Area of Composite Figures: Rectangles and Triangles | Grade 7 Math | LevelKite

You already know how to use a formula to find the area of one familiar shape, such as a circle. Many shapes are not that simple, though.

A composite figure is a shape made of two or more simpler shapes joined together. To decompose a composite figure means to cut it into simpler shapes whose areas you can find, such as rectangles and triangles.

Why does cutting help? Area is the amount of flat surface a shape covers. A cut does not add any surface or take any away. So when the parts cover the whole figure with no gaps and no overlaps, and you measure every part in the same square unit, the areas of the parts add up to the area of the whole figure.

Look at the figure below. The horizontal axis is the $x$ -axis and the vertical axis is the $y$ -axis. The grid lines are $1$ apart, and the scale is $1$ grid step to $1$ centimeter (cm). So each grid square is $1$ cm wide and $1$ cm tall and covers $1$ square centimeter ( $\text{cm}^2$ ).

The figure has five corners: $(1, 1)$ , $(7, 1)$ , $(7, 5)$ , $(4, 9)$ and $(1, 5)$ . You have no formula for a shape like this, but you can read some of its lengths from the corners:

A horizontal side has the same $y$ -coordinate at both ends. Its length is the number of grid steps between its ends, so subtract the $x$ -coordinates, larger minus smaller. The bottom side, from $(1, 1)$ to $(7, 1)$ , is $7 - 1 = 6$ cm long.
A vertical side has the same $x$ -coordinate at both ends, so subtract the $y$ -coordinates, larger minus smaller. The right side, from $(7, 1)$ to $(7, 5)$ , is $5 - 1 = 4$ cm long.

Larger minus smaller keeps every length positive. You cannot find the length of a slanted side this way. If you need a slanted length, the problem will give it to you.

The five-sided figure with corners

(1, 1)

,

(7, 1)

,

(7, 5)

,

(4, 9)

and

(1, 5)

. Each grid step is

1

cm.

Worked example

Example 1: Decompose the five-sided figure

Decompose the five-sided figure above into a rectangle and a triangle. Its corners are $(1, 1)$ , $(7, 1)$ , $(7, 5)$ , $(4, 9)$ and $(1, 5)$ , and the scale is $1$ grid step to $1$ cm. Find the lengths you need for the area of each part.

Look for a straight cut that leaves shapes you know. The corners (1, 5) and (7, 5) have the same y-coordinate, so the segment joining them is horizontal. Cut along it. This cut puts both slanted sides in one part and leaves only horizontal and vertical sides in the other.
Name the part below the cut. Its corners are (1, 1), (7, 1), (7, 5) and (1, 5). Every side is horizontal or vertical, so the sides meet at right angles and the part is a rectangle. Subtract x-coordinates to find how wide it is and y-coordinates to find how tall it is.
$7 - 1 = 6 \text{ cm wide}, \qquad 5 - 1 = 4 \text{ cm tall}$
Name the part above the cut. It has three corners, (1, 5), (7, 5) and (4, 9), so it is a triangle. Choose the cut as its base. The cut is horizontal, so its length is the difference of the x-coordinates of its ends.
$\text{base} = 7 - 1 = 6 \text{ cm}$
Find the height for that base. Go from the opposite corner (4, 9) straight down to the point (4, 5) on the base. This path is vertical and the base is horizontal, so they meet at a right angle and the path is perpendicular to the base. The path is not a side of the triangle, so find its length by subtracting the y-coordinates of its ends.
$\text{height} = 9 - 5 = 4 \text{ cm}$
Check the parts. The rectangle and the triangle share only the cut, so they do not overlap. Together they fill the whole five-sided figure, so they leave no gaps.

Answer

The figure decomposes into a rectangle $6$ cm wide and $4$ cm tall and a triangle with a base of $6$ cm and a perpendicular height of $4$ cm. The next figure shows the two parts.

The same figure cut from

(1, 5)

to

(7, 5)

: a rectangle below the cut and a triangle above it. Each grid step is

1

cm.

Key concept

Decompose, find each area, then add

To find the area of a composite figure, decompose it into rectangles and triangles, find the area of each part, and add those areas.

The sum is the area of the whole figure when all of these are true:

The parts cover the whole figure with no gaps and no overlaps.
Every length is positive and in one unit, such as centimeters, so every area is in the matching square unit.
For each triangle, the height is perpendicular to the base. A side that is not perpendicular to the base is not the height.

On a grid, try cuts along horizontal or vertical lines through corners. Then you can read the lengths you need from the coordinates.

\text{area of the figure} = \text{sum of the areas of its parts}

Worked example

Example 2: Find the area of the five-sided figure

Find the area of the five-sided figure from Example 1. Its parts are a rectangle $6$ cm wide and $4$ cm tall and a triangle with a base of $6$ cm and a perpendicular height of $4$ cm.

Find the area of the rectangle. Its 6 cm side and its 4 cm side meet at a corner, so multiply them. On the grid, the rectangle covers 4 rows of 6 squares, and each square is 1 square centimeter.
$A_{\text{rectangle}} = 6 \text{ cm} \times 4 \text{ cm} = 24 \text{ cm}^2$
Find the area of the triangle. The 4 cm height is perpendicular to the 6 cm base, so the triangle formula applies. Multiply the base by the height to get the area of a 6 cm by 4 cm rectangle, then take half of it.
$A_{\text{triangle}} = \frac{1}{2} \times 6 \text{ cm} \times 4 \text{ cm} = \frac{1}{2} \times 24 \text{ cm}^2 = 12 \text{ cm}^2$
Add the two areas. You can add them because the rectangle and the triangle cover the whole figure with no gaps and no overlaps, and both areas are in square centimeters.
$A_{\text{figure}} = 24 \text{ cm}^2 + 12 \text{ cm}^2 = 36 \text{ cm}^2$

Answer

The area of the five-sided figure is $36$ square centimeters.

A six-sided figure cut from

(5, 1)

to

(5, 4)

and from

(1, 6)

to

(5, 6)

into two rectangles and a triangle. Each grid step is

1

cm.

Worked example

Example 3: A figure with three parts

The six-sided figure above has corners $(1, 1)$ , $(10, 1)$ , $(10, 4)$ , $(5, 4)$ , $(5, 6)$ and $(1, 9)$ . The scale is $1$ grid step to $1$ cm. Two cuts split the figure: one from $(5, 1)$ to $(5, 4)$ and one from $(1, 6)$ to $(5, 6)$ . Find its area.

The vertical cut drops from the corner (5, 4) to the bottom side. The horizontal cut runs from the corner (5, 6) across to the left side. They leave a tall rectangle on the left, a low rectangle on the right and a triangle on top. The parts share only the cuts and fill the figure, so there are no overlaps and no gaps.
Find the area of the tall rectangle, with corners (1, 1), (5, 1), (5, 6) and (1, 6). Use its own corners: subtract x-coordinates to find how wide it is and y-coordinates to find how tall it is.
$(5 - 1) \text{ cm} \times (6 - 1) \text{ cm} = 4 \text{ cm} \times 5 \text{ cm} = 20 \text{ cm}^2$
Find the area of the low rectangle, with corners (5, 1), (10, 1), (10, 4) and (5, 4). Its left side is at x = 5, not at x = 1, so subtract 5 to find how wide it is. Its top is at y = 4 and its bottom is at y = 1.
$(10 - 5) \text{ cm} \times (4 - 1) \text{ cm} = 5 \text{ cm} \times 3 \text{ cm} = 15 \text{ cm}^2$
Find the area of the triangle, with corners (1, 6), (5, 6) and (1, 9). Its base runs from (1, 6) to (5, 6), so it is 5 - 1 = 4 cm long. The vertical side from (1, 6) to (1, 9) meets the base at a right angle, so that side is the perpendicular height: 9 - 6 = 3 cm. You do not need the slanted side.
$\frac{1}{2} \times 4 \text{ cm} \times 3 \text{ cm} = \frac{1}{2} \times 12 \text{ cm}^2 = 6 \text{ cm}^2$
Add the three areas. All three are in square centimeters, and the parts cover the figure with no gaps and no overlaps.
$20 \text{ cm}^2 + 15 \text{ cm}^2 + 6 \text{ cm}^2 = 41 \text{ cm}^2$

Answer

The area of the six-sided figure is $41$ square centimeters.

Check with a different cut. You can often decompose a figure in more than one way. In Example 3, remove the cut from $(5, 1)$ to $(5, 4)$ and cut from $(1, 4)$ to $(5, 4)$ instead. Keep the cut under the triangle.

The bottom rectangle now has corners $(1, 1)$ , $(10, 1)$ , $(10, 4)$ and $(1, 4)$ . It is $10 - 1 = 9$ cm wide and $4 - 1 = 3$ cm tall, so its area is $9 \times 3 = 27$ square centimeters.
The rectangle above it has corners $(1, 4)$ , $(5, 4)$ , $(5, 6)$ and $(1, 6)$ . It is $5 - 1 = 4$ cm wide and $6 - 4 = 2$ cm tall, so its area is $4 \times 2 = 8$ square centimeters.
The triangle does not change. Its area is still $6$ square centimeters.

The total is $27 + 8 + 6 = 41$ square centimeters, the same as before. The parts changed, but the surface they cover did not. Both sets of parts cover the same figure with no gaps and no overlaps, so they must give the same total. If your two totals ever differ, at least one of them has a mistake.

Takeaway: You may not have a formula for a whole composite figure, but you do have formulas for its parts. Decompose the figure into rectangles and triangles that cover it with no gaps and no overlaps. Read each part's own lengths from the grid in one unit, use a height that is perpendicular to the base for every triangle, and add the areas in square units.

Next, you will use the same idea to find the area of composite figures that include semicircles and quarter circles.