This method will only work if you are given the area and length of the rectangle. You might also see the formula written as A=(h)(w){\displaystyle A=(h)(w)}, where h{\displaystyle h} equals the height of the rectangle and is used instead of length. [2] X Research source These two terms refer to the same measurement.
For example, if you are trying to find the width of a rectangle that has an area of 24 square centimeters, and a length of 8 centimeters, your formula will look like this:24=8w{\displaystyle 24=8w}
For example, in the equation 24=8w{\displaystyle 24=8w}, you would divide each side by 8. 24=8w{\displaystyle 24=8w}248=8w8{\displaystyle {\frac {24}{8}}={\frac {8w}{8}}}3=w{\displaystyle 3=w}
For example, for a rectangle with an area of 24cm2{\displaystyle 24cm^{2}} and a length of 8cm{\displaystyle 8cm}, the width would be 3cm{\displaystyle 3cm}.
This method will only work if you are given the perimeter and length of the rectangle. You might also see the formula written as P=2(w+h){\displaystyle P=2(w+h)}, where h{\displaystyle h} equals the height of the rectangle and is used instead of length. [5] X Research source The variables l{\displaystyle l} and h{\displaystyle h} refer to the same measurement, and the distributive property dictates that these two formulas, although arranged differently, will give you the same result.
For example, if you are trying to find the width of a rectangle that has a perimeter of 22 centimeters, and a length of 8 centimeters, your formula will look like this:22=2(8)+2w{\displaystyle 22=2(8)+2w}22=16+2w{\displaystyle 22=16+2w}
For example, in the equation 22=16+2w{\displaystyle 22=16+2w}, you would subtract 16 from each side, then divide by 2. 22=16+2w{\displaystyle 22=16+2w}6=2w{\displaystyle 6=2w}62=2w2{\displaystyle {\frac {6}{2}}={\frac {2w}{2}}}3=w{\displaystyle 3=w}
For example, for a rectangle with a perimeter of 22cm{\displaystyle 22cm} and a length of 8cm{\displaystyle 8cm}, the width would be 3cm{\displaystyle 3cm}.
This method will only work if you are given the length of the diagonal and the length of the side of the rectangle. You might also see the formula written as D=w2+h2{\displaystyle D={\sqrt {w^{2}+h^{2}}}}, where h{\displaystyle h} equals the height of the rectangle and is used instead of length. [8] X Research source The variables l{\displaystyle l} and h{\displaystyle h} refer to the same measurement.
For example, if you are trying to find the width of a rectangle that has a diagonal length of 5 centimeters, and a side length of 4 centimeters, your formula will look like this: 5=w2+42{\displaystyle 5={\sqrt {w^{2}+4^{2}}}}
For example:5=w2+42{\displaystyle 5={\sqrt {w^{2}+4^{2}}}}52=w2+42{\displaystyle 5^{2}=w^{2}+4^{2}}25=w2+16{\displaystyle 25=w^{2}+16}
For example, in the equation 25=16+w2{\displaystyle 25=16+w^{2}}, you would subtract 16 from each side. 25=16+w2{\displaystyle 25=16+w^{2}}9=w2{\displaystyle 9=w^{2}}
For example:9=w2{\displaystyle {\sqrt {9}}={\sqrt {w^{2}}}}3=w{\displaystyle 3=w}
For example, for a rectangle with a diagonal length of 5cm{\displaystyle 5cm} and a side length of 4cm{\displaystyle 4cm}, the width would be 3cm{\displaystyle 3cm}.
If you do not know the area or the perimeter, or the relationship between the length and the width, you cannot use this method. The formula for area is A=(l)(w){\displaystyle A=(l)(w)}. [10] X Research source The formula for perimeter is P=2l+2w{\displaystyle P=2l+2w}. [11] X Research source For example, you might know that the area of a rectangle is 24 square centimeters, so you would set up the formula for area of a rectangle.
The relationship might be given by stating how many times bigger one side is than the other, or how many units more or less it is. For example, you might know that the length is five centimeters longer than the width. Your expression for the length is then l=w+5{\displaystyle l=w+5}.
For example, if you know that area is 24 square centimeters, and that l=w+5{\displaystyle l=w+5}, your formula will look like this:A=(l)(w){\displaystyle A=(l)(w)}24=(w+5)(w){\displaystyle 24=(w+5)(w)}
For example, you can simplify 24=(w+5)(w){\displaystyle 24=(w+5)(w)} to 0=w2+5w−24{\displaystyle 0=w^{2}+5w-24}.
You might need to use addition or division to solve, or you might need to factor a quadratic equation or use the quadratic formula to solve. For example, 0=w2+5w−24{\displaystyle 0=w^{2}+5w-24} can be factored as follows:0=w2+5w−24{\displaystyle 0=w^{2}+5w-24}0=(w+8)(w−3){\displaystyle 0=(w+8)(w-3)}You will then have two possible solutions for w{\displaystyle w}: w=3{\displaystyle w=3} or w=−8{\displaystyle w=-8}. Since a rectangle cannot have a negative width, you can eliminate -8. So your solution is w=3{\displaystyle w=3}.
