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Surface Area of Pyramids - Concept

Teacher/Instructor Brian McCall
Brian McCall

Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

Surface area is a two-dimensional property of a three-dimensional figure. Pyramids have polygonal bases and triangular faces, which are congruent if the base is regular. To find the total surface area of a pyramid, first find the area of the base and then find the area of each triangular faces. The height of each triangle is called the slant height. Related topics include regular polygonal area and triangle area.

You can calculate the surface area of a pyramid by breaking it apart into pieces and adding up the sum of those areas. So if we start with our base, we have a equilateral pentagon so I'm going to draw that in and in order to calculate the area of that we need to know its side length and the apothem. So the area of this is going to be the apothem times the side length times the number of sides all divided by 2. And since this is a regular pentagon we can assume that the 5 triangles that are formed are all going to be congruent to each other. So once you calculate the area of the base you're going to have to add in the area of this triangle times however many triangles you have. So let's say you had a heptagon you're going to have 7 triangles that are all going to be congruent to each other as long as your heptagon is regular. So notice that the height here of this triangle is not called h.
In a 3 dimensional figure we're going to call that l which stands for your slant height. So I'm going to label this as l and notice that the base of this triangle is also s your side length. So the area of this triangle is going to be base times height divided by 2 or s times l divided by 2. So if you want to calculate the surface area which I will write over here of a regular polygon with a base that's a pyramid, you're going to need to add up the area of bases which is a times s times n divided by 2 and then you're going to need to add in however many triangles you have that are congruent. So that's going to be s times l divided by 2 times n, where n is the number of sides. So you can use this formula, I'm going to erase this down here and extend that. This will calculate the surface area of any pyramid with a regular polygon as a base.

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