Triangular Pyramid

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

Triangular Pyramid Definition

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.

How Triangular Pyramids Get Their Name

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.

Triangular Pyramid Faces, Edges, and Vertices

A triangular pyramid has:

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

Triangular Pyramids Faces Edges And Vertices

Regular Triangular Pyramid

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.

Regular And Irregular 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]

Table Of Contents

  1. Triangular Pyramid
  2. Surface Area of a Triangular Pyramid
  3. Volume of a Triangular Pyramid

Surface Area of a Triangular Pyramid

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.

Triangular Pyramid Types Of Area

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.

Surface Area of a Triangular Pyramid Formula

SA = Base Area + 12 (Perimeter × Slant Height)

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.

How to Find the Surface Area of a Triangular Pyramid

Suppose you have this triangular pyramid:

Surface Area Of A Triangular Pyramid Formula

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

A = 34 a2 

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

A = 34 102  43.3 square cubits (cubits2)

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

SA = Base Area + 12 Perimeter × Slant Height

SA = 43.3 cubits2 + 12 30 cubits × 14 cubits

SA = 43.3 cubits2 + 12 420 cubits2

SA = 43.3 cubits2 + 210 cubits2

SA = 253.3 cubits2

Area is always measured in square units, whether they are cm2, m2, ft2, or cubits2.

How to Calculate Lateral Surface Area of a Triangular Pyramid

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 = 12 (Perimeter × Slant Height)

Lateral Surface Area Of A Triangular Pyramid Formula

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 of a Triangular Pyramid

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.

Triangular Pyramid Volume Formula

V =13Ah

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

Triangular Pyramid Volume Formula

For our pyramid with a base 10 cubits and slant height of 14 cubits, the height, h, works out to 13.0767 cubits. 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 = 13Ah

V = 13 (43.3 cubits2 × 13.0767 cubits)

V = 13 (566.2211 cubits3)

V  188.75 cubits3

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.

Next Lesson:

Surface Area of a Rectangular Prism

What you learned:

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.

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