# Triangular Pyramid: Definition, Formula & Examples Video

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• 1:04 Volume Formula for a…
• 1:43 Surface Area Formula…
• 3:32 Examples
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Lesson Transcript
Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

Learn what a triangular pyramid is and how to find its volume and surface area. We will use examples and definitions to familiarize ourselves with these types of pyramids.

## Definition Of A Triangular Pyramid

To see a pyramid in real life, just think about the Great Pyramids of Egypt. A pyramid is a three dimensional shape that has a polygon as its bottom and triangles as its sides, all meeting at a common point. The triangular sides are called faces, and the bottom polygon is called the base of the pyramid. The number of sides the base polygon of a pyramid has is equal to the number of triangular faces on the pyramid. The common point where all the triangular faces meet is called the apex.

A pyramid can have any polygon as a base. We are going to concentrate on when that base is a triangle. A pyramid with a triangle as a base is a triangular pyramid. Since the base is a triangle and a triangle has three sides, a triangular pyramid has three triangular faces. Triangular pyramids show up in architecture, art, design, and other areas.

There are two types of triangular pyramids - regular and non-regular. A regular triangular pyramid has a base with sides that are equal in length. A non-regular triangular pyramid has a base with sides that have different lengths.

## Volume Formula For a Triangular Pyramid

The volume of an object is how much space there is inside an object, so the volume of a triangular pyramid is how much space there is inside the pyramid. The volume of a triangular pyramid can be found using the formula V = 1/3AH where A = area of the triangle base, and H = height of the pyramid or the distance from the pyramid's base to the apex.

For example, if we had a triangular pyramid with height 12 units and the area of the base was 24 square units, then the volume of the pyramid would be V = (1/3)(24)(12) = 96 cubic units.

## Surface Area Formula For a Triangular Pyramid

The surface area of an object is the total area of the object's surface. Thus, the surface area of a triangular pyramid is the area of all of its faces and base combined. When we have a regular triangular pyramid, all of the faces of the pyramid have the same area. Therefore, the surface area of a regular triangular pyramid can be found by adding the area of the base to 3 times the area of one of the faces. That is, SA = A + 3a where A is the area of the pyramid's base, and a is the area of one of the pyramid's faces.

To find the area of a triangle, we use the formula A = 1/2bh, where b is the base of the triangle, and h is the triangle's height.

Therefore, our surface area formula becomes SA = A + 3(1/2bh) = A + 3/2bh, where b is the base of one of the pyramid's faces, which is also the length of one of the sides of the pyramid's base, and h is the height of one of the pyramid's faces.

For example, If we have a regular triangular pyramid with faces having height = 10 units and base = 6 units, and the area of the pyramid's base is 16 square units, then the surface area of the pyramid is SA = 16 + (3/2)(6)(10) = 106 square units. This tells us that if you add up all of the areas of the pyramid's faces and the pyramid's base, you get 106 square units.

When we have a non-regular pyramid, we just calculate the areas of each of the faces of the pyramid and the base of the pyramid individually and then add them all up. That is SA = Area of base + Area of face 1 + Area of face 2 + Area of face 3.

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