# Math Models and Geometry

## Solve Geometry Applications: Volume and Surface Area

In this section, we will finish our study of geometry applications. We find the volume and surface area of some three-dimensional figures. Since we will be solving applications, we will once again show our Problem-Solving Strategy for Geometry Applications.
2. Read the problem and make sure you understand all the words and ideas. Draw the figure and label it with the given information.
3. Identify what you are looking for.
4. Name what you are looking for. Choose a variable to represent that quantity.
5. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
6. Solve the equation using good algebra techniques.
7. Check the answer in the problem and make sure it makes sense.
8. Answer the question with a complete sentence.

### Find Volume and Surface Area of Rectangular Solids

A cheerleading coach is having the squad paint wooden crates with the school colors to stand on at the games. (See ). The amount of paint needed to cover the outside of each box is the surface area, a square measure of the total area of all the sides. The amount of space inside the crate is the volume, a cubic measure.
Each crate is in the shape of a rectangular solid. Its dimensions are the length, width, and height. The rectangular solid shown in  has length  units, width  units, and height  units. Can you tell how many cubic units there are altogether? Let’s look layer by layer.
Altogether there are  cubic units. Notice that  is the 
The volume,  of any rectangular solid is the product of the length, width, and height.
We could also write the formula for volume of a rectangular solid in terms of the area of the base. The area of the base,  is equal to 
We can substitute  for  in the volume formula to get another form of the volume formula.
We now have another version of the volume formula for rectangular solids. Let’s see how this works with the  rectangular solid we started with. See .
To find the surface area of a rectangular solid, think about finding the area of each of its faces. How many faces does the rectangular solid above have? You can see three of them.
Notice for each of the three faces you see, there is an identical opposite face that does not show.
The surface area  of the rectangular solid shown in  is  square units.
In general, to find the surface area of a rectangular solid, remember that each face is a rectangle, so its area is the product of its two dimensions, either length and width, length and height, or width and height (see ). Find the area of each face that you see and then multiply each area by two to account for the face on the opposite side.

### Volume and Surface Area of a Cube

A cube is a rectangular solid whose length, width, and height are equal. See Volume and Surface Area of a Cube, below. Substituting, s for the length, width and height into the formulas for volume and surface area of a rectangular solid, we get:
So for a cube, the formulas for volume and surface area are  and 

### Find the Volume and Surface Area of Spheres

A sphere is the shape of a basketball, like a three-dimensional circle. Just like a circle, the size of a sphere is determined by its radius, which is the distance from the center of the sphere to any point on its surface. The formulas for the volume and surface area of a sphere are given below.
Showing where these formulas come from, like we did for a rectangular solid, is beyond the scope of this course. We will approximate  with 


### Find the Volume and Surface Area of a Cylinder

If you have ever seen a can of soda, you know what a cylinder looks like. A cylinder is a solid figure with two parallel circles of the same size at the top and bottom. The top and bottom of a cylinder are called the bases. The height  of a cylinder is the distance between the two bases. For all the cylinders we will work with here, the sides and the height,  , will be perpendicular to the bases.
Rectangular solids and cylinders are somewhat similar because they both have two bases and a height. The formula for the volume of a rectangular solid,  , can also be used to find the volume of a cylinder.
For the rectangular solid, the area of the base,  , is the area of the rectangular base, length × width. For a cylinder, the area of the base,  is the area of its circular base,   compares how the formula  is used for rectangular solids and cylinders.
To understand the formula for the surface area of a cylinder, think of a can of vegetables. It has three surfaces: the top, the bottom, and the piece that forms the sides of the can. If you carefully cut the label off the side of the can and unroll it, you will see that it is a rectangle. See .
The distance around the edge of the can is the circumference of the cylinder’s base it is also the length  of the rectangular label. The height of the cylinder is the width  of the rectangular label. So the area of the label can be represented as
To find the total surface area of the cylinder, we add the areas of the two circles to the area of the rectangle.
The surface area of a cylinder with radius  and height  is

### Find the Volume of Cones

The first image that many of us have when we hear the word ‘cone’ is an ice cream cone. There are many other applications of cones (but most are not as tasty as ice cream cones). In this section, we will see how to find the volume of a cone.
In geometry, a cone is a solid figure with one circular base and a vertex. The height of a cone is the distance between its base and the vertex.The cones that we will look at in this section will always have the height perpendicular to the base. See .
Earlier in this section, we saw that the volume of a cylinder is  We can think of a cone as part of a cylinder.  shows a cone placed inside a cylinder with the same height and same base. If we compare the volume of the cone and the cylinder, we can see that the volume of the cone is less than that of the cylinder.
In fact, the volume of a cone is exactly one-third of the volume of a cylinder with the same base and height. The volume of a cone is
Since the base of a cone is a circle, we can substitute the formula of area of a circle,  , for   to get the formula for volume of a cone.
In this book, we will only find the volume of a cone, and not its surface area.

### Summary of Geometry Formulas

The following charts summarize all of the formulas covered in this chapter.

### Key Concepts

1. Volume and Surface Area of a Rectangular Solid
     

2. Volume and Surface Area of a Cube
     
3. Volume and Surface Area of a Sphere
     
4. Volume and Surface Area of a Cylinder
     
5. Volume of a Cone
     

### Practice Makes Perfect

Find Volume and Surface Area of Rectangular Solids 
In the following exercises, find ⓐ the volume and ⓑ the surface area of the rectangular solid with the given dimensions.
In the following exercises, solve.
In the following exercises, find ⓐ the volume and ⓑ the surface area of the cube with the given side length.
In the following exercises, solve.
Find the Volume and Surface Area of Spheres 
In the following exercises, find ⓐ the volume and ⓑ the surface area of the sphere with the given radius. Round answers to the nearest hundredth.
In the following exercises, solve. Round answers to the nearest hundredth.
Find the Volume and Surface Area of a Cylinder 
In the following exercises, find ⓐ the volume and ⓑ the surface area of the cylinder with the given radius and height. Round answers to the nearest hundredth.
In the following exercises, solve. Round answers to the nearest hundredth.
Find the Volume of Cones 
In the following exercises, find the volume of the cone with the given dimensions. Round answers to the nearest hundredth.
In the following exercises, solve. Round answers to the nearest hundredth.

### Everyday Math


### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.


ⓑ After reviewing this checklist, what will you do to become confident for all objectives?
