Volume and Area of Proportional Shapes
Finding Volume, Area and Length
of Similar Shapes
Note: This is a difficult topic because students really focus on what to do without trying to understand why. In response, understanding why steps are performed is a great focal point.
Big Idea
Similar shapes have corresponding parts that are proportional. Given the dimension of a pair of corresponding parts, like heights of cylinders, or areas, or volumes, set up a proportion that can be solved to find the requested information.
Key Knowledge
Prerequisite knowledge includes the ability to setup and solve simple proportions, a working understanding of the difference between area and volume, the ability to manipulate exponents and an understanding of similar two-dimensional shapes.
Pro – Tip
(for students)
When setting up the proportion to be solved, write the shape that contains your unknown parameter in the numerator.
Q: What does similar mean in mathematics?
In mathematics similar is a very important word. Similar means that corresponding parts are proportional. Consider two similar cylinders. The ratio of their radii will be proportional to the ratio of their heights.
A proportion is a set of equal ratios. A ratio compares two things. So radius 1/radius 2 is a ratio. That ratio being equal to the ratio of the heights is a proportion. Since proportions are equations they can be solved with inverse operations.
Q: Why is this true 1,000,000 cm3 = 1m3?
The conversions for dimension change depending on the exponent of the dimension (basically). Distance is an exponent of one. So, 12 meters is 1,200 centimeters. But area is squared. So a square with 1 meter sides has an area of 1m2. It is true that it is 12 m2, but we never write 12 because that’s just one. However, 1002 is not just 100, it is 10,000. Let’s look at a diagram.
Let’s put that all together now.
If you have two similar shapes, you’ll be given a set of information that allows you to write a ratio of known parts. For example, you’ll know a pair of lengths, or the area of both, or the volume of both. You’ll then be given one piece of information about one shape and asked to find the corresponding information from the other.
These problems:
- Provide corresponding information that is known
- Provide information on one part of one shape, ask you for the corresponding information on the other.
Here’s an example.
Two cones are mathematically similar. The height of cone A is 12 cm, the height of cone B is 7 cm. The volume of cone A is 125 cm3. What is the volume of cone B?
Do you see that the corresponding information was the heights of each cone? The information for one is the volume and you’re asked to find the volume of the second.
How:
Q: What if we were given a ratio of areas and asked for volume?
Example: Cylinder A is mathematically similar to cylinder B. Cylinder A has a surface area of 42 cm2 and cylinder B has a surface area of 101 cm2. Cylinder B has a volume of 812 cm3. What is the volume of cylinder A?
We are given the area of each cylinder and the volume of cylinder B. We are asked for the volume of cylinder A. Since cylinder A is smaller, we expect a smaller number. We will write our scale factor with A in the numerator.
If you fail to write your units, you’ll likely forget that the scale factor is volume (squared) and the thing being sought is volume (cubed).
We cannot use the scale factor for area to find volume directly.
We need to change scale factor’s dimension from cm2 into cm3. Here’s how. Take the square root of the scale factor to get the cm2 to be cm. Then, cube the scale factor. You could do that all at once by taking the scale factor and raising it to the 3/2 power.
From that point forward, if you’re efficient with your calculator you should have no problems.
Let’s look at one pair of last situations. What if you’re given the volumes of corresponding parts and asked for the area? First thing to do is set up a scale factor of volumes. Take the cube root, then square that and you have a scale factor in dimension two.
What if you were given area and needed to find a length? After you set up the scale factor of areas you’d need to take its square root to be in dimension one. That’s really it.
Summary: Similar means corresponding parts are proportional. To solve problems involving area and volume of similar shapes, set up a scale factor. Then, write a proportion. Make sure the units between the scale factor and unknown value are identical. Use exponents and square or cube roots (exponent of one – half and one – third) to manipulate the dimensions to be identical. Then, solve the proportion.
Similar Shapes
Volume, Area, Length
Lesson Plan
Note: This is a difficult topic because student really focus on what today without trying to understand why. In response, understanding why steps are performed is a great focal point.
Big Idea
Similar shapes have corresponding parts that are proportional. Given the dimension of a pair of corresponding parts, like heights of cylinders, or areas, or volumes, set up a proportion that can be solved to find the requested information.
Key Knowledge
Prerequisite knowledge includes the ability to setup and solve simple proportions, a working understanding of the difference between area and volume, the ability to manipulate exponents and an understanding of similar two-dimensional shapes.
Pro – Tip
(for students)
When setting up the proportion to be solved, write the shape that contains your unknown parameter in the numerator.
To download the PowerPoint, click the icon.
|
Time |
Notes |
Slide # |
|
2 – 3 |
Introduce the lesson for the day and explain what is expected from students. |
1 – 2 |
|
5 |
Explore and explain why there are 1,000,000 cm3 in one cubic meter. Be sure to have students doing the cognitive work here. Perhaps have them start by sketching a rectangle that can be measured in different units and then calculating the area in each unit |
3 |
|
5 |
Students should know what similar and corresponding mean. They should also know what ratios and proportions are. But if not, we address that here. This also gives students a global of view of what is happening. The idea here is that by helping students to see the big picture (what they’re given and how they’ll use it), that they’ll not get too focused on procedure and will understand how their steps fit with their purpose. |
4 – 5 |
|
10 |
Walk students through the first example here. Focus on getting the ratio written with the unknown in the numerator, then how inverse operations are used to solve it. |
6 |
|
10 |
This example can be done through guided practice as well. It is here they must tie-in what was learned in slide #3. In order to solve for a parameter, volume, or area, the units between the scale factor and the ratio with the unknown value must be identical. |
7 |
|
10 – 15 |
Have students work through this problem. You can offer help on converting the units, but they should have the knowledge to come up with the method on their own. Encourage this through collaboration and academic risk taking…if they try and make a mistake, they’ll benefit from the feedback great than just being told what is right. |
8 |
|
10 |
The last example here should be tried by students. Just like with the previous example, they may need a little help adjusting the scale factor and ratio to have matching units. |
9 |
|
5 – 10 |
Closure: Summarize and compress the learning through guided discussion. |
10 – 11 |
|
|
Homework |
12 |
Scale Factor, Area and Volume
- Two spheres are similar. The radius of sphere A is 6 cm, the radius of sphere B is 12 cm. How much larger is volume of sphere B compared to sphere A. Write your answer as an integer.
- There are two cubiods. The larger cuboid has a surface area of 121 cm2. The smaller cuboid has a surface area of 100 cm2.
- How much larger is the volume of the larger cuboid?
- If the length of the smaller cuboid is 4.5 cm, what is the length of the larger cuboid?
- How much larger is the volume of the larger cuboid?
- A purple cone has a volume of 729 cm3. A blue cone, which is mathematically similar, has a volume of 343 cm3.
- What is the ratio of their radii?
- If the surface area of the blue cone is 212 cm2, what is the surface area of the purple cone?
- What is the ratio of their radii?
With a $1.00 purchase of the packet you help support this website. What you get in this packet is:
- Topic Reference Sheet
- Lesson Guide
- PowerPoint
- Assignment and Key
