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Area and Volume of Similar Figures

Updated: Jun 22, 2021

Dear Secondary Math students, Math Lobby will be covering the topic on Area and Volume of Similar Figures today. This is usually tested together with Similar Triangles in the O level Elementary Math and N Level NA Math examinations. Without further ado, let’s begin!

In this note, you will learn:

1) What are similar figures?

2) How to use ratio in the formula for calculating area of similar figures?

3) How to use ratio in the formula for calculating volume of similar figures?

1) What are similar figures?

When two shapes are said to be similar, it means that their corresponding angles are congruent and all the corresponding sides are proportional in ratio. (This topic is related to the topic on similarity in “Similar Triangles”, if you are unsure on the topic of similar triangles, click on the link learn more!)

However, when two solids are said to be similar, it means that they are of the same type with proportional corresponding radii, heights, lengths, widths, etc. Examples of similar solids are shown below:

For similar figures, there is a law applicable for both its surface area and volume, and it’s called the square-cube law. This law will be explained in the next few point down below.

2) How to use ratio in the formula for calculating area of similar figures?

When two solids are similar with a scale factor of a/b, then their surface areas are said to be in the ratio of the initial scale factor squared, or (a/b)². Let’s take a look at an example below for a clearer visual understanding:

Let’s say we are given a square that is 2 cm by 2 cm, and another larger square that is 4 cm by 4 cm.

The initial scale factor of the length of the green and blue squares is 2 / 4 = 1 / 2, meaning that for every 1 cm of the green square, it will be 2 cm on the blue square.

However, for the surface areas of the squares, things are a little different. If we apply the ratio for the formula to calculate the area of the similar figures, it would be (1 / 2)² = 1 / 4. But why?

As you can see from the diagram above, the area of the green square is 2 cm x 2 cm = 4 cm², whereas the blue square would have an area of 4 cm x 4 cm = 16 cm². Therefore, the ratio would be 4 / 16, which can be simplified to give 1 / 4, which proves the validity of the ratio application!

3) How to use ratio in the formula for calculating volume of similar figures?

Similar to the ratio application for surface area, volumes of similar solids have a relationship with the scale factor of similar solids, which is given by (a/b)³. This basically means that for the volumes of similar solids, the scale ratio will be cubed of the initial scale ratio.

So, taking the example from above, if both the two-dimensional green and blue squares become three-dimensional cubes, the ratio of the volume for the green cube to the blue cube will be (1 / 2)³= 1 / 8!

Another thing to note for the cube law is that this is also applicable for weight and masses as well because density of the figures does not change!

Now it is time to put on your thinking caps and start solving some questions!

1. Given two similar cones, Cone A has a radius of 5 cm and Cone B has a radius of 15 cm. The volume of cone A is 200cm³. Find the volume of cone B.

2. Given two similar prisms, Prism A has a cross-sectional area of 5cm² and Prism B has a volume of 150cm³. The scale factor of prism A to prism B is 1 / 5. Find the cross-sectional area of prism B and volume of prism A.

3. Given two similar cuboids, Cuboid A has volume of 10cm³ and Cuboid B has a volume of 640cm³. Given that the surface area of cuboid A is 250cm², find the surface area of cuboid B.

And that’s all we have for today, students! Math Lobby hopes that after reading and practicing questions with this article, you are able to have a clear understanding on the concepts of the square-cube law involving the area and volume of similar figures.

For any questions, feel free to contact us on Facebook or Instagram! And as always: Work hard, stay motivated and we wish all students a successful and enjoyable journey with Math Lobby!

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