Properties of Congruent & Similar Solids

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  • 0:01 Solids
  • 0:40 Congruent Solids
  • 1:13 Similar Solids
  • 2:50 Volumes and Areas of…
  • 3:58 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Watch this video lesson and you will learn the one characteristic that all congruent solids have and you will learn how the volumes of similar solids are related to their similarity ratio.


Mathematical solids are everywhere! Bricks are mathematical solids. Stuffed animals are mathematical solids. In math, we define a solid as a three-dimensional object. This covers pretty much any object that we see in the world around us. The world we live in is three-dimensional, so everything in it is a mathematical solid. Our own bodies are classified as solids!

In this lesson, we take a look at these solids, and we will see what makes solids congruent and what makes solids similar.

Congruent Solids

When we say that two solids are congruent, we are saying that they are equal to each other. This means that they are exactly the same in every detail. Let's visualize this: Imagine that you are playing with building blocks. When you have two congruent building blocks, these building blocks will be exactly the same. They have the same height, the same width, and the same length. If you put them next to each other, you won't be able to tell them apart. They don't have any differences between them. So, being congruent is the same as being exactly the same in every detail.

Similar Solids

When two solids are similar, it means that they are the same shape but different sizes. Think of it as one being a model of the other. So, one will be either the larger version or the smaller version of the other. They are the same in every other detail except their size. Because they are the same in every other detail, every single measurement of one solid is the same amount larger or smaller than every corresponding measurement of the other solid.

For example, take a look at these two pyramids. One is larger, and the other is smaller. It looks like the smaller one is a toy-size version of the larger. Isn't it cute? Both have square bases. The height of the larger one is 8 inches, and the height of the smaller one is 4 inches. Because the height of the larger pyramid is two times as large as the height of the smaller pyramid (4 * 2 = 8), then all the other measurements of the larger pyramid will be two times as large as those of the smaller pyramid.

So, if the bottom edge of the smaller pyramid measures 3 inches, then the bottom edge of the larger pyramid will measure 3 * 2 = 6 inches (twice as large as the smaller). Mathematically, we can write this as a ratio a:b, where a and b are your objects. If we compare the height of the larger pyramid to the smaller pyramid, we write 8:4. Simplifying this, we get 2:1. This tells us that when something measures 1 in the smaller pyramid, it will measure 2 in the larger pyramid.

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