Introduction to Geometry Word Problems
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How do you solve geometry word problems?
In order to solve geometric word problems, you will need to have memorized some geometric formulas for at least the basic shapes (circles, squares, right triangles, etc). You will usually need to figure out from the word problems which formulas to use — and many times you will need more than one formula for one exercise.
So make sure you have memorized any formulas that are used in the homework, because you may be expected to know them on the test.
Some problems are just straightforward applications of basic geometric formulas.
- The radius of a circle is 3 centimeters. What is the circle's circumference?
The formula for the circumference C of a circle with radius r is:
C = 2πr
...where π (in the formula above) is of course the number approximately equal to or 3.14159.
They gave me the value of r and asked me for the value of C, so I'll just "plug-n-chug":
C = 2π(3) = 6π
Then, after re-checking the original exercise for the required units (so my answer will be complete):
circumference: 6π cm
Can I just round π?
Unless you are told to use one of the approximations for π, or are told to round to some number of decimal places (from having used the [π] button on your calculator), you are generally supposed to keep your answer in exact form, as shown above.
If you're not sure if you should use the π form or the decimal form, use both. In the above case, this would mean listing both "6π cm" and also "≈18.85 cm" as answers, where the symbol "≈" means "approximately equal to".
- A square has an area of sixteen square centimeters. What is the length of each of its sides?
The formula for the area A of a square with side-length s is:
A = s2
They gave me the area, so I'll plug this value into the area formula, and see where this leads:
16 = s2
±4 = s
s = 4
(While −4 is a valid solution to the equation 16 = s2, the context of that equation is finding a length, which must be positive. That's why I threw out the minus solution.)
After re-reading the exercise to find the correct units, my answer is:
side length: 4 cm
Most geometry word problems are a bit more involved than the examples above. For most exercises, you will be given at least two pieces of information, such as a statement about a square's perimeter and then a question about its area.
To find the solutions, you will need to know the equations related to the various pieces of information; you will then probably solve one of the equations for a useful new bit of information, and then plug the result into another of the equations.
In other words, geometry word problems often aren't simple one-step exercises like the ones shown above. But if you take all the information that you've been given, write down any applicable formulas, try to find ways to relate the various pieces, and see where this leads, then you'll almost always end up with a valid answer.
- A cube has a surface area of fifty-four square centimeters. What is the volume of the cube?
The formula for the volume V of a cube with edge-length e is:
V = e3
To find the volume, I need the edge-length. Can I use the surface-area information to get what I need? Let's see...
A cube has six sides, each of which is a square; and the edges of the cube's faces are the sides of those squares. The formula for the area of a square with side-length e is:
A = e2
There are six faces on a cube, so there are six squares that form its surface area. Then the cube's total surface area SA must be:
SA = 6e2
Plugging in the value they gave me, I get:
54 = 6e2
9 = e2
3 = e
Since the volume is the cube of the edge-length, and since the units on this cube are centimeters, then:
volume: 27 cc
(The common abbreviation for "cubic centimeters" is "cc's", as you've no doubt heard on medical TV dramas, and one cc is equal in volume to one milliliter, abbreviated as mL. This metric-unit fact may be sprung on you seemingly out of nowhere. For instance, the above exercise could have asked specifically for the volume in terms of milliliters, and you'd just have to know that one cc is the same as one mL.)
- A circle has an area of 49π square units. What is the length of the circle's diameter?
The formula for the area A of a circle with radius r is:
A = πr2
I know that radius r is half of the length of the diameter d, so:
49π = πr2
49 = r2
7 = r
Then the radius r has a length of 7 units, and:
diameter length: 14 units
Do I have to memorize all the geometric formulas?
While you probably won't ever encounter some geometric formulas, you can *not* assume that you will always be given all the necessary geometric formulas on your tests. At some point, you will need to know at least some of them by heart.
Which geometric formulas should I memorize?
The basic formulas you should know include the formulas for the area and perimeter (or circumference) of squares, rectangles, triangles, and circles; and the surface areas and volumes of cubes, rectangular solids (that is, brick-shaped objects), spheres, and cylinders. Depending on the class, you may also need to know the formulas for cones and pyramids.
If you're not sure what your instructor expects, ask now.
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