Missing Dimensions and Quadrilateral Area Reasoning | Grade 6 Math | LevelKite

You already know how to find the area of a parallelogram, a trapezoid and a rhombus when you know its lengths. All three are quadrilaterals, which are figures with four straight sides. In this lesson you use the same area formulas in three new ways.

Work backward. You know the area and find a missing dimension. A dimension is a length that sets the size of a figure, such as a base, a height or a diagonal.
Solve a word problem. You pick the lengths you need out of a story.
Find a mistake. You check someone's area work and show where it went wrong.

The table below lists the formulas you will use. In each one, $A$ is the area in square units. Every other letter is a length in units.

Area formulas for this lesson

Figure	Lengths you need	Area formula
Parallelogram (rectangles, rhombuses and squares are parallelograms too)	A base $b$ and the height $h$ , measured at a right angle to that base	$A = b \cdot h$
Trapezoid (a quadrilateral with at least one pair of parallel sides)	A pair of parallel sides $b_1$ and $b_2$ , and the height $h$ between them, measured at a right angle to both	$A = \frac{1}{2}(b_1 + b_2) \cdot h$
Rhombus, when you know its diagonals	The diagonals $d_1$ and $d_2$ , the segments that join opposite corners	$A = \frac{1}{2} \cdot d_1 \cdot d_2$

Key concept

An area formula is an equation you can use three ways

An area formula is an equation: it says the area is equal to an expression built from the figure's lengths. It is true only when you put in the lengths it names. A height must be measured perpendicular to its base, which means at a right angle to it. A slanted side, one that does not meet the base at a right angle, is not a height.

Because the two sides of the equation are equal, you can use it three ways:

Put in the lengths to get the area.
Put in the area and every length but one. Then undo the operations one at a time. Do the same thing to both sides each time, and never divide by $0$ .
Compare someone's work with the equation, part by part, to find a mistake.

Start with what a missing-dimension problem asks. For a parallelogram, $A = b \cdot h$ says the area is the base times the height. If you know the area and the base, then the height is the number you multiply by the base to get the area. Division finds that number, because division undoes multiplication.

Why is that allowed? An equation says two amounts are equal. If you divide both amounts by the same number, as long as it is not $0$ , the results are still equal. The same holds when you multiply both amounts by the same number, or subtract the same number from both. A length is greater than $0$ , so dividing by a known length never means dividing by $0$ .

You may also replace an expression with its value. For example, $\frac{1}{2} \cdot 8$ is the number $4$ , so writing $4$ in its place does not change what the equation says.

So here is the plan: write the equation, put in what you know, and undo one operation at a time.

Worked example

A missing height in a parallelogram

A parallelogram has an area of $24$ square units. Its base is $6$ units long. What is its height $h$ , measured at a right angle to that base?

Write the area formula for a parallelogram. A is the area in square units, b is the base in units, and h is the height in units, measured at a right angle to the base.
$A = b \cdot h$
Put in the numbers you know: the area is 24 and the base is 6. Keep the letter h, because the height is the missing dimension.
$24 = 6 \cdot h$
Divide both sides by 6. This undoes the multiplication by 6. Dividing both sides by the same number, when that number is not 0, keeps them equal.
$24 \div 6 = (6 \cdot h) \div 6$
Do the arithmetic. On the left, 24 divided by 6 is 4. On the right, multiplying h by 6 and then dividing by 6 leaves h.
$4 = h$
Check by putting 4 back in for h. The base times the height must give the area you started with, and it does.
$6 \cdot 4 = 24$

Answer

The height of the parallelogram is $4$ units.

One parallelogram with a base of

6

units and a height of

4

units.

The figure shows one parallelogram that fits the example. The horizontal axis is $x$ and the vertical axis is $y$ . Each grid square is $1$ unit on a side. The vertices, or corners, are $(1, 1)$ , $(7, 1)$ , $(10, 5)$ and $(4, 5)$ .

The base lies on the line $y = 1$ , from $x = 1$ to $x = 7$ . Its length is $7 - 1 = 6$ units.
The opposite side lies on the line $y = 5$ , from $x = 4$ to $x = 10$ . Both of these sides are horizontal, so they are parallel.
Each of the other two sides goes $3$ units right and $4$ units up, so those two are parallel as well. The figure is a parallelogram.
Vertical grid lines meet the horizontal base at right angles. So the height is the vertical distance from $y = 1$ to $y = 5$ , which is $5 - 1 = 4$ units.

The grid agrees with $h = 4$ . Notice the order of the work. You named the height $h$ and solved for it first. The plot only confirmed your answer.

Worked example

A missing base in a trapezoid

A trapezoid has a pair of parallel sides, $b_1$ and $b_2$ . Its area is $40$ square centimeters, and $b_1$ is $6$ centimeters long. The height $h$ , measured at a right angle between those two sides, is $5$ centimeters. How long is $b_2$ ?

Write the area formula for a trapezoid: the area is one half of the sum of the two parallel sides, times the height measured at a right angle between them. A is the area in square centimeters, and the lengths are in centimeters.
$A = \frac{1}{2}(b_1 + b_2) \cdot h$
Put in the numbers you know: the area is 40, one parallel side is 6 and the height is 5. Keep the letter for the missing side.
$40 = \frac{1}{2}(6 + b_2) \cdot 5$
Multiply both sides by 2. This undoes the one half, because 2 times one half is 1. On the left, 2 times 40 is 80. Multiplying both sides by the same number keeps them equal.
$80 = (6 + b_2) \cdot 5$
Divide both sides by 5 to undo the multiplication by 5. On the left, 80 divided by 5 is 16. The right side is now just the sum of the two parallel sides.
$16 = 6 + b_2$
Subtract 6 from both sides to undo the addition of 6. On the left, 16 minus 6 is 10.
$10 = b_2$
Check by putting 10 back in for the missing side. The sum of the parallel sides is 16, half of 16 is 8, and 8 times 5 is 40, the given area.
$\frac{1}{2}(6 + 10) \cdot 5 = 8 \cdot 5 = 40$

Answer

The other parallel side, $b_2$ , is $10$ centimeters long.

The same plan finds any missing length. Each height is perpendicular to its base.

What you know	Equation	Undo	Missing dimension
Parallelogram: area $35$ square meters, height $5$ meters	$35 = b \cdot 5$	Divide both sides by $5$ : $35 \div 5 = 7$	Base $b = 7$ meters
Trapezoid: area $30$ square feet, parallel sides $4$ feet and $8$ feet	$30 = \frac{1}{2}(4 + 8) \cdot h$ . Since $\frac{1}{2}(4 + 8) = 6$ , this is $30 = 6 \cdot h$	Divide both sides by $6$ : $30 \div 6 = 5$	Height $h = 5$ feet
Rhombus: area $24$ square inches, one diagonal $8$ inches	$24 = \frac{1}{2} \cdot 8 \cdot d_2$ . Since $\frac{1}{2} \cdot 8 = 4$ , this is $24 = 4 \cdot d_2$	Divide both sides by $4$ : $24 \div 4 = 6$	Other diagonal $d_2 = 6$ inches

A word problem hides the dimensions inside a story. Before you calculate, answer three questions.

What shape is it? The shape tells you which area formula to use.
Which lengths does that formula need? For a parallelogram, find a base and the height perpendicular to it. For a trapezoid, find a pair of parallel sides and the perpendicular distance between them. Look for the words perpendicular and right angle.
What does the question ask for? It may ask for the area, for a missing length, or for something you work out from the area, such as how many bags of soil to buy.

A story may also give a length that the formula does not use, such as a slanted side. Leave that number out.

Worked example

Soil for a trapezoid-shaped flower bed

A flower bed is shaped like a trapezoid. Its two parallel edges are $6$ meters and $9$ meters long. A third edge is perpendicular to both parallel edges and is $4$ meters long. The fourth edge is slanted and is $5$ meters long. One bag of soil covers $5$ square meters. How many bags cover the whole bed?

Name the shape and the lengths it needs. The bed is a trapezoid, so you need its pair of parallel edges and the height between them. The parallel edges are 6 meters and 9 meters. The 4-meter edge is perpendicular to both, so the height is 4 meters. The slanted 5-meter edge is not used.
Write the trapezoid area formula with these lengths. A is the area of the bed in square meters.
$A = \frac{1}{2}(6 + 9) \cdot 4$
Add the parallel edges first: 6 plus 9 is 15. Then multiply in the easiest order. Half of 4 is 2, and 15 times 2 is 30. Changing the order of the factors does not change a product.
$A = \frac{1}{2} \cdot 15 \cdot 4 = 15 \cdot 2 = 30$
Answer the question that was asked. Each bag covers 5 square meters, so find how many groups of 5 square meters fit in 30 square meters. That is a division.
$30 \div 5 = 6$

Answer

The flower bed has an area of $30$ square meters, so $6$ bags of soil cover it.

The last job is to find a mistake in someone's area work. Redoing the problem can give you the right answer, but it does not show what went wrong. Instead, read the work one part at a time and ask four questions in this order.

Formula. Does the formula match the shape, with every part in place? The trapezoid formula needs its $\frac{1}{2}$ .
Lengths. Is each number in the right place? The two lengths added for a trapezoid must be a pair of parallel sides. The height must be perpendicular to the base, so a slanted side cannot be the height.
Arithmetic. Is every sum, product and half worked out correctly?
Units. Is the area given in square units?

A "no" marks a mistake. Fix that part, then redo everything that comes after it. The figure below shows the trapezoid for the next example. Its horizontal axis is $x$ and its vertical axis is $y$ .

A trapezoid with vertices

(1, 1)

,

(13, 1)

,

(10, 5)

and

(4, 5)

, split into a rectangle with a right triangle on each side.

Worked example

Find the mistake in a trapezoid's area

The figure above shows a trapezoid with vertices $(1, 1)$ , $(13, 1)$ , $(10, 5)$ and $(4, 5)$ , drawn as a rectangle with a right triangle on each side. Each grid square is $1$ centimeter (cm) on a side. The parallel sides are $12$ cm and $6$ cm long, and each of the other two sides is $5$ cm long. Ravi wrote: $A = \frac{1}{2}(12 + 6) \cdot 5 = 45$ square centimeters. Find Ravi's mistake and give the correct area.

Check the formula. Ravi used one half, times the sum of two sides, times one more length. That is the trapezoid formula, and the shape is a trapezoid. The formula passes.
$A = \frac{1}{2}(\text{sum of parallel sides}) \cdot \text{height}$
Check the two lengths that are added. The 12 cm side lies on the line y = 1 and the 6 cm side lies on the line y = 5. Both are horizontal, so they are a pair of parallel sides. Their lengths are correct too. These numbers pass.
$13 - 1 = 12 \quad \text{and} \quad 10 - 4 = 6$
Check the height. Ravi used 5, but 5 cm is the length of a slanted side. Look at the middle piece in the figure. Its sides lie along grid lines, so it is a rectangle and its upright sides are perpendicular to both parallel sides. Each upright side runs from y = 1 to y = 5, so the height is 4 cm. This is the mistake.
$h = 5 - 1 = 4$
Put in the correct height and redo the arithmetic. 12 plus 6 is 18, half of 18 is 9, and 9 times 4 is 36. The unit is square centimeters, which is right for an area.
$A = \frac{1}{2}(12 + 6) \cdot 4 = 9 \cdot 4 = 36$
Check with the pieces in the figure. The middle rectangle is 6 cm by 4 cm, so its area is 24 square centimeters. Each right triangle is half of a 3 cm by 4 cm rectangle, so its area is 6 square centimeters. The pieces add up to the same area.
$24 + 6 + 6 = 36$

Answer

Ravi used a slanted side, $5$ cm, as the height. The height is $4$ cm, so the correct area is $36$ square centimeters. His $45$ was too large because the slanted side is longer than the height.

Other mistakes on the same trapezoid, whose correct area is 36 square centimeters

Work shown	Check that fails	What went wrong
$A = (12 + 6) \cdot 4 = 72$ square centimeters	Formula	The $\frac{1}{2}$ is missing, so the result is twice the area.
$A = \frac{1}{2}(5 + 5) \cdot 4 = 20$ square centimeters	Lengths	The two $5$ cm sides lean in opposite directions, so they are not parallel and are not the lengths to add.
$A = \frac{1}{2} \cdot 18 \cdot 4 = 9 \cdot 2 = 18$ square centimeters	Arithmetic	The half was taken from both $18$ and $4$ . Take it from one factor only: $9 \cdot 4 = 36$ .
$A = \frac{1}{2}(12 + 6) \cdot 4 = 36$ centimeters	Units	The number is right, but area is measured in square centimeters.

You now have one tool that does three jobs. An area formula is an equation that links a figure's area to the lengths it names: a base and the height perpendicular to it, a pair of parallel sides and the perpendicular height between them, or the diagonals of a rhombus.

To find a missing dimension, put in what you know and undo one operation at a time. Do the same thing to both sides, and never divide by $0$ .
In a word problem, choose the lengths the formula needs, leave out the ones it does not, and answer the question that was asked.
To find a mistake, check the formula, the lengths, the arithmetic and the units.

Each time, put your result back into the equation. If the two sides are not equal, look for a slip in your work.