A **triangular pyramid** is a three-dimensional solid – a polyhedron – with a triangular base and three triangular faces meeting at the pyramid’s apex.

The base of a pyramid can be any two-dimensional geometric shape:

- Triangle
- Rectangle
- Square
- Hexagon
- Octagon

There are many types of pyramids, and all pyramids are named by the shape of their bases.

Just as you can have a triangular pyramid, you can also have a rectangular pyramid, a pentagonal pyramid, etc.

The Great Pyramids of Egypt in Giza, for example, is a square pyramid because its base (bottom) is a square. A triangular pyramid is a pyramid with a triangular base.

A triangular pyramid has:

- Triangular base
- 3 triangular faces
- 6 edges
- 4 vertices

A pyramid with an equilateral triangle base is a **regular triangular pyramid**. If a scalene or isosceles triangle forms the base, then the pyramid is a **non-regular triangular pyramid**.

No rule requires the base of a triangular pyramid to be an equilateral triangle, though constructing scalene or isosceles triangular pyramids is far harder than constructing an equilateral triangular pyramid.

*[insert accurate drawing based on this reference of a triangular pyramid net diagram]*

Two different surface area measurements can be taken for any 3D solid: the **lateral surface area** and the **surface area**.

Lateral surface area, $LSA$, does not include the base for our pyramid. The surface area of a pyramid, $SA$, includes the base.

The surface area of a triangular pyramid with three congruent, visible faces is the area of those three triangular faces, plus the area of the triangular base.

The formula for calculating the surface area involves the area of the base, the perimeter of the base, and the slant height of any side.

$SA=BaseArea+\frac{1}{2}(Perimeter\times SlantHeight)$

This formula works because you are adding the base area to the area of all three slanted faces. The perimeter gives you the sum of all three bases. You multiply that sum times the slant height of the triangular pyramid as though you had one big rectangle, and then you take one-half of that as the area of the three triangles.

Suppose you have this triangular pyramid:

The base of the pyramid is an equilateral triangle since all three of its sides are $10cubits$. To find the area of the base triangle, use this formula for the area of an equilateral triangle with sides $a$:

$A=\frac{\sqrt{3}}{4}{a}^{2}$

For this particular triangular pyramid, the formula works out as:

$A=\frac{\sqrt{3}}{4}{10}^{2}\approx \mathbf{43.3}\mathbf{square}\mathbf{cubits}\mathbf{\left(}{\mathbf{cubits}}^{\mathbf{2}}\mathbf{\right)}$

We have now found the area of the base. We already know the perimeter of the base is $30cubits$ (the three sides are each $10cubits$), and we are given the slant height, $14cubits$.

$SA=BaseArea+\frac{1}{2}\left(Perimeter\times SlantHeight\right)$

$SA=43.3cubit{s}^{2}+\frac{1}{2}\left(30cubits\times 14cubits\right)$

$SA=43.3cubit{s}^{2}+\frac{1}{2}\left(420cubit{s}^{2}\right)$

$SA=43.3cubit{s}^{2}+210cubit{s}^{2}$

$SA=253.3cubit{s}^{2}$

Area is always measured in square units, whether they are $c{m}^{2}$, ${m}^{2}$, $f{t}^{2}$, or $cubit{s}^{2}$.

You may have needed to take your time getting through all that, finding the area of the base, finding the perimeter, adding everything.

To find the area of * just* the slanted sides – the lateral surface area ($LSA$) – you need to do a lot less work:

$LSA=\frac{1}{2}(Perimeter\times SlantHeight)$

These formulas only work for regular pyramids. If you have a non-regular triangular pyramid, calculate the area of each of the four faces individually (three slanted faces and the base) and add them together.

Volume is the amount of space a 3D solid takes up, so, with a triangular pyramid, we are finding how much room it has inside it. It is always measured in cubic units. Though the pyramid rapidly diminishes to an apex, the calculation is not hard.

$V=\frac{1}{3}Ah$

In the volume of a triangular pyramid formula, $A$ is the area of the base and $h$ is the height from base to apex

For our pyramid with a base $10cubits$ and slant height of $14cubits$, the height, $h$, works out to $13.0767cubits$. We already know the area from our earlier calculations, so we can plug the know numbers in to get the volume in cubic cubits:

$V=\frac{1}{3}Ah$

$V=\frac{1}{3}(43.3cubit{s}^{2}\times 13.0767cubits)$

$V=\frac{1}{3}(566.2211cubit{s}^{3})$

$V\approx 188.75cubit{s}^{3}$

Please note that, with the fraction as a factor in our multiplication, we do not have a precise decimal answer, so we have an approximate value.

After working your way through this lesson and video, you should know:

- What is a triangular pyramid?
- How many faces, edges, and vertices does a triangular pyramid have?
- What is a regular triangular pyramid?
- How to calculate the surface area of a triangular pyramid
- How to find the volume of a triangular pyramid
- The formulas to find volume, lateral surface area, and surface area

Instructor: **Malcolm M.**

Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

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