# Surface Area of a Pyramid and Cone

This article presents an intuitive approach to find the surface area of pyramids and cones. Check out the SWT lesson Introduction to Surface Area if you would like a refresher on the definition.

## 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.

### A Smidge of Calculus

There is a subtle connection to the concept of limit, from calculus, within this lesson. 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.

## Introduction

### Finding the Surface Area of a Pyramid: 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 surface area works. Which equations for the areas of 2-dimensional shapes might be useful? Can you create your own general algorithm? 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!

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

Now that you have tried to find the relationship, take a look at this app. Please leave the hint buttons (bottom left corner) alone for now. You can play with the full-screen button, action button, and dials as much as you like. You can also tip the figure by dragging it and zoom in and out (using the pinching gesture on mobile devices and the mouse scroll wheel otherwise).

What do you see? What does the dial do? What does the action button do?

The real trouble in these pyramid situations is that often you don't get the measurements you need, and must find them using tricks of geometry. In the app the base is always a regular n-gon. How might you go about finding the height of the triangular sides, if you had the height of the overall pyramid, and the distance from the center of the base to the midpoint of one of the edges of the regular n-gon?

There is another way to calculate the surface area of a pyramid. Can you think of it?

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

Great! Now you know how to find the surface area of pyramids. Next, now can you think of a way to extend your knowledge to find the surface area of a cone? Can you think of a way to leverage what you just learned about pyramids? Give it a try. When you have a conjecture (or decide you have made a good enough effort) have a look at the next section.

## Finding the Surface Area of a Cone: An Algebraic Approach

Now that you've had a chance to try to find the relationship, return to the app located in the section 2 previous to this one. What happens when the number on the dial gets large?

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

Have a look at the app in this section. Please play around with the the dials and action button, but not the hint button. What do you see?

## Finding the Surface Area of a Pyramid and Cone: A Hands on Approach (Nets)

The image of a net of a triangular pyramid, in this section, links to a PDF containing 7 nets. The PDF is accessible if you click the image (then the details button, if you are using the mobile version of the website) and then click the big version of the image again on the next page. The idea is to cut along the dotted lines and fold along the solid ones, except for one solid line in the cone net. This special line indicates how much overlap to use to attach the curved side of the cone to itself. The rest are faces of the pyramids and tabs to glue the shapes together (tape can be used too). You should see lots of similarities to the apps. There are many ways to layout these nets. They just need to fold up into the shape you desire. You will find nets for a triangular pyramid, square pyramid, pentagonal pyramid, hexagonal pyramid, and cone. All the pyramids have regular n-gons as bases. Pyramids can of course be made with less symmetric bases, but no such example is provided here. Hands on learners will benefit greatly from building these shapes. If you had trouble understanding what was going on in the apps, this is you chance to go through the hints again and compare to the nets. Sometimes holding and folding the shapes yourself can yield better geometric intuition.