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.
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?
Digital Manipulative: Surface Area of a Pyramid, Visually
There is a pyramid displayed when the app starts. What does the dial do?
The dial starts at 3. When it changes the pyramid changes. What is the correlation? Use the first (left most) hint button, once you have a guess (or need help making a guess).
When the dial is set to 3 the pyramid is a triangular pyramid. When the dial is set to 4 the pyramid is a square pyramid. When the dial is set to 5 the pyramid is a pentagonal pyramid. When the dial is set to 6 the pyramid is a hexagonal pyramid. Do you see a pattern?
It appears that the dial setting dictates the number of sides on the base of the pyramid (3 = tri, 5 = pent, 6 = hex, etc.). What does the action button do?
The action button seems to (un/re)fold the pyramid, by moving the faces. How does this help?
The goal is to find the surface area of the pyramid. How does this unfolded, flattened shape help?
Areas are calculated for 2-dimensional shapes. The shape shown is now 2-dimensional and has the same area as the surface area of the original pyramid (the faces just got moved around). If you find the area of this shape you also find the surface area of the original pyramid. How might you do that?
The flattened 2-dimensional shape is a composite shape, made up of the faces of the pyramid. Does this help?
To find the area of the composite shape you can just add the areas of the simpler shapes it is made from (the faces of the pyramid). This is as good as it gets for general pyramids. Writing the procedure as an equation doesn't really improve the situation. There is an optimization for the special case where the base of the pyramid has a centered peak and a regular n-gon for a base (like in the app). Can you spot it?
In the special case described, the triangular sides have the same dimensions and thus the same area. You only need to calculate that area once, then you can multiply it by the number of sides to get the area of all of them together.
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?
Try sketching it on paper. Does this help?
Try the second (middle) hint button. Does that help?
Sometimes the Pythagorean Theorem
can be useful when finding the measurements you need.
There is another way to calculate the surface area of a pyramid. Can you think of it?
Let's agree on a convention. All the triangular faces will be uniquely labeled with the numbers
up to the total number of faces (call it
). For the octagonal pyramid, that would be
. This can be written more concisely as
. Then it can be generalized to work for any number of triangular faces (
) by writing
. You can then write the area for each face as
. The sum of these areas plus the area of the base is the surface area. This can be written
. Can you think of a way to simplify this equation?
How about substituting the equation for the area of a triangle in for the triangular sides? What would that look like?
It would look like this
. Where the height of each triangle (slant height of the pyramid) is given by
and the length of each triangle's base is given by
. Please note that the numbering system matches the height and base with the triangle labeled with the same number. Can this equation be simplified?
The triangles are all the same. What does that mean for their measurements?
All the heights are the same length. All of the bases are the same length. How does that affect the equation for the surface area of a pyramid?
The equation becomes
. Can you simplify this equation?
Sure! Collect like terms. What would that look like?
It might look like this
. Some collection of like terms did not happen. This is because you are looking for a new method of finding the surface area of a pyramid (collecting the base value would bring us to the result in hint 10). What might you do instead?
Try pressing the third (right most) hint button. What does this suggest?
The perimeter of the base of the pyramid becomes highlighted in red. The sum of the bases of the triangles is the perimeter of the base of the pyramid. How does this simplify the expression for the pyramid's surface area?
The equation further simplifies to
. This is our final version of the equation for the surface area of a regular n-gonal pyramid with the peak at the center.
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?
Look at the base of the pyramid. Do you see anything there?
When the number of sides of the regular n-gon gets large, the regular n-gon
begins to look like a circle. What does this mean for the pyramid?
The pyramid begins to look like cone. How does this help you?
Recall the equation derived for the pyramid in the final hint (hint 25), in the section on the visual approach to finding the surface area of a pyramid. How can that equation be modified to fit the cone?
What is the equation for the area of the circle? What is the equation for the circumference of a circle?
Substituting in the equation for the area of a circle and the equation for the circumference of a circle produces the equation
. Can this equation be simplified?
The twos cancel to give
. This is a pretty good equation, but it can be further simplified, if you like. Can you see how?
It's possible to common factor out
. What does that leave you with?
It leaves you with the final equation for the surface area cone,
. Some people like to use the height of the pyramid instead of the slant height, because the height is considered a more fundamental measurement. If you have heard of cylindrical coordinates, that is what is being referred to. How could the height be brought into this equation?
Have another look at the pyramid app. Make sure hint 2 (the middle one is active). Does this give you any ideas?
Using the Pythagorean Theorem
. The equation then can be written as
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?
Digital Manipulative: Surface Area of a Cone, Visually
It appears that the app is displaying a net that's been unfolded much like in the previous app. The regular n-gonal base has been removed, however. For a cone we know that this is going to end up being circle, so the focus is on finding the area of the rolled upside instead. What does the blue (top most) dial do?
Just as in the previous app, the blue dial controls the number of sides of the regular n-gon and thus the number of triangular faces of the pyramid. What does the white (bottom most) dial do?
The white dial appears to control the height of the triangles (or equivalently the slant height of the pyramid). This in turn would change the height of the pyramid. What does the action button do?
The action button changes the arrangement of the triangles from a star shape pattern, to a more compact pattern where the point of each triangle that otherwise would have been at the peak of the pyramid, now touch in a central location. Use the action button to return to the star shaped pattern, set the blue dial to 100. What do you see?
The regular n-gonal hole looks like a circle. This means that the overall pyramid now looks like a cone. What happens if you hit the action button again and vary the slant height with the white dial?
A sector of a circle appears. The fraction of the circle varies with changes in the slant height. What is the radius of the circle?
The radius is equivalent to the slant height. How will you find the fraction that describes the amount of the circle that is present?
The fraction is equivalent to the fraction of the circumference that the arc of this partial circle represents. How will you find this value?
What is the length of the arc?
Try using the hint button now. Make sure to use the action button to see both orientations. Does this help?
The length of the arc is equivalent to the circumference of the circular base of cone. Use the circumference equation to find what fraction of the larger circle this represents. Can you write this fraction?
The fraction would look something like
. Can you simplify this fraction?
The fraction simplifies to
, by canceling the
. How will you use this to find the area of the sector?
The area of the sector is equivalent to this fraction times the area of the whole circle
. Substitute in for the fraction and the equation for the area of a circle. Keep in mind the radius of this circle is the slant height. What do you get?
The equation becomes
. How can you simplify this equation?
There is some cancellation among the slant height values yielding
. You then add the area of the circular base of the cone to this result to get the whole surface area of the cone. That is
, as you've already seen. As in the algebraic approach to finding the surface area of a cone, it is possible to simplify the equation to
remains a popular way to rewrite the equation.
Finding the Surface Area of a Pyramid and Cone: A Hands on Approach (Nets)
Pyramid and Cone Nets (click to view)
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.