Lesson 5: The Volume of 3-D Objects
Explore 3
Read
Take a look at the following example, which compares the relationship between the volume of a right rectangular pyramid and the volume of a right rectangular prism.
Foundations and Pre-calculus Mathematics 10 (Pearson)
Read “Example 2: Determining the Volume of a Right Rectangular Pyramid” on page 39 to see how the formula for the volume of a right pyramid is used to solve a problem.
Then read “Example 1: Determining the Volume of a Right Square Pyramid Given Its Slant Height” on page 38 to see how the volume of a pyramid is determined if the slant height (as opposed to the height) of the pyramid is given. Pay attention to how the Pythagorean theorem is used in the solution.
Tip
Whenever you come across a formula with a fraction, there are two ways that you can evaluate it. For example, if you want to enter the formula 
into your calculator, you can enter either of the following:
-------------------------------------------------------------------
Self-Check
SC 3. Find the volume of the square pyramid shown in the diagram.
Spheres
Retrieve your analysis from Math Lab: Comparing the Volume of a Cylinder and a Sphere that you saved to your course folder.
The height of the can is equal to twice the radius of the ball.
To determine the formula for the volume of a sphere, you can do what you did for the cone. First, what was the ratio of the volume of the tennis ball compared to the volume of the juice container? Did you observe that the volume of the tennis ball was about 
that of the container?
This means that the volume of the sphere would be 
.
This formula is correct, but there’s a way to simplify the formula by finding another way to express the can’s height.
Think about the fact that the can and the ball have the same radius and the same height. Does it make sense to you that the height of the can would be equal to twice the radius of the ball?
So you could write the volume of a sphere as 
or, more simply, 
.
Read
Read your textbook for “Example 3: Determining the Volume of a Sphere” on page 49 to see how the formula for the volume of a sphere is used to solve a problem involving the volume of the sun.
Then read part b) of “Example 4: Determining the Surface Area and Volume of a Hemisphere” on page 50 to see how to modify the formula to determine the volume of a hemisphere.
Try This
Complete TT 2 in your course folder ( binder) to practise applying the formula for the volume of the sphere. You may have to do something more than apply the formula for the context-based questions.
Foundations and Pre-calculus Mathematics 10 (Pearson)
TT 2. Complete “Exercises” questions 13.c), 13.d), and 19 on pages 51 and 52.
Use the link below to check your answers to Try This 2.
The following table summarizes the different types of 3-D objects you have examined in this lesson. Included are the formulas that have been developed for these objects.
|
Cylinder |
Cone |
Rectangular |
Sphere |
|
|
|
|
|
Self-Check
SC 4. Sheila is excavating the basement for her house on a small lot in town. The dimensions of the basement excavation need to be 40-ft long by 30-ft wide and 9-ft deep. The excavated soil is placed in a circular area beside the excavation. The radius of this area is 20 ft. As more soil is added, the soil pile forms the shape of a cone. The highest the excavator can lift the soil is 24 ft.
Backhoe © ownway/shutterstock
Will the excavator be able to put all the soil from this excavation into this one cone-shaped pile? (Show your calculations to the nearest whole number.)