3D Maths
Rectangular Solid
A rectangular solid is a prism with a rectangular base and edges that are perpendicular to its base.
All you need to know is: the length (l), width (w) and height (h). These can be used to determine the surface area, volume and diagonal length in the following way:
Volume
The formula for the volume of a rectangular solid just adds on width to the volume of a rectangle:
Volume=lwh
Calculate the volume of the figure below in terms of x:
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Just plug the values into the volume formula and you’re good to go: V = (3x)(2x)(x) = 6x3.
Surface Area
The surface area of a solid is the area of its outermost skin. The surface area of a rectangular solid is made up of 6 rectangles: the sum of the areas of these six rectangles is the surface area of the box. To make things even easier, the six rectangles come in three congruent pairs. In the image below: One pair is transparent, one pair is light gray, and one pair is a darker gray.
w, two faces have areas of l 
h, and two faces have areas of w 
h. The surface area of the entire solid is the sum of the areas of the congruent pairs:
The diagonal of a rectangular solid 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:
Cubes
A cube is a square brought into 3-D. The length, width, and height of a cube are equal, and each of its six faces is a square.
Volume
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 formula for the volume of a cube is:
Volume= s3
... 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 the formula for finding the surface area of a rectangular solid, expect the s:
The formula for the diagonal of a cube is also adapted from the formula for the diagonal length of a rectangular solid, with s substituted for l, w, and h.
Right Circular Cylinders
A right circular cylinder has two connected congruent circular bases and looks like this:
Volume of a Cylinder
The volume of a cylinder is the product of the area of its base and its height:
It’s important to remind you that you will be given some geometry formulas (volumes and areas, etc.) on the SAT test, so you don’t need to spend a lot of time memorizing them. However, you should definitely go over this lesson a few times and do a lot of practice questions until you’re comfortable with them.
Great lesson, detailed but straight to the point!
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