Geometry Word Problems (page 1 of 6) 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 problem which formula 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 formulae.

• 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 = 2(pi)r

...where "pi" (above) is of course the number approximately equal to 22 / 7 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 = 2(pi)(3) = 6pi

Then, after re-checking the original exercise for the required units (so my answer will be complete):

the circumference is 6pi cm.

Note: Unless you are told to use one of the approximations for pi, or are told to round to some number of decimal places (from having used the "pi" 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 "pi" form or the decimal form, use both: "6pi cm, or about 18.85 cm".

• 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

After re-reading the exercise to find the correct units, my answer is:

The length of each side is 4 centimeters.

Most geometry word problems are a bit more involved than the example 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 solution, 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 bit of new 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 one 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 so there are six squares, and the cube's total surface area SA must be:

SA = 6e2

Plugging in the value they gave me, I get:

54 = 6e2
54 / 6 = (6e2) / 6

9 = e2

3 = e

Since the volume is the cube of the edge-length, and since the units on this cube are centimeters, then:

the volume is 27 cubic centimeters, or 27 cc, or 27 mL

(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. So all three versions of the answer above are equivalent.)     Copyright © Elizabeth Stapel 2000-2011 All Rights Reserved

• A circle has an area of 49pi 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 = (pi)r2

I know that radius r is half of the length of the diameter d, so:

49pi = (pi)r2
(49pi) / pi = [(pi)r2] / pi

49 = r2

7 = r

Then the radius r has a length of 7 units, and:

the length of the diameter is 14 units

Warning: You can not assume that you will always be given all the geometric formulae on your tests. At some point, you will need to know at least some of them "by heart". The basic formulae you should know include the formulae for the area and perimeter (or circumference) of squares, rectangles, triangles, and circles; and the surface areas and volumes of cubes, rectangular solids, spheres, and cylinders. Depending on the class, you may also need to know the formulae for cones and pyramids. If you're not sure what your instructor expects, ask now.

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 Cite this article as: Stapel, Elizabeth. "Geometry Word Problems: Introduction." Purplemath. Available from     https://www.purplemath.com/modules/perimetr.htm. Accessed [Date] [Month] 2016

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