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Calculating the Area
of 2-D Composite Shapes
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1. Our Training Videos
Skills Application: Calculating the Area of Triangles
Step-by-Step: Calculating the Area of Triangles
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Sheet metal workers often custom-fabricate square-to-round transitions to joint ductwork of different pipes. They start with a flat sheet of metal on which they carefully lay out the square and round ends and the locations of the fold lines. They also calculate the area of sheet metal used for pricing out the finished product

## Running with Pi

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3. Our Training Worksheets

Area can be calculated by breaking up a composite shape into simple shapes and adding up their areas. Watch the video to see calculating the area of a feature wall.

Parallelograms, Trapezoids, Hexagons, Octagons and other geometric shapes can be broken up into triangles and rectangles. Their areas can also be calculated by adding up the areas of those triangles and rectangles. See Episodes 19 & 21 on area of rectangles and triangles.

Sometimes the above lengthy approach can be shortened. You can find formulas below that condense the area calculations into 1 short formula.

Parallelogram
A parallelogram is a skewed rectangle. Base (B) is measured along its long side, height (h) is measured inside or outside the shape perpendicularly to the base.

A = B ∙ h

Example:
The area (A) of a parallelogram with a base B = 5 cm, and a height h = 4 cm is

A = 5  x  4 = 20 cm2

Trapezoid
A trapezoid has a shorter base (b) and a longer base (B) and a height (h) that is measured inside or outside the shape perpendicularly to the base.

$\text{A = }\frac{B + b}{2}\cdot{h}$

Example:
The area (A) of a trapezoid with a long base B = 6 cm, a short base b = 4 cm and a height h = 3 cm is

$\text{A = }\frac{6 + 4}{2}\times{3}\text{ = 15 cm}{^2}$

Hexagon
A hexagon is any six-sided closed figure. The hexagons used most often in building construction are regular hexagons.

Area of a Regular Hexagon with side "a":

${A =}\frac{3 \sqrt3}{2}{a^2}$
or  A ≈ 2.598a2

Example:
The area of a hexagon with side = 4 centimeters is

$\text{A = }\frac{3 \sqrt3}{2}{ 4^2}$
= 2.598 x 16 ≈ 41.57 cm2

Octagon
An octagon is any eight-sided closed figure. The octagons used most often in building construction are regular octagons. A simple example of a regular octagon is STOP sign. The area of a regular octagon can be found by breaking it into eight identical isosceles triangles; however, we will use one of the following shortcut formula.

Regular Octagon
Area of a Regular Octagon with side “a”:

$\text{A = 2(1 +}\sqrt2\text{)}{a^2}$

Example
The area of an octagon with side 2 inches is:

$\text{A = 2(1 + }\sqrt2\text{)}{a^2}$
= 2(1+
$\sqrt2$
)×4 ≈ 19.2 in2

Note, that
$\sqrt2$
≈ 1.41

Check out the ADVANCED tab. The exercise will further your understanding and help with building a bigger mental picture. It builds links between numbers and algebra. Take your time and make the comparisons. It is important to cross-link the concepts of area and numbers.

### A Short Review

Composite shapes include:

 parallelogram A = Base x Height (same as a rectangles Area) trapezoid A = $\frac{B + b}{2}$ x Height (2 bases are the 2 parallel sides) regular hexagon A = 2.598a2 (a = side length of hexagon) regular octagon A = 4.828a2 (a = side length of octagon)

Other complex shapes can be broken down into simple shapes such as triangles, circles and rectangles. (See those tabs on this site)

These all need to be calculated separately, then the relevant numbers for area all added up.

Due to the many steps involved, some complex shape calculations can take 25 minutes or an hour.

### Worksheet: Level 1

The operations used are clearly specified. Only one type of mathematical operation is used in a task.

### Worksheet: Level 1 Sample Questions

From each of the pictures below write an addition sentence with the given measurements.

This addition sentence represents the whole area of the composite shape.

Example:

Further Example:

Draw a circle with a square inside it anywhere. Cross-hatch or color in the area that shows the Area of the remaining complex shape if the circle is made of paper and the square is cut out from the circle.

### Worksheet: Level 2

Tasks involve one or two types of mathematical operation. Few steps of calculation are required.

### Worksheet: Level 2 Sample Questions

Calculate the following:

Remember: Area of a composite shape is calculated by adding up the area subtotals of their sub-shapes.

Example:

The composite of a square and half a circle on it:

Half a circle‘s area = 10
+ Square area = 25

Area of composite shape = ___

Further Example:

Calculate the area of a rectangle with an Area of 10 ft² from which 6 circles were removed, each with an Area of 5 in². Watch the units of Area.

### Worksheet: Level 2 Sample Answer Key

The composite of a square and half a circle on it:

Half a circle‘s area = 10
+ Square area = 25

Area of composite shape = ___

### Worksheet: Level 3

Tasks require a combination of operations. Several steps of calculation are required.

### Worksheet: Level 3 Sample Questions

Calculate the following:

Remember: Rectangles: L ⋅ w = A

Example:

1. A counter top needs to be tiled. It is rectangular, 32 in wide and 65 in long. It has a sink in it which does not get tiled. The hole in the counter for the sink is 14in x 18in. Calculate the area to be tiled in ft² because tiles are sold per ft²

Remember: 144 in² = 1 ft²

Further Example:

Calculate the Area of a flange seal with an outside diameter = 30 cm, inside diameter 20 cm, 8 holes are punched into the flange seal for bolting, each hole with a radius of 5mm.

### Worksheet: Level 3 Sample Answer Key

Step 1 - Full counter top area:

L ⋅ w = A

length × width = area

32 in × 65 in = 2080 in²

Step 2 - Sink cut- out area:

L ⋅ w = A

length × width = area

14 in × 18 in = 252 in²

Step 3 - Counter with sink area:

2080 in² - 252 in² = 1830 in²

1830 in² ÷ 144² = 12.7 ft²

Tasks involve multiple steps of calculation. Advanced mathematical techniques may be required.

Calculate the area of wall to be painted, height = 22 feet, width = 10 feet, less a fireplace with a height of 4 feet, width = 6 feet, and a TV inset height = 3 feet, width = 4 feet.

One way to get the answer:

Layout your calculations neatly, so you can review, track changes, correct or learn from them. One way a layout can look is like this:

Calculate the area of wall to be painted, height = 22 feet, width = 10 feet, less a fireplace with a height of 4 feet, width = 6 feet, and a TV inset height = 3 feet, width = 4 feet.

Another way to get the answer:

1.   Identify that the area of a rectangle is to be calculated.

2.   Make a sketch of the rectangle:

3.   Label the sides of the rectangle. Use either full words or abbreviate.

The idea is to visualize the problem before any math is done.

4.   Write the formula : h  x  w  = A

5.   Re-write the formula: change all of the letters into words, according to their meaning:

height x width = area

6.   Re-write the formula: change all of the words you can for actual given numbers:

Leave out units of measure words. (eg. feet, cm)

22  10  = A

7.   Calculate: 22  x  10   =  220

8.   Take a pencil and write down  “feature wall area = 220”

9.   Look at the units of measure. ft was multiplied with ft. The unit for area is ft².

Write ft² after the 220

Yes, weve got the area of the WHOLE wall. But wait. Not everything gets painted. The TV inset wont. Neither will the fireplace.

10.  Look at the given measurements for the fireplace. The area of it is not given, only its size. The area of the fireplace needs to be calculated.

11.  Write up the problem:

heightwidth = Fireplace area

12.  Re-write the formula: change all of the words you can for actual given numbers:

Leave out units of measure words. (eg. feet, cm)

4   x   6   = A

13.  Calculate: 4  x  6  =  24

14.  Take a pencil and write down  “fireplace area = 24”

15.  Look at the units of measure. ft was multiplied with ft. The unit for area is ft².

Write ft² after the 24

Done with the fireplace, now the TV inset.

16.  Look at the given measurements for the TV inset. The area of it is not given, only its size. The area of the TV inset needs to be calculated.

17.  Write up the problem:

heightwidth = TV inset area

18.  Re-write the formula: change all of the words you can for actual given numbers:

Leave out units of measure words. (eg. feet, cm)

3  x  4  = A

19.  Calculate: 3  x  4  =  12

20.  Take a pencil and write down  “TV inset area = 12”

21.  Look at the units of measure. ft was multiplied with ft. The unit for area is ft².

Write ft² after the 1²

22.  Now recognize that the areas of the fireplace and TV inset are to be subtracted from the total area of the feature wall.

23.  Write down the procedure for the problem:

area of feature wallarea of fireplacearea of TV inset = area for paint

24.  Re-write the set-up: change all of the words you can for actual given numbers:

Leave out units of measure words. (eg. feet, cm)

220  –  24   –  12  =  area for paint

25.  Calculate:  220  –  24  –  12  =  184

26.  Take a pencil and write down  “area for paint = 184”

27.  Look at the units of measure. ft was multiplied with ft. The unit for area is ft².

Write ft² after the 184 Now youre done with the math.

28.  The last step: check your work, make sure everything was copied and written correctly then determine that the correct way to write the answer is 184 ft²

Compare the area of a rectangle calculations in episodes 19 & 22.

Area of a rectangle is said to be calculated by

“length x width”         in one and

“height x width”         in the other.

One version includes “height” the other does not. The formulas look different. But they are really not.

Why? Is there still a way that explains how these 2 different formulas are the same, consistent and valid? Look for clues in episode 09.