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High School Geometry: Homework Help Resource13 chapters | 142 lessons

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Lesson Transcript

Instructor:
*Jennifer Beddoe*

A pyramid is a 3-dimensional geometric shape formed by connecting all the corners of a polygon to a central apex. This lesson will discuss the different types of pyramids and formulas surrounding them.

A **pyramid** is a 3-dimensional shape whose base is a polygon. Each corner of a polygon is attached to a singular apex, which gives the pyramid its distinctive shape. Each base edge and the apex form a triangle.

There are many types of pyramids. Most often, they are named after the type of base they have. Let's look at some common types of pyramids below.

**Triangular pyramid** has a triangle as its base:

**Square pyramid** has a square as its base:

**Pentagonal pyramid** has a pentagon as its base:

This list could go on and on (hexagonal pyramid, heptagonal pyramid, etc). There are also a few special names for pyramids that you should know.

**Right pyramid** - the apex of the pyramid is directly above the center of its base:

**Oblique pyramid** - the apex of the pyramid is not directly above the center of its base:

**Regular pyramid** - the base of this pyramid is a regular polygon:

**Irregular pyramid** - this type of pyramid has an irregular polygon as its base:

There are formulas that can be used to find both the surface area and the volume of a pyramid. The **surface area** of a pyramid is the total area of all the surfaces that the pyramid has. To that end, the formula for finding the surface area when all of the side faces are the same is:

SA = (base area) + (1/2) * (perimeter) * (slant height)

The **base area** is the area of the base and can be determined based on what figure the base is. For example, if the base of the pyramid is a square, the formula for finding the area is:

A = *s*^2

The **perimeter** is the distance around the base of the pyramid. The **slant height** is the diagonal height from the center of one of the base edges to the apex.

If the pyramid has side faces that differ from each other (like in the case of an irregular pyramid), then the surface area equation is:

SA = (base height) + (lateral area)

In this case, you must take each side of the pyramid separately (including the base), find the areas, and then just add them together.

The volume of a pyramid can be found using this formula:

V = (1/3) * (base area) * (height)

The base area is, again, just the area of the base of the pyramid. However, in this case, the height is the length of a line from the apex that makes a right angle with the base.

Let's take a look at some examples.

A square pyramid has a height of 9 meters. If the side of the base measures 4 meters, what is the volume of the pyramid?

Since the base is a square, area of the base = 4 * 4 = 16 m^2.

Volume of the pyramid = (1/3) * (base area) * (height)

V = (1/3) * (16) * (9) = 48 m^3

What is the surface area of the pyramid discussed in the first example?

In order to find the surface area, we must first find the slant height of the pyramid. Since we know the height and the base length, we can use the Pythagorean Theorem to find the slant length.

You can see that the blue and red lines create a right triangle. The length of the long leg of the triangle is 9 meters, or the height of the triangle. The length of the short leg of the triangle is 2 meters because it is half the length of the base of the triangle (4 m). The red line is the hypotenuse of the triangle and can be calculated using the formula:

*a*^2 + *b*^2 = *c*^2

2^2 + 9^2 = *c*^2

81 + 4 = *c*^2

85 = *c*^2

*c* = 9.2 m

Now that you know the slant height, you can solve for the surface area of this triangle.

SA = (base area) + (1/2) * (perimeter) * (slant height)

SA = 16 + (1/2) * (16) * (9.2) = 89.6 m^2

In geometry, a **pyramid** is a three-dimensional shape that can have any polygon as its base. The corners of the polygon all connect at the apex, or point, of the pyramid. There are formulas that can be used to determine the surface area and the volume of any pyramid. The formula for finding the surface area when all of the side faces are the same is:

SA = (base area) + (1/2) * (perimeter) * (slant height)

The **base area** is the area of the base and can be determined based on what figure the base is. For example, if the base of the pyramid is a square, the formula for finding the area is:

A = *s*^2

The **perimeter** is the distance around the base of the pyramid. The **slant height** is the diagonal height from the center of one of the base edges to the apex.

If the pyramid has side faces that differ from each other (like in the case of an irregular pyramid), then the surface area equation is:

SA = (base height) + (lateral area)

The volume of a pyramid can be found using this formula:

V = (1/3) * (base area) * (height)

- pyramid: a three-dimensional shape that can have any polygon as its base
- triangular pyramid: a pyramid with a triangle as its base
- square pyramid: a pyramid with a square as its base
- pentagonal pyramid: a pyramid with a pentagon as its base
- right pyramid: a pyramid in which the apex of the pyramid is directly above the center of its base
- oblique pyramid: a pyramid in which the apex of the pyramid is not directly above the center of its base
- regular pyramid: a pyramid in which the base of the pyramid is a regular polygon
- irregular pyramid: a pyramid in which the base of the pyramid is an irregular polygon
- surface area: the total area of all the surfaces
- base area: the area of the base
- perimeter: the distance around the base of the pyramid
- slant height: the diagonal height from the center of one of the base edges to the apex

Upon reviewing this lesson, you should be able to:

- Identify different types of pyramids
- Calculate the volume, area, perimeter, and slant height using pyramid formulas

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High School Geometry: Homework Help Resource13 chapters | 142 lessons

- Area of Triangles and Rectangles 5:43
- Perimeter of Triangles and Rectangles 8:54
- How to Identify Similar Triangles 7:23
- Angles and Triangles: Practice Problems 7:43
- Triangles: Definition and Properties 4:30
- Classifying Triangles by Angles and Sides 5:44
- Interior and Exterior Angles of Triangles: Definition & Examples 5:25
- Constructing the Median of a Triangle 4:47
- Median, Altitude, and Angle Bisectors of a Triangle 4:50
- Constructing Triangles: Types of Geometric Construction 5:59
- Properties of Concurrent Lines in a Triangle 6:17
- Perfect Numbers: Definition, Formula & Examples 6:15
- Pyramid in Math: Definition & Practice Problems 5:31
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