Finding Missing Dimensions and Solving Word Problems | Grade 7 Math | LevelKite

You already know how to use a formula (a rule written with letters and symbols) to find the perimeter or the area of a polygon, a closed flat shape with straight sides. The perimeter is the total distance around the shape. The area is the amount of flat surface inside it.

In this lesson you work the other way. You know the area or the perimeter, and one dimension is missing. A dimension is a length that describes the size of a shape, such as a length, a width, a base or a height.

Here is the idea with small numbers. A rectangle is $8\text{ cm}$ long and has an area of $40\text{ cm}^2$ . Let the variable $w$ , a letter that stands for a number, be its width in centimeters. The area of a rectangle is its length times its width, with both in the same unit, so $8 \times w = 40$ . You are asking, 8 times what number gives 40? Division undoes multiplication, so it answers that question: $40 \div 8 = 5$ . The width is $5\text{ cm}$ . Finding the area goes forward with multiplication. Finding the missing width goes backward with division.

\text{Forward: } 8 \times 5 = 40 \qquad \text{Backward: } 40 \div 8 = 5

Formulas you already know. Each one needs positive measures and one consistent unit of length.

Shape	Measure	What the letters mean	Formula
Any polygon	Perimeter	$P$ is the perimeter	$P$ = all the side lengths added together
Rectangle	Perimeter	$P$ is the perimeter, $l$ is the length, $w$ is the width	$P = 2l + 2w$
Rectangle	Area	$A$ is the area, $l$ is the length, $w$ is the width	$A = lw$
Triangle	Area	$A$ is the area, $b$ is the base (one side), $h$ is the height, perpendicular (at a right angle) to the base	$A = \frac{1}{2}bh$
Trapezoid	Area	$A$ is the area, $b_1$ and $b_2$ are the two parallel sides (bases), $h$ is the perpendicular distance between them	$A = \frac{1}{2}(b_1 + b_2)h$

Key concept

A formula is an equation you can solve backward

An area or perimeter formula is an equation, a statement that two amounts are equal. It links the dimensions of a shape to its area or perimeter. When you know every quantity in the formula except one, substitute: replace each letter with its known value. Then undo the operations one at a time (for example, divide to undo a multiplication) until the unknown dimension stands alone.

This works for every formula in the table of formulas when all measures are positive, all lengths are in one consistent unit, and any height is perpendicular to its base. The dimension you find is a length, so it must be positive too.

Use these five steps to find a missing dimension. They hold under the conditions already stated: positive measures, one consistent unit, and a height perpendicular to its base.

Name the unknown. Choose a variable. Say what it measures and in which unit.
Write the formula that links the measures you know to the unknown.
Substitute the known values.
Undo the operations. Use inverse operations, which are operations that undo each other: subtraction undoes addition, and division undoes multiplication. Do the same thing to both sides.
Check. Put your answer back into the formula. It must return the given area or perimeter, and it must be a positive length.

Why is step 4 allowed? The two sides of an equation are equal amounts. If you subtract the same number from both sides, or divide both sides by the same nonzero number, the results are still equal. Every number you divide by in this lesson is positive, so it is never zero.

Worked example

Missing width from a perimeter

A rectangle has a perimeter of $46\text{ cm}$ and a length of $15\text{ cm}$ . Find its width.

Name the unknown. Let w be the width of the rectangle in centimeters. A width is a length, so w must be positive.
$w > 0$
Write the perimeter formula for a rectangle, then substitute 46 for the perimeter P and 15 for the length l. Both are in centimeters, so the units are consistent. Each letter is replaced by the number it stands for, so the equation still describes this rectangle.
$P = 2l + 2w \quad\Rightarrow\quad 46 = 2(15) + 2w$
Multiply 2 by 15 to get 30. Writing 30 in place of 2 times 15 does not change the value of the right side. The 30 is the part of the perimeter made by the two lengths.
$2 \times 15 = 30 \quad\Rightarrow\quad 46 = 30 + 2w$
Subtract 30 from both sides to undo the addition. Subtracting the same number from both sides keeps them equal. The 16 that is left is the total of the two widths.
$46 - 30 = 30 + 2w - 30 \quad\Rightarrow\quad 16 = 2w$
Divide both sides by 2 to undo the multiplication. Because 2 is not zero, dividing both sides by it keeps them equal. Each width is half of 16.
$16 \div 2 = 2w \div 2 \quad\Rightarrow\quad 8 = w$
Check. Substitute 8 for w in the formula, with both lengths in centimeters. It returns the given perimeter, and 8 is positive, so it can be a width.
$2(15) + 2(8) = 30 + 16 = 46$

Answer

The width of the rectangle is $8\text{ cm}$ .

Worked example

Missing height from the area of a triangle

A triangle has an area of $54\text{ cm}^2$ . One side, chosen as the base, is $12\text{ cm}$ long. Find the height of the triangle that is perpendicular to this base.

Name the unknown. Let h be the height in centimeters, measured at a right angle to the 12 cm base. A height is a length, so h must be positive.
$h > 0$
Write the area formula for a triangle, then substitute 54 for the area A and 12 for the base b. The area is in square centimeters and the base is in centimeters, so the units are consistent.
$A = \frac{1}{2}bh \quad\Rightarrow\quad 54 = \frac{1}{2}(12)h$
Multiply one half by 12 to get 6. Writing 6 in its place does not change the value of the right side. The equation now says that 6 times the height is 54.
$\frac{1}{2} \times 12 = 6 \quad\Rightarrow\quad 54 = 6h$
Divide both sides by 6 to undo the multiplication. Because 6 is not zero, the two sides stay equal.
$54 \div 6 = 6h \div 6 \quad\Rightarrow\quad 9 = h$
Check. Substitute 9 for h and 12 for b, both in centimeters. The formula returns the given area, and 9 is positive, so it can be a height. Centimeters times centimeters gives square centimeters, the unit of the given area.
$\frac{1}{2}(12)(9) = \frac{1}{2}(108) = 54$

Answer

The height perpendicular to the $12\text{ cm}$ base is $9\text{ cm}$ .

A word problem describes a situation and leaves it to you to decide which measure it is about. Decide by asking what is being measured.

The total distance around the whole edge of a flat shape is its perimeter. The answer is in linear units.
The amount of flat surface inside the edge is its area. The answer is in square units.

Base your choice on what the situation measures, not on a single word in the problem. Then use the same five steps, with every length positive and in one unit. Some problems give every dimension and ask for one measure. Others need two formulas: one measure gives you a missing dimension, and that dimension gives you the other measure.

One shape, two different questions

Situation	What is measured	Measure	Unit of the answer
Trim that goes all the way around a rectangular window	Total distance around the edge	Perimeter	Linear, such as cm
Glass that fills the same rectangular window	Flat surface inside the edge	Area	Square, such as $\text{cm}^2$
Ribbon sewn along every edge of a flat triangular flag	Total distance around the edge	Perimeter	Linear, such as cm
Cloth that makes up the same triangular flag	Flat surface inside the edge	Area	Square, such as $\text{cm}^2$

Worked example

A fence around a garden

A rectangular vegetable garden covers $72\text{ m}^2$ of ground and is $6\text{ m}$ wide. A fence will go all the way around the garden, with no gaps and no overlap. How many meters of fence are needed?

Decide what each fact measures. The ground the garden covers is its area, 72 square meters. The fence goes around the whole edge, so its total length is the perimeter. The perimeter formula needs the length of the garden, which is missing.
Let the letter l (for length) be the length of the garden in meters. It is a length, so l must be positive. Write the area formula for a rectangle, then substitute 72 for the area A and 6 for the width w. The width is in meters and the area is in square meters, so the units are consistent.
$A = lw \quad\Rightarrow\quad 72 = l \times 6$
Divide both sides by 6 to undo the multiplication. Because 6 is not zero, the two sides stay equal. The garden is 12 m long, a positive length.
$72 \div 6 = l \times 6 \div 6 \quad\Rightarrow\quad 12 = l$
Now find the perimeter P in meters. Use the rectangle perimeter formula with 12 for the length and 6 for the width, both in meters. Multiply first, then add.
$P = 2(12) + 2(6) = 24 + 12 = 36$
Check. The length you found gives back the given area. The answer 36 is positive and in meters, a linear unit, which fits a length of fence.
$12 \times 6 = 72$

Answer

The garden needs $36\text{ m}$ of fence.

Takeaway. An area or perimeter formula works in both directions. When one dimension is missing, name it with a variable, substitute what you know, and undo the operations while you keep both sides equal. In a word problem, first decide what the situation measures: the distance around the edge is a perimeter in linear units, and the surface inside is an area in square units. These steps hold when all measures are positive, all lengths share one unit, and any height is perpendicular to its base. Finish every problem by putting your answer back into the formula.