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.
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.
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
If you are missing two values in a formula, let x equal one of those values and try to write the other value in terms of x, so that only one variable is used. Remember, you cannot find the value of two variables if you have only one equation.
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.
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:
- Asquare = s2, where s is the length of one side of the square
- Arectangle = lw, where l is the length and w is the width of the rectangle
- Atriangle = , where b is the base and h is the height of the triangle
- Acircle = πr2, where r is the radius of the circle
- Vcube = e3, where e is the length of one edge of the cube
- Vrectangular prism = lwh, where l is the length, w is the width, and h is the height of the prism
- Vcylinder = πr2h, where r is the radius and h is the height of the cylinder
- Vcone = , where r is the radius and h is the height of the cone
- Vsphere = , where r is the radius of the sphere
If the length of one side of a square is (x + 8) units, what is the area of the square in terms of x?
The formula for area of a square is Asquare = s2, so we must square the length of one side of the square: (x + 8)2 = (x + 8)(x + 8). Use FOIL to find the area of the square:
- First: (x)(x) = x2
- Outside: (x)(8) = 8x
- Inside: (8)(x) = 8x
- Last: (8)(8) = 64
- x2 + 8x + 8x + 64 = x2 + 16x + 64.
The area of the square is x2 + 16x + 64 square units.
The height of a cone is twice its radius. If the volume of the cone is 18π cm3, what is the height of the cone?
Let x equal the radius of the cone. Because the height of the cone is twice the radius, the height is 2x. The formula for volume of a cone is Vcone = πr2h, so substitute 18π for V, x for r, and 2x for h:
- 27 = x3
- 3 = x
The radius is 3 centimeters. Because the height is twice the radius, the height of the cone is 6 centimeters.
Find practice problems and solutions for these concepts at Using Algebra in Geometry Practice Questions.
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