Using Algebra in Geometry Study Guide (page 2)

Introduction to Using Algebra in Geometry

A circle is the longest distance to the same point.

—Tom Stoppard (1937– ) British Playwright

In this lesson, you'll learn how to use algebra to find the perimeter, area, and volume of two– and three–dimensional figures.

Geometry is filled with formulas—perimeter of a square, area of a triangle, volume of a sphere—and any time you have a formula and an unknown value you can use algebra, even in geometry, to find that value.

Perimeter

The perimeter of a square is equal to 4 times the length of one side of a square: P = 4s. If we know the length of one side, we can substitute it for s to find P. If we know the perimeter, we can divide by 4 to find the length of one side. A square that has a perimeter of 56 inches has sides that each measure 56 ÷ 4 = 14 inches.

If the length of one side of a square is an algebraic expression, we can express its perimeter by multiplying that expression by 4. A square whose sides each measure 2x inches has a perimeter of 4(2x) = 8x inches.

Example

A square has a perimeter of 100 yards. If the length of one side is equal to (6x + 1) yards, what is the value of x?

We use the formula P = 4s, and replace P with 100 and s with 6x + 1:

100 = 4(6x + 1)
100 = 24x + 4
96 = 24x
x = 4

The perimeter of a rectangle is equal to twice its length plus twice its width: P = 2l + 2w. The length and width of a rectangle might be given to us as algebraic expressions.

Example

The length of a rectangle is 5 centimeters more than 3 times its width. If the perimeter of the rectangle is 82 centimeters, what is the length of the rectangle?

We don’t know the value of either the width or the length of the rectangle. But, if we let x represent the width, we can represent the length as 3x + 5, because the length is 5 more than 3 times the width. Now, we can substitute these values into the formula for the perimeter of a rectangle:

P = 2l + 2w
82 = 2(3x + 5) + 2x
82 = 6x + 10 + 2x
82 = 8x + 10
72 = 8x
9 = x

Because x represents the width of the rectangle, the width of the rectangle is 9 centimeters. The length is 5 centimeters more than 3 times the width: 3(9) + 5 = 27 + 5 = 32 centimeters.

Area and Volume

We can perform the same sort of substitutions for area and volume formulas to find the length of a side or an edge of a solid. The following are some common area and volume formulas:

A_square = s², where s is the length of one side of the square

A_rectangle = lw, where l is the length and w is the width of the rectangle

A_triangle = , where b is the base and h is the height of the triangle

A_circle = πr², where r is the radius of the circle

V_cube = e³, where e is the length of one edge of the cube

V_{rectangular prism} = lwh, where l is the length, w is the width, and h is the height of the prism

V_cylinder = πr²h, where r is the radius and h is the height of the cylinder

V_cone = , where r is the radius and h is the height of the cone

V_sphere = , where r is the radius of the sphere