The Purplemath Forums
Geometric Formula ReviewThere are many geometric formulas, and they relate height, width, length, or radius, etc, to perimeter, area, surface area, or volume, etc. Some of the formulas are rather complicated, and you hardly ever see them, let alone use them. But there are some basic formulas that you really should memorize, because it really is reasonable for your instructor to expect you to know them.
For instance, it is very easy to find the area A of a rectangle: it is just the length l times the width w:
Arect = lw Copyright © Elizabeth Stapel 2000-2011 All Rights Reserved
If you look at a picture of a rectangle, and remember that "perimeter" means "length around the outside", you'll see that a rectangle's perimeter P is the sum of the top and bottom lengths l and the left and right widths w:
Prect = 2l + 2w
Squares are even simpler, because their lengths and widths are identical. The area A and perimeter P of a square with side-length s are given by:
Asqr = s2
Psqr = 4s
You should know the formula for the area of a triangle;
it's easy to memorize, and tends to pop up unexpectedly in the middle of word problems. Given the
measurements for the base b and the height h of the triangle, the
area A is:
Atri = (1/2)bh
Of course, the perimeter P of the triangle will just be the sum of the lengths of the triangle's three sides.
You should know the formula for the circumference C and area A of a circle, given the radius r:
Acir = (pi)r2
Ccir = 2(pi)r
("pi" is the number approximated by 3.14159 or the fraction 22/7)
Remember that the radius of a circle is the distance from the center to the outside of a circle. In other words, the radius is just halfway across. If they give you the diameter, the length of a line going all the way across, then you'll first have to divide it in half in order to apply the above formulas.
The above are all "flat", two-dimensional shapes. Sometimes you will have to deal with three-dimensional shapes, such as cubes or cones.
The formula for the volume V of a cube is easy, since the length, the width, and the height are all the same value s:
Vcube = s3
The formula for the surface area (the area you would measure if you needed to paint the ouside of the cube) is fairly easy, too, since all the sides have the same square area of s2. There are six sides (top, bottom, left, right, front, and back), so the surface area SA is:
The formulas get a bit more complicated for a "rectangular prism", which is the technical term for a brick. The volume V is still fairly simple, being length times width times height:
Vrect = lwh
The surface area formula is a bit more ornate. (Try to follow the reasoning that I'm going to use, because you'll probably forget the formula, but it's easy to recreate if you just take a little time and think about it.) The top and bottom of the "brick" have the same area, being length times width. The left and right sides of the brick have the same area, being width times height. And the front and back of the brick have the same area, being length times height. (Draw a picture, labelling the dimensions, if you're not sure of this.) Then the formula for the surface area SA of a brick is:
SArect= 2lw + 2wh + 2lh
Cylinders (which are like tubes, but with caps on the ends) also come up occasionally. The volume V of a cylinder is easy: it's the area of the end (which is just the area of the circle) times the height h:
Vcyl = (pi)r2h
The surface area SA is the area of the ends (which are just circles), plus the area of the side, which is a circle's circumference times the height h of the cylinder:
SAcyl = 2(pi)r2 + 2(pi)rh
Depending on the class you're taking, you might also need to know the formula for the volume V of a cone with base radius r and height h:
Vcone = (1/3)(pi)r2h
...or the volume V of a sphere (a ball) with radius r:
Vsph = (4/3)(pi)r3
You may notice other formulas cropping up in your homework or classroom exercises. You may need to memorize these other formulas (there are many!), so be sure to check with your instructor before the test to learn which you will be expected to know.
Some instructors provide all of the geometric formulas, so your test will have a listing of anything you might need. But not all instructors are this way, and you can't expect every instructor, every department, or "common", department-wide, or otherwise standardized tests to give you all this information. Ask your instructors for their policies, but remember that there does come a point (high school? SAT? ACT? college? "real life"?) at which you will be expected to have learned at least some of these basic formulas. Start memorizing now!