve these problems. If you invent one for yourself, which is not covered here, that is awesome!

### Before Using the Digital Manipulative

Before using the digital manipulative (in the next section), take a moment to review the following lessons (if you haven't already).

It is of particular importance that you are aware of the approach taken in the manipulatives and the final equations found. You will use these ideas to find the equation for the surface area of a sphere (in the next section).

## Finding the Surface Area of a Sphere: A Visual Approach

Now that you have tried to find the relationship, take a look at this app. Leave the action button (beside the full-screen button) and hint button alone for now. You can use the full-screen button if you like. Please play with the dial. You can also tip the figure by dragging it and zooming in and out (using the pinching gesture on mobile devices and the mouse scroll wheel otherwise). As a note, setting the dial to 128 or above, will slow the frame rate while panning and zooming (as well as the animation when you are asked to use the action button), quite a bit on mobile devices. Likewise for a dial setting of 512 and above on any device without a discrete graphics card.

What do you see? What does the dial do?

Please note, that when the hints ask you to use the action button, you will see an animation. If you are particularly eagle-eyed, you might notice an imperfection in the animation. If you don't notice it, then don't worry, it doesn't change anything. The animation is perfect at the beginning and at the end, but in between there is some minor overlapping of shapes. The shapes are meant to be solid, and thus they should not be capable of passing through each other. The potential changes to the code I considered (to remedy this imperfection) either would slow the animation significantly or make it a great deal less pretty, so I decided to leave it as is. Hopefully, this warning suffices to prevent any confusion among those who spot the issue.

## A Brief Discussion of Other Approaches (Extra Credit)

There are many other ways to derive the equation for the surface area of a sphere. The most obvious is to learn Calculus and then use it directly (rather than waving hands). Can you think of any variations on the approach discussed? When you are done thinking, two will be discussed.

One option would be to use Cavalieri's Principle to dodge that nasty sum. You could then smash the pyramids together into one big pyramid with a height equal to the radius and a base with an area equal to the surface area of the sphere and do the algebra. There are many options, but the base might look like a sinusoidal map projection, if certain choices were made. It would be a lot of geometric manipulation to keep track of and the animation would grind any machine without a discrete GPU to a halt, even with only 8 pyramids.

The other option is the same idea, but you get there by peeling spherical shells off of the sphere and mushing each layer into a shape similar to the sinusoidal map projection (or any other choice that respects the rules of Cavalieri's Principle), then stacking them to make a pyramid. The same reasons for not doing it apply (complexity and compute power).