1 section 3.1 introduction — an exploration into: area and perimeter patterns section 3.1...
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Section 3.1 introduction — an exploration into:
Area and Perimeter Patterns
Section 3.1 introduction — an exploration into:
Area and Perimeter Patterns
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Area and Perimeter PatternsArea and Perimeter PatternsExploration
3.1
You can draw shapes which have the same area, but different perimeters.
You’ll also look at shapes that have the same perimeters, but different areas.
In this Exploration, you’ll look at how to maximize the perimeter for a given area.
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Area and Perimeter PatternsArea and Perimeter PatternsExploration
3.1
You can find the area of a shape by counting up the number of unit squares.
The perimeter is calculated by finding the sum of all the side lengths.
Area = 4 units2
Perimeter = 10 units
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Area and Perimeter PatternsArea and Perimeter Patterns
Example
Solution follows…
Exploration
3.1
Find the area and the perimeter of this shape.
Solution
6
2
6
2
Area = 12 square units
Perimeter = 6 + 2 + 6 + 2 = 16 units
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Area and Perimeter PatternsArea and Perimeter PatternsExploration
3.1
When given a set area, you can draw shapes with different perimeters — like these:
These shapes both have an area of 10 square units.
This one has a perimeter of 22 units...
...but this one has a perimeter of 14 units.
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Area and Perimeter PatternsArea and Perimeter PatternsExploration
3.1
And when given a set perimeter, you can draw shapes with different areas — like these:
These shapes both have a perimeter of 12 units.
This one has an area of 5 square units...
...but this one has an area of 8 square units.
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Area and Perimeter PatternsArea and Perimeter Patterns
Solution follows…
Exploration
3.1
Exercises
1. Find the area and perimeter of each shape.
A
B
C
Area = 4 square unitsPerimeter = 1 + 4 + 1 + 4 = 10 units
Area = 4 square unitsPerimeter = 2 + 2 + 2 + 2 = 8 units
Area = 6 square unitsPerimeter = 1 + 6 + 1 + 6 = 14 units
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Area and Perimeter PatternsArea and Perimeter Patterns
Solution follows…
Exploration
3.1
Exercises
1. Find the area and perimeter of each shape.
D
E
F
Area = 6 square unitsPerimeter = 2 + 3 + 2 + 3 = 10 units
Area = 8 square unitsPerimeter = 1 + 8 + 1 + 8 = 18 units
Area = 8 square unitsPerimeter = 2 + 4 + 2 + 4 = 12 units
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Area and Perimeter PatternsArea and Perimeter Patterns
Solution follows…
Exploration
3.1
Exercises
1. Find the area and perimeter of each shape.
GI
H
Area = 16 square unitsPerimeter = 4 + 4 + 4 + 4 = 16 units
Area = 16 square unitsPerimeter = 2 + 8 + 2 + 8 = 20 units
Area = 16 square unitsPerimeter = 1 + 16 + 1 + 16 = 34 units
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Area and Perimeter PatternsArea and Perimeter Patterns
Solution follows…
Exploration
3.1
Exercises2. Look at the areas and perimeters of the following sets of shapes. What do you notice about them? Which type of shape maximizes the perimeter in each set?
A and B C and D
E and F G, H and I
The longest, thinnest shapes have the greatest perimeters.
The shapes in each set have the same area but different perimeters.
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Area and Perimeter PatternsArea and Perimeter Patterns
Solution follows…
Exploration
3.1
Exercises
3. Draw three different rectangles with areas of 12 square units that all have different perimeters. What are the dimensions of the rectangle that has the largest perimeter?
4. Draw the rectangle with an area of 20 square units that has the largest perimeter possible.
This shape has the largest perimeter. It’s dimensions are 1 unit by 12 units.
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Area and Perimeter PatternsArea and Perimeter Patterns
Solution follows…
Exploration
3.1
Exercises
5. Draw two rectangles that have different areas but both have perimeters of 14 units.
6. Draw a rectangle that has a perimeter of 20 units and has the largest area possible.
Any two of:
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Area and Perimeter PatternsArea and Perimeter Patterns
Round UpRound Up
Exploration
3.1
A rectangle with a big difference between its length and width measurement will have a large perimeter for its area.
It works the other way for maximizing the area of a rectangle with a fixed perimeter — the closer the shape is to a square, the bigger the area will be.