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### Congruent and Similar Solids

- Scale factor: the ratio of corresponding measures of two solids
- Two solids are congruent if:
- The corresponding angles are congruent
- Corresponding edges are congruent
- Areas of corresponding faces are congruent
- The volumes are congruent
- Congruent solids have:
- The same size and same shape
- A scale factor of 1:1
- If two solids are similar with a scale factor of a:b, then
- The ratio of the surface areas is a
^{2}:b^{2} - The ratio of the volumes is a
^{3}:b^{3}

### Congruent and Similar Solids

Two congruent solids must have same number of edges.

Two congruent solids always have same surface area and volume

- 3
^{3}:7^{3}

All the spheres are congruent.

For two cones, if their bases are congruent, and they have the same height, then they are congruent.

All the cones are similar.

the two prisms are similar, the corresponding sides are 7 m and 3 m, find the surface of the larger prism.

- The scale factor is 7:3
- The ratio of surface area is 49:9
- [49/9] = [SA/21]
- SA = 21*[49/9] = 114.3

If the volume of two cubics are the same, then the two cubics are congruent.

If two cylinders have the same area of base and the same height, then they are congruent.

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

### Congruent and Similar Solids

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro
- Scale Factor
- Congruent Solids
- Similar Solids
- Extra Example 1: Determine if Each Pair of Solids is Similar, Congruent, or Neither
- Extra Example 2: Determine if Each Statement is True or False
- Extra Example 3: Find the Scale Factor and the Ratio of the Surface Areas and Volume
- Extra Example 4: Find the Volume of the Larger Prism

- Intro 0:00
- Scale Factor 0:06
- Scale Factor: Definition and Example
- Congruent Solids 1:09
- Congruent Solids
- Similar Solids 2:17
- Similar Solids
- Extra Example 1: Determine if Each Pair of Solids is Similar, Congruent, or Neither 3:35
- Extra Example 2: Determine if Each Statement is True or False 7:47
- Extra Example 3: Find the Scale Factor and the Ratio of the Surface Areas and Volume 10:14
- Extra Example 4: Find the Volume of the Larger Prism 12:14

### Geometry Online Course

### Transcription: Congruent and Similar Solids

*Welcome back to Educator.com.*0000

*For the next lesson, we are going to go over congruent and similar solids.*0001

*Whenever we have two solids that are either similar or congruent, there is a scale factor.*0010

*A scale factor is just the ratio that compares the two solids; it is the ratio of the corresponding measures (it has to be corresponding).*0018

*If we are going to use this side for the scale factor, then we have to use the corresponding side of the other solid.*0028

*So then again, the scale factor is the ratio of the two similar solids.*0039

*Here, the scale factor...since this is 2 and this is 4, we are going to say that it is 2:4; simplified, this is 1:2.*0046

*The scale factor is 1:2, or we can say 1 to 2, like that; it is just the ratio between the two similar solids.*0060

*For congruent solids, these have to be true: the corresponding angles are congruent; the corresponding edges are congruent*0072

*(we have to have congruency between the two solids); the areas of the faces are congruent; and the volumes have to be congruent.*0082

*And congruent solids have the same size and same shape.*0094

*Remember that, for congruent solids, it is same size and same shape; for similar solids, it is going to be different sizes, but same shape.*0100

*Remember: whenever we have something similar, it has to be the same exact shape, but then just a different size.*0113

*Congruent solids will have the same shape and the same size.*0120

*And the scale factor is going to be 1:1, because obviously, the corresponding sides and the corresponding parts are going to be the same.*0124

*So, it is going to be a ratio of 1:1.*0132

*Looking at similar solids, if the scale factor is a:b, then the ratio of the surface areas is going to be a ^{2}:b^{2},*0139

*and the ratio of the volumes is going to be a ^{3}:b^{3}.*0153

*Let's say we have two solids, and the scale factor between the two is 2:3;*0161

*then the ratio of the surface is going to be 2 ^{2}:3^{2}, so it is going to be 4:9.*0168

*If that is the scale factor (that is the ratio between the two solids), their surface areas are going to be 4:9.*0184

*And then, the ratio of the volumes is going to be 2 ^{3}:3^{3}; 2 cubed is 8; 3 cubed is 27.*0194

*That is going to be the ratio of their volumes.*0209

*The first example is to determine whether each pair is similar, congruent, or neither.*0218

*Looking at these two, this pair: here, this is a cube, because we know that all of the sides are going to be congruent.*0225

*So, this is a cube; this is also a cube with all of the edges measuring 5.*0241

*So, in this case, because they are the same shape, but just different sizes, this is similar.*0248

*And we always know that all cubes are going to be similar, because cubes have the same shape.*0258

*No matter how big or how small, all cubes are the same shape; they can just be different sizes.*0267

*If they are the same shape, but same size, then they would be congruent; if all of these were also 8 inches, then they would be congruent.*0273

*But since they just have the same shape and different sizes, they are just similar.*0282

*And then, these two: let's see, here we have 24; that is diameter; from here to here is 26; we don't know the height.*0289

*Remember: for these two to be congruent, they have to have congruent corresponding parts.*0303

*To find the height here (because I don't know the height), I know the height here; this would be the height for this one,*0315

*because it is just the cylinder that is turned sideways; so if we say that that is the height,*0320

*then I need to find the height of this, so that I can compare.*0327

*The diameter is 24 here; the radius is 12 there; so the radius here will also be 12, because the diameter is twice the length of the radius.*0331

*To find the height here, I am going to use this triangle; and this is a right triangle, so then I can just use the Pythagorean theorem.*0344

*It is going to be...if I name that h...h ^{2} + 24^{2} is going to equal the hypotenuse (26) squared.*0352

*So, 24 squared is 576; and then, 26 squared is 676; so, if I subtract them, I am going to get 100, which makes my height 10 centimeters.*0365

*This is 10; this is also 10; so then, their heights are congruent; their radius is congruent.*0393

*So, if I were to find the area of the base, then it is going to be π times 12 squared (the radius is 12, so it is 12 squared).*0402

*Here, it is also going to be π times 12 squared; so the area of the bases will be the same.*0414

*To find the volume, it is going to be the area of the base...that is π, r, squared, times the height.*0422

*The same thing happens here: the radius is the same, and the height is the same, and we know that π is always the same.*0432

*So then, their volumes are going to be exactly the same.*0444

*Well, if we have two solids with the same exact volume, same shape, same size, same corresponding parts, we know that this has to be congruent.*0447

*And again, because they are going to have the same volume, they are congruent.*0460

*Determine if each statement is true or false: All spheres are similar.*0469

*Spheres always have the same shape; no matter what, all spheres are the same shape.*0477

*Now, sizes could vary; we could have a large sphere; we could have a small sphere.*0483

*But they are always going to be similar, because they always have the same shape.*0488

*So, any time two solids have the same shape, they are always going to be similar; so this is true.*0492

*The next one: If two pyramids have square bases, then they must be similar.*0504

*Well, even if they have a square base, yes, squares are always similar, because squares always have the same shape.*0511

*A square is a square, whether it is large or small; they are always going to be similar.*0524

*But for pyramids, we can have a tall pyramid, or we can have a short pyramid.*0531

*So, it doesn't always mean that they are going to be similar--these do not have the same shape.*0550

*So, even if their square bases are exactly the same (they are congruent), because we don't know the height, this is false.*0557

*If two solids are congruent, then their volumes are equal.*0572

*Well, let's say that we have exactly the same rectangular prism; it is congruent to this.*0578

*If they are exactly the same, then isn't the space inside also going to be the same?*0592

*So, all of this is going to be the same as all of this; so this is true.*0601

*Congruent solids have congruent volumes, the same volume.*0608

*For this example, we are going to find the scale factor and the ratio of the surface areas and the volume.*0617

*The scale factor between this and this prism that are similar is going to be 4:6.*0624

*We are going to use corresponding parts to determine the scale factor; it is going to be 4:6.*0634

*We need to simplify this, and it is going to be 2:3; that is the scale factor between these two prisms.*0641

*Then, to find the ratio of the surface area, for surface area, it is a ^{2}:b^{2}.*0649

*And then, for volume, it is going to be a ^{3}:b^{3}.*0660

*For surface area, the ratio is going to be 2 squared to 3 squared, which is 4 to 9.*0668

*Now, it doesn't mean that the surface area of this is 4 and the surface area of this is 9; it is just the ratio between this one and this one.*0684

*So, when we find the surface area, if we were to find the surface area of both this prism and this prism,*0695

*and then we simplify it, it is going to become 4:9.*0701

*And then, for volume, it will be 2 ^{3} to...what is that one, 3?...3^{3}, so it is going to be 8:27.*0710

*And again, that does not mean that the volume of this is 8, and that the volume of this is 27.*0727

*Let's actually find the volume, given two corresponding sides right here.*0736

*This is similar, so the ratio between these two prisms is 3:2.*0745

*And make sure that you keep the ratio the same; if you are going to keep it at 3:2, that means that you are listing out this one first.*0755

*You are naming this first; so it is this one to that one.*0761

*If you want to go the other way, that is fine; but then, you are going to have to make the scale factor 2:3, instead of 3:2.*0765

*Always keep in mind which one this is: this number refers to the larger prism.*0771

*Now, the volume of the smaller one, the second one, is given; it is 50 inches cubed.*0779

*So, to find the ratio of the volumes, it is 3 cubed to 2 cubed; that is 27 to 8; that is the ratio of the volumes.*0785

*This is the larger one, over the smaller one; that means that, if I want to find the actual volume,*0802

*the volume of this one to the volume of this is going to become 27/8, simplified.*0810

*Then, I just know that I can make a proportion: this ratio is going to equal the volume of that*0820

*(because that one applies to the larger one), so let's say V for volume, over...what is the volume of this smaller one? 50.*0827

*That is how I make my proportion, because the volume of the larger to the volume of the smaller, simplified, is going to become 27/8.*0838

*Use the volume of the larger over the actual volume of the smaller prism.*0848

*So then, here I am going to solve out this proportion; this becomes 8V (cross-products: 8 times V) equals 27 times 50.*0853

*Using your calculator, 27 times 50 equals 1350; divide the 8; your volume is 168.75.*0866

*And then here, our units are inches (and for volume, it has to be) cubed--units cubed.*0887

*And that would be the volume of this larger prism.*0897

*OK, so again, to make your proportion, we know that this ratio has to equal this ratio.*0901

*They both are the ratios of their volumes; you have the volume of the larger prism to the volume of the smaller prism;*0909

*that is going to become 27/8; so this is simplified, but then their volumes have to equal 27/8; that is the ratio of the prisms.*0923

*You just make the two ratios equal to each other; set it equal.*0934

*Make sure that you keep the larger prism as your numerator; so it has to be 27 over the smaller prism.*0939

*And this is going to be V/50, the larger over the smaller.*0949

*If you do it the other way, if you do the smaller over the larger, then you have to make sure that you flip this one, also: 50/V.*0953

*Find cross-products, and then just solve it out.*0960

*That is it for this lesson; thank you for watching Educator.com.*0962

0 answers

Post by Shahram Ahmadi N. Emran on August 4, 2013

Thanks