Geometric Formula Review

For instance, it is very easy to find the area A of a rectangle: it is just the length l times the width w:

Which geometry variables should I be able to recognize?

Some variables being fairly standard, you should expect your instructor and your textbook to be using " A " for "area", " SA " for "surface area", " P " for "perimeter", and " V " for "volume".

Linear measures are " w " for "width", " d " for "depth" [being the distance from the front to the back of a 3-d objects, " h " for "height", and " l " for "length". However, because the l can look a lot like the number 1, sometimes it's wise to use L instead, especially when you're writing stuff down.

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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:

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:

You should know the formula for the area of a triangle; it's easy to memorize, and it 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 of the triangle is:

Of course, the perimeter P of the triangle will just be the sum of the lengths of the triangle's three sides.

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You should know the formula for the circumference C and area A of a circle, given the radius r:

("π" 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 length of the diameter, being the length of a line through the middle going all the way across the circle, then you'll first have to divide that value in half in order to apply the above formulas.

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The above all relate to "flat", two-dimensional shapes. Sometimes you will have to deal with three-dimensional shapes, such as cubes or cones. For these kinds of shapes, you'll be finding the surface area (if you were painting the object, this is the area that you'd have to cover) and the volume (being the interior space that you could fill, were the shape hollow).

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:

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:

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:

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:

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:

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:

...or the volume V of a sphere (a ball) with radius r:

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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.

Do I really have to memorize all the formulas?

It's not necessary to memorize all the formulas you come across, but there are some others that you really should memorize. You should know how to find the area of a rectangle or the circumference of a circle; you probably don't need to memorize the formulas for, say, the volume of a torus or the surface area of a regular tetrahedron. (There are, by the way, loads of other formulas that you probably won't need to memorize. Be grateful for this!)

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 final exam, 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!