Rearrange a Parallelogram into a Rectangle | Grade 6 Math | LevelKite

A parallelogram is a four-sided shape with two pairs of parallel sides. Parallel sides always stay the same distance apart. A rectangle is one kind of parallelogram. Many other parallelograms lean to one side and have no right angles.

Area is the number of unit squares that cover a shape with no gaps and no overlaps. A unit square is $1$ unit long on each side. You already know that the area of a rectangle is its length times its width. In a leaning parallelogram, the slanted sides cut through the unit squares, so the squares are hard to count.

In this lesson you will cut a parallelogram into two pieces and slide one piece to make a rectangle. The pieces do not change, so the area does not change. That idea gives you a rule for the area of a parallelogram.

The figure below shows a parallelogram on a coordinate grid. The $x$ -axis is horizontal and the $y$ -axis is vertical. Each grid square is $1$ unit long on each side, so its area is $1$ square unit.

The parallelogram has corners at $(1, 1)$ , $(6, 1)$ , $(8, 4)$ and $(3, 4)$ . The figure shows it as two pieces that share an edge.

The bottom side runs from $(1, 1)$ to $(6, 1)$ . The top side runs from $(3, 4)$ to $(8, 4)$ . Both lie on horizontal grid lines, so they are parallel.
The left slanted side runs from $(1, 1)$ to $(3, 4)$ . The right one runs from $(6, 1)$ to $(8, 4)$ . Each goes $2$ units right and $3$ units up, so they are parallel too.

Choose the bottom side as the base. Its length is $b = 6 - 1 = 5$ units.

The shared edge runs along a vertical grid line from $(3, 4)$ on the top side down to $(3, 1)$ on the base. A vertical grid line is perpendicular to a horizontal one, so the length of this edge is the height: $h = 4 - 1 = 3$ units.

The piece on the left is a triangle with corners at $(1, 1)$ , $(3, 1)$ and $(3, 4)$ . The other piece has four sides.

Before the slide: a triangle and a four-sided piece make a parallelogram with base

5

units and height

3

units.

Now slide the triangle $5$ units to the right. That distance is the length of the base. The triangle's corners move from $(1, 1)$ , $(3, 1)$ and $(3, 4)$ to $(6, 1)$ , $(8, 1)$ and $(8, 4)$ .

A slide does not stretch or shrink a shape. The triangle still has a $2$ -unit side on a horizontal grid line and a $3$ -unit side on a vertical grid line. Those sides are perpendicular, so its area is still $\tfrac{1}{2} \times 2 \times 3 = 3$ square units.

The next figure shows the result. The triangle's slanted side now runs from $(6, 1)$ to $(8, 4)$ . That is also the slanted side of the four-sided piece. The two pieces lie on opposite sides of it, so they fit together with no gap and no overlap.

After the slide: the same two pieces make a rectangle

5

units long and

3

units wide.

Together the two pieces now make a shape with corners at $(3, 1)$ , $(8, 1)$ , $(8, 4)$ and $(3, 4)$ . Two of its sides lie on horizontal grid lines and two lie on vertical grid lines, so every corner is a right angle. The shape is a rectangle.

The rectangle's length is $8 - 3 = 5$ units and its width is $4 - 1 = 3$ units. Its area is $5 \times 3 = 15$ square units.

You did not add or remove any area. The same two pieces make up the parallelogram, so its area is also $15$ square units.

Notice the match, and why it happens. The slide took $2$ units off the left end of the bottom edge and added $2$ units to the right end, so the bottom edge is still $b = 5$ units long. The top and bottom edges stayed on the same grid lines, so the distance between them is still $h = 3$ units.

Write $A$ for the area of the parallelogram. Then $A = b \times h = 5 \times 3 = 15$ square units.

Key concept

Area of a parallelogram

A parallelogram has the same area as a rectangle whose length is the parallelogram's base and whose width is the parallelogram's height. So the area is the base times the height.

Here $A$ is the area, $b$ is the length of the side you choose as the base, and $h$ is the height. The height must be the perpendicular distance between that base and the opposite side. Measure $b$ and $h$ in the same unit of length. Then $A$ is in square units of that length.

The rule works for every parallelogram. A parallelogram that leans very far may need more than one cut, but it still makes a rectangle with the same base and height. A rectangle is a parallelogram whose height is one of its sides, so the rule gives length times width.

A = b \times h

Worked example

Use a given base and height

A parallelogram has a base of $9$ cm. Its height, measured perpendicular to that base, is $4$ cm. What is its area?

Name the two measurements. The base is 9 cm. The height is 4 cm, and it is perpendicular to the base, so the area rule applies.
$b = 9 \text{ cm}, \quad h = 4 \text{ cm}$
Write the rule. The parallelogram has the same area as a rectangle 9 cm long and 4 cm wide, so multiply the base by the height.
$A = b \times h$
Replace each letter with its value and multiply. Centimeters times centimeters gives square centimeters.
$A = 9 \text{ cm} \times 4 \text{ cm} = 36 \text{ cm}^2$

Answer

The area of the parallelogram is $36$ square centimeters.

The parallelogram for the next example has corners at

(1, 1)

,

(7, 1)

,

(10, 5)

and

(4, 5)

, and its two pieces share an edge on a vertical grid line.

Worked example

Read the base and height from a grid

The figure above shows a parallelogram with corners at $(1, 1)$ , $(7, 1)$ , $(10, 5)$ and $(4, 5)$ . Its two pieces share the edge from $(4, 1)$ to $(4, 5)$ . Each grid square is $1$ unit long on each side, and each slanted side is $5$ units long. Find the area of the parallelogram.

Choose the bottom side as the base. It lies on a horizontal grid line from (1, 1) to (7, 1), so subtract the x-coordinates, which are the first numbers in each pair.
$b = 7 - 1 = 6$
Find the height. The shared edge runs along a vertical grid line from (4, 5) on the top side down to (4, 1) on the base. Vertical grid lines are perpendicular to horizontal ones, so the length of this edge is the height. Subtract the y-coordinates, which are the second numbers.
$h = 5 - 1 = 4$
Leave out the 5. It is the length of a slanted side. A slanted side is not perpendicular to the base, so its length is not the height.
Multiply the base by the height, because the parallelogram has the same area as a rectangle 6 units long and 4 units wide. Units times units gives square units.
$A = b \times h = 6 \times 4 = 24$

Answer

The area of the parallelogram is $24$ square units.

Do not use a slanted side as the height. In the last example, you might multiply the two side lengths, $6 \times 5 = 30$ , because that works for a rectangle. But the area is $24$ square units, not $30$ .

Here is why that fails. Slide the triangle with corners at $(1, 1)$ , $(4, 1)$ and $(4, 5)$ to the right by $6$ units. The two pieces make a rectangle with corners at $(4, 1)$ , $(10, 1)$ , $(10, 5)$ and $(4, 5)$ . The two slanted sides now lie together inside the rectangle, so neither one is a side of it. The rectangle's side lengths are the base, $6$ units, and the height, $4$ units. In a parallelogram, the lengths of two sides that meet at a corner multiply to give the area only when those sides are perpendicular, as in a rectangle.

Worked example

Match the height to its base

A parallelogram has two sides that are $7.5$ m long and two sides that are $4.6$ m long. The perpendicular distance between the two $7.5$ m sides is $4$ m. Find the area of the parallelogram.

Match the height to its base. The 4 m distance runs at a right angle between the two 7.5 m sides, so a 7.5 m side is the base and 4 m is the height. The 4.6 m sides are longer than that distance, so they are slanted and you do not use their length.
$b = 7.5 \text{ m}, \quad h = 4 \text{ m}$
Multiply the base by the height. Split 7.5 into 7 + 0.5. Splitting does not change its value, so multiplying each part by 4 and adding gives the same product.
$7.5 \times 4 = (7 \times 4) + (0.5 \times 4) = 28 + 2 = 30$
Write the area with its unit. Meters times meters gives square meters.
$A = 7.5 \text{ m} \times 4 \text{ m} = 30 \text{ m}^2$

Answer

The area of the parallelogram is $30$ square meters.

Takeaway. You can cut a parallelogram and rearrange the pieces into a rectangle with the same base and the same height. Rearranging does not change area, so the area of a parallelogram is its base times its height: $A = b \times h$ . Use the height that is perpendicular to the base you chose, not the length of a slanted side.

Next, you will find the areas of trapezoids, which are four-sided shapes with at least one pair of parallel sides, and rhombuses, which are parallelograms with four equal sides.