Prisms and Pyramids

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Prisms and pyramids. The test could also ask about prisms and pyramids. These 3D shapes appear less frequently on the test, but they could appear. So we have an example of each here. So first of all, let's talk about prisms. A prism is a 3D shape formed by two congruent triangles in parallel planes, connected by rectangles.

So here we have a 3-4-5 triangle in the front, in the back we have a parallel 3-4-5 triangle, and they're connected by rectangles. Now, of course, it could be any kind of triangle, but most often, what you're gonna see on the test is that the triangle itself is a right triangle. That's gonna be most frequent on the test. The length of the triangle connecting the two congruent triangles is called the height, regardless of whether the prism is lying down or standing up on end.

We call that length the height. It is unlikely that the test will expect you to have memorized the volume of a prism. It is much more likely they will give you the formula, which is volume equals area of the base times the height. In other words, area of the triangle times that long length.

The test has to give us prisms with easy triangles, triangles in which we can find the lengths and areas that we need. So for example, here's a practice problem. Pause the video, and then we'll talk about this. In a prism with the dimensions shown here, the volume is the area of the triangle times the length of the rectangles.

What is the volume, in cubic centimeters of this prism. Well, first of all, we need to find the area of that base. The area of that base, that's just a triangle, area equals one half base times height, three times four. So that's half of 12 which is 6, so that's the area of the triangle. And then that length is 10.

So the volume is 6 times 10, and that's 60. So that's the answer. Now a different practice problem, pause the video and then we'll talk about this one. Okay, a prism with dimensions given here in centimeters is shown, what is the total surface area of that prism?

Well, the total surface area, let's think about this. We have a triangle in the front, triangle on the back, and then we have these three rectangles here. So the bottom rectangle, 3 times 10 is 30. The back rectangle 4 times 10 is 40. We know that hypotenuse is 5, it's a 3, 4, 5 triangle.

So that slant rectangle is 50. So those are all the rectangles, three rectangles. We've already figured out that the area of one triangle is 6. So two triangles is 12. So now we just add all this together, 30 + 40 + 50 + 12, that's 132, and so the area is 132.

A pyramid is a 3D shape with a square or a rectangle at the base, and from each edge, triangles slant up to join at a single point. It may help to visualize the pyramid that Mr. Khufu built over four millennia ago in Egypt. That's what meant by a pyramid shape. Now in the general broad picture of geometry, of course the base could be anything.

It might be a triangle, it might be a rectangle, it might be a square, it might be a hexagon or a pentagon or something like that. You're not gonna have to worry about exotic pyramids like that, you're only gonna have to worry about squared based pyramids on the ACT. So the vertex, at the top of the pyramid, is directly above the middle of the base. In this pyramid, M is the middle of the base, T is the top, and the segment TM perpendicular to the base is the height, that's what's meant by the height of the pyramid.

It is unlikely the test will expect you to memorize the formula for the volume of a pyramid. If the test wants you to calculate that, it most likely will give you the formula. And the formula equals one third times the area of the base time the height. Most often the base is a square, so we only need one side to figure out its area. The test could also ask us for the total surface of a prism, the total surface area.

So the total surface area equals the square base plus the 4 triangles. The square base is easy, to get the area of one of the slanting triangle faces, we know the side of the base is the base of the triangle, we need the height of the triangle, its altitude. Most typically the test will give us the side of the square base, and the height of the pyramid.

The height of the pyramid is not the altitude of the slanting triangle face. Let K be the mid point of AD. And draw the right triangle TMK. So TM is the height of the pyramid. KM is half the base of the pyramid. So the test will give us TM, which is the height.

It will give us the base, we divide that by two, we get MK, which is half the length of the side of the square base. Then we use the Pythagorean theorem to find TK, the hypotenuse and TK, the hypotenuse of that triangle, is the altitude. Notice that it runs up the middle of the triangle, so that is the altitude of the triangle.

And that's what we would need to find the area of triangle ATD. Here's a practice problem. Pause the video, and then we'll talk about this. The pyramid shown has a square base FGHJ, with FJ, the side of the square is 20. Point M is the middle of the square.

Segment MP is perpendicular to the base and has a length of 24, the volume of the pyramid is given by volume = 1/3 ( area of base) *h what is the volume of this pyramid in cubic cm? Well let's think about this, the base is a square, the side is 20. So the area of the base is 20² which is 400, the height is 24, so it's going to be 1/3 (400)(24), I'm going to actually multiply first the 1/3(24) because a third of 24 is 8.

And then 8 times 400 is 3,200. So the volume is 3,200. Here's another practice problem. Pause the video. And then we'll talk about this. Okay, again the pyramid has a square base with 20 cm on the side, M is the middle of the square, MP is perpendicular, has a length of 24, what is the total surface area?

So now we're looking for the surface area. Well, first of all, the square base, that's easy, the square base we already figured out is 20 squared or 400. Let x be the midpoint of FJ. We're gonna have to think about this. You know that MX is 10.

Because side of the square is 20. So MX is half of the side of the square, that's 10. And MP is given in the problem and that's 24. Well, we have 10 and 24, and we have a hypotenuse. Well notice really what this is, is a 5,12,13 triangle, multiplied by 2. This is why it's very important to know these Pythagorean triplets.

So 5, 12, 13 multiplied by 2 would be 10, 24, 26. We have the legs of 10, 24, so the hypotenuse must be 26. And so that means, this is the altitude of that slanted triangle FPJ, the altitude is PX, which is 26. Now, we can find the area of that triangle, one half base times height. One half twenty times 26 or in other words, 10 times 26 which is 260.

So that's the area of one of the triangles. So now, we keep the area and the base, four triangles would be 4 times 260, which is 1040. Then just add 400, and we get 1440, that's the total surface area. So the total surface area is 1440.

The volume of prism, most likely that would be given on the test, you will not need to memorize this. The surface area of a prism are the two triangles ends and the three rectangles. The volume of prism most likely given by the test, the surface area the prism is the one square at the bottom, plus the four triangles.

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