### Unit 10: Surface Area and Volume

We frequently need to calculate the amount of space that exists inside an object. For example, we use the concept of volume to follow a recipe, but we also use it when we need to fill a moving box with books or packages, fill a moving van with boxes or furniture, fill a simple container with water, or fill a gas tank with gasoline.

Since containers come in so many shapes and sizes, in addition to simple square and rectangular boxes, learning how to calculate the amount of space that is available inside a three-dimensional object not only comes in handy, but can save you time, money, and create less waste.

Similarly, you may also need to calculate the amount of space that exists on the outside of an object, such as when you decide to paint your house or office. Making a few key calculations can ensure you buy the right amount of paint you need, the first time.

In this unit, we will investigate three dimensional shapes, and learn to calculate their surface areas and volumes.

**Completing this unit should take you approximately 4 hours.**

Upon successful completion of this unit, you will be able to:

- find surface area and volume for prisms, cylinders, pyramids, cones, and spheres; and
- find surface area and volume of composite solids.

### 10.1: Polyhedrons

We first need to learn about three-dimensional figures, also known as polyhedrons.

Read this article and watch the video which define different common types of polyhedrons. These also cover Euler's theorem, which defines the number of faces and vertices in a given polyhedron.

Read this article, which explains how three-dimensional solids are represented on paper and how we can map three-dimensional solids to two dimensions.

### 10.2: Prisms

Now, let's investigate the volume and surface area of different types of polyhedrons. The first type of polyhedron we study is the prism.

Read this article and watch the videos. Focus on the definitions for surface area and volume in the text. Carefully read the examples of finding surface area and volume.

Then, complete review questions 9–12 and check your answers.

### 10.3 Cylinders

The next shape we study is the cylinder.

Read this article and watch the videos. Make note of the formulas for the surface area of a right cylinder and the volume of a cylinder.

Then, complete review questions 3–6 and check your answers.

### 10.4: Pyramids

A pyramids is another type of three-dimensional polyhedron.

Read this article and watch the video. Note the formulas for surface area and volume of pyramids.

Then, complete review questions 9–12 and check your answers.

### 10.5: Cones

In this section we investigate cones.

Read this article and watch the videos. Pay attention to the formulas for surface area and volume.

Then, complete review questions 1–3 and check your answers.

### 10.6: Spheres

The last simple shape we are going to investigate are spheres.

Read this article and watch the video. Focus on the formulas for surface area and volume.

Then, complete review questions 4, 5, 10, 12, and 13 and check your answers.

### 10.7: Composite Solids

Finally, we will use what we learned about regular polyhedrons to determine the volume and surface area of non-standard polyhedrons, or composite solids.

Read this article and watch the video. Focus on the examples of how we can break composite solids down into standard shapes.