To practice, try to find the surface area of the rectangular solid we used as an example for volume. Here’s the figure again:
All you have to do is enter the given values into the formula for surface area:
Diagonal Length of a Rectangular Solid
The diagonal of a rectangular solid, d, is the line segment whose endpoints are opposite corners of the solid. Every rectangular solid has four diagonals, each with the same length, that connect each pair of opposite vertices. Here’s one diagonal drawn in:
The formula for the length of a diagonal is:
where l is the length, w is the width, and h is the height.
You can look at this formula as the Pythagorean theorem in three dimensions. In fact, you can derive this formula using the Pythagorean theorem. First, find the length of the diagonal along the base. This is
. Then use the Pythagorean theorem again, incorporating height to find the length of the diagonal from one corner to the other:
. Thus,
and
. A Math IC question might ask you:
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What is the length of diagonal AH in the rectangular solid below if AC = 5, GH = 6, and CG = 3? |
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The question gives the length, width, and height of the rectangular solid, so you can just plug those numbers into the formula:
The length of the diagonal
AH (as well as
BG,
CF, and
DE) is
.
Cubes
Just as a square is a special kind of rectangle, a cube is a special kind of rectangular solid. A cube is a rectangular solid whose edges are each the same length. In other words, the length, width, and height are equal, and each of its six faces is a square.
Volume of a Cube
The formula for finding the volume of a cube is essentially the same as the formula for the volume of a rectangular volume. However, since a cube’s length, width, and height are all equal, the cube volume formula is:
where s is the length of one edge of the cube.
Surface Area of a Cube
Since a cube is just a rectangular solid whose sides are all equal, the formula for finding the surface area of a cube is the same as that for a rectangular solid, except with s = l = w = h:
where s is the length of one edge of the cube.
Diagonal Length of a Cube
The same is true for measuring the diagonal length of a cube. The formula for the diagonal of a cube is simply adapted from the formula for the diagonal length of a rectangular solid, with s = l = w = h:
where s is the length of one edge of the cube.
Cylinders
A cylinder is a prism with circular bases.
The height of a cylinder, h, is the length of the line segment whose endpoints are the centers of the bases. The radius of a cylinder, r, is the radius of its base. If you know the height and radius of a cylinder, you can easily calculate its volume and surface area.
