Print-friendly page

Geometry Word Problems (page 1 of 4)

Sections: Introduction, Basic shapes, The Pythagorean Theorem, Max/min problems

The trick to these problems is to note that, unless it's a simple application of basic geometric formulae, they will almost always give you two pieces of information, such as a statement about perimeter and then a question about area. Then you need to write the two equations related to these two pieces of information, solve one of the equations for one of the variables, and then plug this into the other equation. Here are some examples:

• Three times the width of a certain rectangle exceeds twice its length by three inches, and four times its length is twelve more than its perimeter. Find the dimensions of the rectangle.

The first statement compares the length L and the width W. Start by doing things orderly, with clear and complete labelling:

three times the width: 3W
twice its length: 2L
exceeds by three inches, meaning "is three inches greater than": + 3
equation: 3W = 2L + 3

Now I have the second statement, which compares the length L and the perimeter P. I will be complete with my labelling:

four times its length: 4L
perimeter: P = 2L + 2W     (this is just the perimeter formula for rectangles)
twelve more than: + 12
equation: 4L = P + 12, or 4L = (2L + 2W) + 12  (by substitution)

So now I have my two equations:   Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

3W = 2L + 3
4L = 2L + 2W + 12

There are various ways of solving this; the way I do it (below) just happens to be what I thought of first. I'll take the first equation and solve for W:

3W = 2L + 3
W = ( ²/₃ )L + 1

Now I'll simplify the second equation, and then plug in this above expression for W:

4L = 2L + 2W + 12
2L = 2W + 12
2L = 2[ ( ²/₃ )L + 1 ] + 12    (by substitution from above)
2L = ( ⁴/₃ )L + 2 + 12
2L = ( ⁴/₃ )L + 14
2L – ( ⁴/₃ )L = 14
( ⁶/₃ )L – ( ⁴/₃ )L = 14
( ²/₃ )L = 14
L = (14)×( ³/₂ ) = 21

Then:

W = ( ²/₃ )L + 1
    = ( ²/₃ )×(21) + 1
    = 14 + 1 = 15

Now, remember that the question didn't ask "Find the values of the variables L and W". It asked you to "Find the dimensions of the rectangle," so the actual answer is:

The length is 21 inches and the width is 15 inches.

(Always be sure to check for the appropriate units, which was "inches" in this case.)

Some problems are just straightforward applications of basic geometric formulae.

• Suppose a water tank in the shape of a right circular cylinder is thirty feet long and eight feet in diameter. How much sheet metal was used in its construction?

What they are asking for here is the surface area of the water tank. The total surface area of the tank will be the sum of the surface areas of the side (the cylindrical part) and of the ends. If the diameter is eight feet, then the radius is four feet. Then the surface area of each end is given by the area formula for a circle with radius r: A = (pi)r². (Remember that there are two end pieces, so I will be multiplying this by 2 when I find my surface-area formula.) The surface area of the cylinder is the circumference of the circle, multiplied by the height: A = 2(pi)rh.

	Side view of the cylindrical tank, showing the radius "r".
	An "exploded view" of the tank, showing the three separate surfaces whose areas I need to find.

Then the total surface area of this tank is given by:

2 ×( (pi)r² ) + 2(pi)rh    (the two ends, plus the cylinder)
    = 2( (pi) (4²) ) + 2(pi) (4)(30)
    = 2( (pi) × 16 ) + 240(pi)
    = 32(pi) + 240(pi)
    = 272(pi)

Always remember to put the correct units on your answer. Since the dimensions were given in terms of feet, then the area is in terms of square feet. Then:

the surface area is 272(pi) square feet.

By the way: You can not assume that you will always be given all the geometric formulae. At some point, you will need to learn some of them, because you will expected to know them. The basic formulae you should know include the formulae for the area and perimeter/circumference of squares, rectangles, triangles, and circles, and the surface areas and volumes of cubes, rectangular solids, spheres, and cylinders. Depending on the class, knowing the formulae for cones and pyramids might be a good idea, too.

Top  |  1 | 2 | 3 | 4  |  Return to Index  Next >>

  Copyright © 2006-2008  Elizabeth Stapel   |   About   |   Terms of Use

 Feedback   |   Error?