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High School Trigonometry: Help and Review31 chapters | 240 lessons | 1 flashcard set

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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.

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.

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* = 1/3*A**H* 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.

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* + 3*a* 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* = 1/2*bh*, where *b* is the base of the triangle, and *h* is the triangle's height.

Therefore, our surface area formula becomes *SA* = *A* + 3(1/2*bh*) = *A* + 3/2*bh*, 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.

Consider a Rubik's triangle. A Rubik's triangle is a regular triangular pyramid. There are 3 inches from the pyramid's base to the apex. The area of the Rubik's triangle's base is 6.5 square inches. The height of one of the faces is 3.5 inches, and the base of one of the faces is 4 inches. Given this information, we can find the volume and the surface area of the Rubik's triangle.

Volume = 1/3*AH* = (1/3)(6.5)(3) = 6.5 cubic units

Surface Area = *A* + (3/2)*bh* = 6.5 + (3/2)(4)(3.5) = 27.5 square units

Let's consider one more example. Find the volume and surface area of the following regular triangular pyramid.

First, we will find the volume, so we need the area of the pyramid's base and the height of the pyramid. We see that *A* = 8.7 square centimeters and the *H* = 9 cm. Therefore, we plug these values into our volume formula to get *V* = (1/3)(8.7)(9) = 26.1, so our volume is 26.1 cubic centimeters.

Now, we find our surface area. This is a regular triangular pyramid, so we need the base and height of one of our faces and the area of the pyramid's base. From the picture, we see that *A* = 8.7 square centimeters, *b* = 10 cm, and *h* = 7cm. We plug these values into our formula to get *SA* = 8.7 + (3/2)(10)(7) = 113.7. Therefore, the surface area of our pyramid is 113.7 square centimeters.

**Triangular pyramids** are pyramids with a triangular base. The sides of a triangular pyramid are called **faces**, and the point where the faces all meet is called the **apex**. We have formulas to calculate the volume and surface area of a triangular pyramid.

To find the volume, we use the formula *V* = (1/3)*A**H*, where *A* = area of the pyramid's base and *H* = height of the pyramid. To find the surface area of a regular triangular pyramid, we use the formula *SA* = *A* + (3/2)*b**h*, where *A* = the area of the pyramid's base, *b* = the base of one of the faces, and *h* = height of one of the faces.

**Pyramid** - a three dimensional shape that has a polygon as its bottom and triangles as its sides, all meeting at a common point

**Faces** - the triangular sides of a pyramid

**Base** - the bottom polygon of a pyramid

**Apex** - the common point where all the triangular faces meet on a pyramid

**Triangular pyramid** - a pyramid with a triangle as a base

**Regular triangular pyramid** - a pyramid which has a base with sides that are equal in length

**Non-regular triangular pyramid** - a pyramid which has a base with sides that have different lengths

**Volume** - how much space there is inside an object

**Surface area** - the total area of an object's surface

After this lesson, you should be able to:

- Match the following terms to the appropriate parts of a pyramid: face, base, apex
- Differentiate between a regular and a non-regular triangular pyramid
- Explain how to find the volume and surface area of any triangular pyramid

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High School Trigonometry: Help and Review31 chapters | 240 lessons | 1 flashcard set

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