 Section 13.4 deals with congruent and similar solids. Now these ideas of congruent and similar solids really basically the same idea from previous chapters when we talked about congruent and similar polygons. So you'll want to take out your notes and put this into your table of contents, and then we'll begin. So congruent solids have exactly the same size and shape. Two solids are similar if they have the exact same shape but not necessarily the same size. And so instead we would say the corresponding linear measurements must be proportional. So linear measurements are things like height, length, width, diameter, slant height, etc. So let's look at an example. Let's say we have these two similar prisms and with a smaller of the two, its measurements are 12 by 9 by 6. The larger let's say is 20 by 15 by 10. Now we know since the prisms are similar, all of the corresponding angles have to be congruent. In other words the ratio of the angles must be 1 to 1. All of the corresponding edges or the linear measurements are proportional. And so that's scale factor or the ratio of corresponding edges. Let's take a look at one pair. I've got 12 and 20. And we typically do scale factor by shape on the right divided by left or left divided by right. So in this case we'll do right divided by left. So 20 divided by 12. So that would give us a scale factor of 5 to 3. The ratio of the base perimeters will also follow that same scale factor because all linear measurements will be proportional in that 5 to 3 ratio. Now all corresponding surface areas are also proportional. However, they're not going to follow the same ratio. They're not going to follow 5 to 3. Instead if we take let's say the surface areas, total surface areas, that area ratio, 1300 divided by 468, ends up being 25 to 9. Which is related to 5 to 3. The surface areas are 5 squared to 3 squared. And so any of those surface area measurements, the two dimensional measurements, will follow that squared ratio. So volumes are proportional as well. If we look at the volume of these two prisms and then take their ratio, so 3000 divided by 648, it's quite a large fraction, but that fraction ends up being 125 to 27. Which again is related to 5 to 3. In fact, it's 5 to the third power, to 3 to the third power. And so to recap, if we have two similar solids with a scale factor A to B, the perimeter ratio is also A to B. In other words, any one-dimensional measurements, so perimeter, heights, base measurements, slant heights, those will always follow that scale factor of A to B. Surface area ratio will be A squared to B squared. And volume, A cubed to B cubed. So let's apply that to a couple of examples. Let's say we have a cone-shaped drinking cup, and it is currently 15 centimeters tall with a diameter of 10 centimeters. If we triple the height and diameter, what will be the volume of the new bigger cup? We'll walk through two different ways to solve this problem. So the first method is just figure out what that bigger cup looks like. We have our smaller cup and our bigger cup. The original cone had a diameter of 10 and a height of 15. And if you take and triple all those measurements, then the diameter is now 30. The height is now 45. And so the volume of the red cone will be 1 third pi r squared h, which is 3,375 pi cubic centimeters. The second way to solve this is using proportions. I know that the volume of the new cup divided by volume of the old cup will follow that same scale factor, 3 to 1, but not 3 to 1, 3 cubed to 1 cubed. And then the original cup, sorry, the blue cup, yep, the original cup had a volume of 125 pi. And so if we set that into our proportion, we know x over 125 pi must equal 27 over 1. Again, 27 came from 3 to the third. 1 to the third, of course, is just 1. Now, if you solve that using cross multiplication, we get a volume of 3,375 pi cubic centimeters. So again, two ways to solve the same problem. Next example, sorry, got a bit cut off there. A square pyramid has a slant height of 16 units and a base edge of 20 units. So we've got the square pyramid, and we want the surface area of a similar square pyramid that's been reduced by a scale factor of 2 to 5. So we're going to use that perimeter, sorry, the proportions idea, the surface area of the new divided by surface area of the old must follow 2 squared to 5 squared. And so the surface area of the original we can find using 1 half perimeter temp slant height plus the area of one base. And so that gives us 1,040 square units. We'll sub that into our proportions calculations and solve for the unknown. And so therefore the surface area of the new smaller pyramid will be 166.4 square units. Let's take a look at one more example. So let's say we have a cylinder with a surface area of 800 pi and a volume of 3,000 pi cubic units. What would happen to the surface area and volume of a similar cylinder if we enlarged by a scale factor of 3 to 2? So this is again a proportions problem. We know new over old will follow our scale factor. And so surface area of new over old will be scale factor squared, so 3 squared over 2 squared. Sub in what you know, solve for the unknown. Likewise with volumes, instead of 3 over 2, we want 3 cubed over 2 cubed. And so we'll sub in the known value and solve for the unknown. And there we go. Similar and congruent solids. Enjoy.