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# Word Problem Worksheets 4 (Geometry Formulas)

Lesson Plans > Mathematics > Algebra > Word Problems

## Word Problem Worksheets 4 (Geometry Formulas) Below are several handouts which involve formulas and word problems. The first is a sheet of geometry formulas that students may find useful in solving the word problems in the three worksheets. My purpose in creating these worksheets was not to test students' memory of geometry formulas, but to provide word problem solving practice for algebra students. Armed with the formula handout, these problems are accessible to students who have not yet taken a geometry class. If you are giving this handout to geometry students, you may choose not to give them the formula "cheat sheet."

The first worksheet has no second degree equations to solve, and no systems of equations to solve.

The second worksheet is comprised of systems of two linear equations in two unknowns.

The third worksheet contains problems that have quadratic solutions. Note that question number 5 has a sneaky trick to it. Students can grind out the solution without seeing the trick, but seeing the trick leads to a more elegant solution. A hint to the "trick" is provided in the solution key.

## In This Unit

Lesson by Mr. Twitchell

## Formulas

Below is a listing of several common geometry formulas relating to areas, volumes, perimeters, surface areas, right triangles and polygons.

## Perimeters

Rectangle: P = 2L + 2W
Triangle: P = a + b + c
Circle: C = 2pr

## Areas

Rectangle: A = LW
Triangle: A =
1
2
bh
Circle A = pr2

## Surface Area

Rectangular Prism: S.A. = 2(LW + WH + LH)
Cylinder: S.A. = 2pr2 + 2prh
Sphere: S.A. = 4pr2
Cone: S.A. = pr2prs (s = slant height)

## Volumes

Rectangular Prism: V = LWH
Cone/pyramid: V =
1
3
Ah, where A = the area of the base
Sphere: V =
4
3
pr3
Cylinder: V = pr2h

## Other Formulas

Pythagorean Theorem: c2 = a2 + b2 (c = hypotenuse of a right triangle)
Number of diagonals in an n-gon: D =
n(n - 3)
2

Sum of the measures of the interior angles of an n-gon: S = 180(n - 2)

## Word Problem Worksheet #4.1

Write equations for each problem and solve them. Answer the question.

Example:   In a triangle, the side lengths are x, x + 3, and x + 4. the perimeter is 37. What is the longest side length?

Solution:    x + x + 3 + x + 4 = 37; x = 10; x + 4 = 14

1. The perimeter of a rectangle is 80. One of its side lengths is 32. What is the other side length.

2. The perimeter of a rectangle is three times the length of the longer dimension. The shorter dimension is 12 units. What is the perimeter?

3. The radius of a cylinder is 7 inches, and its surface area (using
22
7
as an approximation for p) is 396 in3. What is the cylinder's height?

4. Polygon X has twice as many sides as Polygon Z. The sum of the measures of the interior angles of X is 1800 degrees. What is the sum of the measures of the interior angles of Y?

5. The volume of a cone is ten times its height. What is the area of the base?

6. The circumference of a circle (measured in inches) is
2
3
the area of the circle (measured in square inches). What is the circle's radius?

7. The perimeter of a rectangle is 25 less than three times the sum of its dimensions. Find the rectangle's perimeter.

8. The shortest side length in a triangle is three less than the middle side length, and the longest side length is 5 more than the middle side length. The perimeter is 74. What are the three side lengths?

9. The middle side length in a triangle is nine less than twice the shortest side length, and the longest side length is 12 less than twice the middle side length. The perimeter is 45. What is the longest side length?

10. The area of a rectangle is 40 square units more than its length (measured in linear units). The width is 6 units. What is the area?

## Word Problem Worksheet #4.1: Answer Key

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## Word Problem Worksheet #4.2

Write equations for each problem and solve them. Answer the question.

Example:   In a triangle, the shortest side lengths is 10 units. The perimeter is 42 units. The sum of the middle side length and twice the longest side length is 49. What are the triangle's side lengths?

Solution:    10 + x + y = 42; x + 2y =49; 10, 15, 17

1. The length of a rectangle is seven more than its width. The perimeter is two more than eight times the width. What are the rectangle's dimensions?

2. One of the legs of a right triangle is seven units longer than the other. The area of the triangle is 6 times the shortest leg. What is the length of the hypotenuse?

3. The perimeter of a rectangle is 86 units. If the height was increased by 4 units and the width was cut in half, the new perimeter would be 94 units. What were the original dimensions of the rectangle?

4. In an n-gon the number of diagonals is equal to 27 less than half the square of the number of sides. What is the value of n?

5. A rectangular prism has sides of length 5, x, y, in order from smallest to largest. The perimeter of the largest face is 34 units. The difference between y and x is 2. What is the volume of the prism?

6. The number of sides in a regular polygon is x + y. The sum of the measures of its interior angles is 2880 degrees.  x is four more than y. Find the ordered pair (x, y).

7. The volume of a sphere (measured in cubic units) is 121 times its surface area (measured in square units). Find the sphere's diamter.

8. Two spheres have radii that differ by 2. The difference between their surface areas is 160p. Find each radius.

9. The slant height and the radius of a cone are reciprocals of one another. The surface area of the cone is 50p. Find the slant height of the cone.

10. The length of a rectangle is 4 units more than the width, and the perimeter is 34 units more than the length. What is the rectangle's area?

## Word Problem Worksheet #4.2: Answer Key

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## Word Problem Worksheet #4.3

Write equations for each problem and solve them. Answer the question.

Example:   The area of a rectangle is 72, and its length is six more than its width. Find the dimensions.

Solution:    LW = 72; L = W + 6; L = 12, W = 6

1. The perimeter of a rectangle is 34 inches, and its area is 72. What are the rectangle's dimensions?

2. The legs of a right triangle have lengths x and x + 3. The hypotenuse has length 89
. What is the triangle's area?

3. The legs of a right triangle have lengths x and x + 2. The hypotenuse's length is x +4. What is the length of the shortest leg?

4. The number of diagonals in an n-gon is 65. How many sides does the polygon have?

5. The area of a rectangle, in square units, is 250. Its perimeter, is 65 units. What is the sum of the squares of the rectangle's length and width?

6. The areas of two circles differ by 35p in2. The sum of the circles' radii is 7 in. What are the two radii?

7. The height of a cylinder is 8 feet, and its surface area is 40p ft2. What is the cylinder's radius?

8. The volume of a sphere is
792
7
. Find the sphere's surface area. Use
22
7
for p.

9. A polygon has 176 more diagonals than sides. What is the sum of the measures of its interior angles?

10. In a cone, the height is
4
3
times the radius, and the surface area is 384p. What is the slant height of the cone?

## Word Problem Worksheet #4.3: Answer Key

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