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High School Trigonometry: Homework Help Resource30 chapters | 203 lessons

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Lesson Transcript

Instructor:
*Karin Gonzalez*

Karin has taught middle and high school Health and has a master's degree in social work.

In this lesson, you will learn the definition of a triangular pyramid. You will also learn the formulas to find the base area, surface area, and volume of a triangular pyramid. Then work through an example calculation of each.

The first thing that comes to mind when thinking of pyramids may be the Great Pyramids of Egypt. If you look at a picture of these historical pyramids, you can see that their base is actually a square. Therefore, they are considered square pyramids, not triangular pyramids.

So what is a **pyramid**? It is a polyhedron or three-dimensional shape with at least three sides and polygonal base. A **triangular pyramid** is a pyramid with a triangle as a base and three triangular faces. It has four vertices or points and six edges.

There are also rectangular, pentagonal, square, and hexagonal pyramids. Can you guess what these are? Yes! They are pyramids with rectangular, pentagonal, square, and hexagonal bases, respectively.

There are three main components of a triangular pyramid. First is the base, which is a triangle, of course. Next are the faces, which are three triangles. Last is the **apex**, which is the point at the top where all of the faces meet. Easy enough! There are some important measurements as well: height, base length, apothem length, and the slant height. Whoa! What are apothem length and slant height, you may say? Here is a diagram to illustrate these parts of a triangular pyramid:

The slant height, base length, and apothem length are indicated in blue.

Height is the perpendicular line going from the point of the triangle to the midpoint of the base.

There are three main formulas for a triangular pyramid. First is the formula to find the base area of the triangular pyramid:

The **surface area** is the area of the outer layer or outer surfaces of a structure. So, the surface area of a triangular pyramid is the area of the surfaces of the three triangle faces and the triangular base. Because each surface is a triangle, to find area is basically multiplying the base times height and dividing it by two.

The formula is this:

In order to find the volume for a triangular pyramid, the area of the base and the height of the triangular pyramid must be know. Once these measurements are calculated, find the volume of a triangular pyramid with this formula:

Find the base area, surface area, and volume of a triangular pyramid with the following measurements:

Apothem Length = 8 cm

Base Length = 14 cm

Height = 20 cm

Slant Height = 22 cm

To find the base area of the triangular pyramid, plug the given values into the formula:

To find the surface area of the triangular pyramid, plug the given values into the formula:

To find the volume of the triangular pyramid, plug in the area that was already calculated into the formula:

**Pyramids** are polyhedrons with a polygonal base and triangular sides that meet at the apex. A** triangular pyramid** is a pyramid with a triangle base and three triangular faces, four vertices, and six edges. Base area, surface area, and the volume of a triangular pyramid can be calculated using the dimensions of the pyramid and the given formulas:

Base Area = (Apothem Length * Base Length) / 2

Surface Area = (Apothem Length * Base Length) / 2 + (3 * Base Length * Slant Height) / 2

Volume = (Base Area * Height) / 3

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Lesson
10 in chapter 22 of the course:

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High School Trigonometry: Homework Help Resource30 chapters | 203 lessons

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