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Calculating the Perimeter of 2-D Complex Shapes
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1. Our Training Videos
Skills Application: Calculating the Perimeter of 2-D Complex Shapes
Step-by-Step: Calculating the Perimeter of 2-D Complex Shapes
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2. Running with Pi

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The surface finish on manufactured items is critical in some applications. Inside an engine the gap between the cylinder wall and the piston is sealed with a springy ring, the perimeter of which is made with great care. It is not smooth. It is beveled and textured, which makes its perimeter quite complex. The reason for texturing is that a microscopic layer of oil can cling to it (and not run off). The ring needs to stay wetted for normal engine operation. Changes to its complex perimeter could destroy an engine.

## Running with Pi

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3. Our Training Worksheets

Perimeter can be calculated by breaking up a composite shape into simple shapes and adding up their lengths. Watch the video to see calculating the perimeter of an ice rink.

Parallelograms, Trapezoids, Hexagons, Octagons and other geometric shapes can be broken up into triangles and rectangles. Their perimeters can also be calculated by adding up the appropriate side lengths of those triangles and rectangles.

Right Triangles

Perimeter calculations involve using the Pythagorean formula. The length of the hypotenuse side (c) must be calculated and added to sides (a) and (b). Labels on the legs or sides (a) and (b) are interchangeable, but side (c) must be the one opposite the right (90°) angle. See more about angles in episode 30.

Perimeter = a + b + c

Length of side (c) calculation uses the Pythagorean formula a2 + b2 = c2, so

$\text{c = }\sqrt{a^2 + b^2}$

EXAMPLE:

What is the perimeter of a triangle with legs a = 7 in,  b = 4 in?

$\text{c = }\sqrt{a^2 + b^2}$
=
$\sqrt{7^2 + 4^2}$
=
$\sqrt{49 + 16}$
=
$\sqrt{65}$
= 8.062 in

Perimeter = a + b + c = 7 + 5 + 8.062 = 20.062 in

### A Short Review

Complex shapes can be broken down into simple shapes such as triangles, circles and rectangles.

These all need to be calculated separately, then the relevant numbers for perimeter 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 Question

Examples:

From the picture below write an addition sentence with the given measurements that represents the whole perimeter.

1.  Half circles circumference = 10, rectangles long side = 15

2. A complex shape is made of joining a square and an equilateral triangle. The side length of the triangle equals the squares side length. Draw the complex shape and show which sides form the Perimeter and which sides are inside the complex shape (not part of the Perimeter).

### Worksheet: Level 1 Answer Key

Answers:

1.  Half circles circumference = 10, rectangles long side = 15

10 + 15 + 10 +15

2. A complex shape is made of joining a square and an equilateral triangle. The side length of the triangle equals the squares side length. Draw the complex shape and show which sides form the Perimeter and which sides are inside the complex shape (not part of the Perimeter).

### Worksheet: Level 2

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

### Worksheet: Level 2 Sample Question

Example:

1.  Calculate the perimeter of the following complex shape:

Remember:
Calculate the perimeter of the simple shape then add up the subtotals.

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

Length of half a circle‘s circumference = 10
+ Length of 3 sides of the square =    5
Perimeter of complex shape = ____

2. Calculate the Perimeter of a complex shape made of a square with an equilateral triangle drawn on each of its 4 sides. The side length of the triangle equals the square's side length. Triangle's side length is 2

### Worksheet: Level 2 Sample Answer

Example:

1.  Calculate the perimeter of the following complex shape:

Length of half a circle‘s circumference = 10
+ Length of 3 sides of the square =    5
Perimeter of complex shape = 15

2. (2 + 2) + (2 + 2) + (2 + 2) + (2 + 2) = 16

### Worksheet: Level 3

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

### Worksheet: Level 3 Sample Question

Example:

Calculate the perimeter of the following composite shape of 2 half circles and a rectangle:

1.  w = 10, L = 15

### Worksheet: Level 3 Sample Answer

Answer:

1.

Circle‘s circumference (shape 1)
C = d ⋅ ?
circumference = diameter × ?
C = 10 × ?

C = 31.4159

### Infosheet: Advanced Level

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

### Additional Information: Another Way to get the Answer (Version 1)

Calculate the perimeter of an ice rink with a width of 20 m, length 50 m.

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

Additional graphics:

### Additional Information: Another Way to get the Answer (Version 2)

Calculate the perimeter of an ice rink with a width of 20 m, length 50 m

Another way to get the answer using the formula d · p + 2L – 2d = P

1. Identify that the perimeter of the rink is to be calculated.
2. Write down the formula d · p + 2L – 2d = P
3. Re-write the formula: change all of the letters into words, according to their meaning:

diameter x  p + 2 x Length – 2 x diameter = perimeter

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

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

20   x  p + 2 x     50      – 2 x      20       = perimeter

1. Calculate:    20   x  p + 2 x   50    – 2 x    20   = 122.83185307

Note: Find and use the p button on your calculator. The calculators memory stores 20 digits for π, so calculations are highly accurate. Using the number 3.14 is much less accurate, often unacceptable in trades and engineering.

Your calculators circuitry may not be programmed with doing multiplications before addition and subtraction, so you may need to bracket the operation, like this: (see info on calculator steps further below)

(20  x  p) + (2 x  50) – (2 x 20)   = 122.83185307

If you calculator does not have brackets, do the multiplications separately, Take a pencil and write down the subtotals, then add and subtract like this:

20        x   p  = 62.83185307
+       2     x  50 = 100
–       2     x  20 = 40

1. Take a pencil and write down  “rink perimeter = 122.8318”
2. Write m after the number 122.8318 Now youre done with the math.
3. 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 122.8318 m
Note: This calculation highlights the math for perimeter. It does not include material cutting and fitting considerations, available possible materials, fastener or other needs.

Get a \$20-or-so dual-display scientific calculator, even if you cant take it with you into the exam hall. A calculator is also a learning tool, Technology Use is one of the 9 Essential Skills you cant do without. Trying out different functions on a calculator will further your understanding and help with building a bigger mental picture. It builds links between numbers and algebra. It is important to try alternative calculations or approaches, to sketch & label angles, tracking changes to units of measure, visualizing to cross-link the concepts of perimeter and numbers.

Using a scientific calculator, do the following:

Press ( button
Enter 20
Press x (multiplication) button (this could be omitted, depending on your calculators circuits)
Press p button or 2ndF button then the p button, depending on how your calculator is laid out.
Press ) button
Press + button
Press ( button
Enter 2
Press x (multiplication) button
Enter 50
Press ) button
Press  button
Press ( button
Enter 2
Press x (multiplication) button
Enter 20
Press ) button
Press =
You should see 122.8318531 on your display.