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Rectangular Solids


Most of the geometry on the test is two-dimensional, but some of it is three-dimensional. In 2D, we could talk about area of shapes. In 3D, we will also talk about the volume of shapes. We will start with the easiest 3-dimensional shape, the cube. So think about what makes up a cube.

A cube has 6 faces. So those are the 6 congruent squares, we call those faces. It has 8 vertices, 8 corners. And it has 12 line segment edges. 4 at the top, 4 going up and down, and 4 at the bottom, 12 all together. The faces meet at right angles at the edges, and three mutually perpendicular edges meet at each vertex.

That defines a cube. If a cube has an edge-length of s, then its volume is given by s cubed. When we raise a number to the third power, we call this cubing the number precisely because this is how we find the volume of a cube. Exactly the same way that squaring we find the area of a square. Each face on this cube is a square with the side of s, and so would have an area of s squared.

And since there are 6 faces, the total surface area would be 6s squared. The cube is a special case of a more general category known as rectangular solids. The sides can be any rectangles, but we still have to have the requirements that the faces meet at right angles at each edge and three mutually perpendicular edges meet at each vertex.

Notice that the opposite rectangles are congruent in a rectangular solid. The volume of a rectangular solid is the product of the three different edge lengths. So we have four edges that are the length of h, four edges that are the length of w, and four edges that are the length of d. And we just take the product with these three numbers, the volume is h times w times d, height times width, times depth.

The surface consists of two h x w faces, two h x d faces, and two w x d faces. Thus, the total surface area is given by that formula. In a rectangular solid, we can consider two different kinds of diagonals. One kind, called a face diagonal, is a diagonal of only one face of the solid. So it's like we're ignoring the rest of the solid and we're just looking at that single rectangle, the diagonal in that single rectangle.

To find that length, we would just use the Pythagorean Theorem with the two edges of that face. In particular here, we have a right angle three and four and one hypotenuse. It has to be a 3, 4, 5 triangle. So AC = 5. The other kind is called a space diagonal does not run along a face it passes through the interior of the rectangular solid for one vertex of the opposite vertex.

So here we see it's going from A to D. For the length of this we can use the 3D version of the Pythagorean theorem. So edge squared plus edge squared plus edge squared equals the length of the diagonal square, the length of the space diagonal square. So here, that's going to be 9 + 16 + 25 = 50. That is what AD squared is, so AD equals the square root of 50.

And of course we could factor out a 25, so that's gonna be 5 root 2. What is the length of the space diagonal of a cube with edge length s? Well, same thing, we use the 3D Pythagorean theorem. (AB) squared = s squared + squared + s squared. And of course, that's 3s squared.

Take a square root, and we get root 3 times s The space diagonal of cube is root three times the edge length. The test loves that little fact. Here's a practice problem, pause the video and then we'll talk about this. Okay, so, we have two lengths, the 4 and the 6, and we have the length of the space diagonal 8.

And we want to find the volume. Well the first thing we'd need is the third length. So we'll just call that d, the depth. We have 4 squared + 6 squared + d squared = 8 52 + d squared = 64. Subtract, we get d squared equals = 12, and then d has to be square root of 12, we take a positive square root because we are looking for a length of course.

Square root of 12 we can simplify, we factor out the 4. So that's 4 times 3. Square root of 4 is 2, and so this is 2 root 3. So that's the length of the depth. Well now we have the height, the width, and the depth. So the volume is just gonna be 4 times 6 times 2 root 3.

And that is 48 root 3. That's the answer. Rectangular solids are 3D shapes with six rectangle sides all meeting perpendicularly. The volume is given by the product of the three different edge lengths. The surface area is the sum of 2hw + 2dw + 2hd.

In other words, we're just summing over the rectangles on the individual phases. The cube is a special case of the rectangular solid, with six congruent square faces. For a cube of edge length s, the volume is s cube, the surface area is 6s squared. We can find face diagonals with the ordinary Pythagorean theorem, and space diagonals with the 3D Pythagorean theorem.

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