Elementary Mathematics Institute

August 23, 2004

Perimeter and Area

3-D Shapes

Perimeter and Area

Wreck-TanglesHow do areas of rectangles with equal perimeters compare?

Complete this activity using the push pins, 30 cm string loop, cardboard and worksheet.

Wreck TanglesLength Width Area Difference

between length and width

Wreck TanglesLength Width Area Difference

between length and width

14 1 14 13

13 2 26 11

12 3 36 9

11 4 44 7

10 5 50 5

9 6 54 3

8 7 56 1

7 8 56 1

6 9 54 3

5 10 50 5

4 11 44 7

3 12 36 9

Conclusion:

As the length decreased by 1, the width increased by 1. The area increased until the difference between the length and width approached zero. This was the largest possible area.

There was a pattern. The differences between the length and width of the rectangles decreased by two and then started to increase by 2 after the largest area was reached.

Is it possible for two shapes to have the same area but different perimeters? Explain your answer by using words and drawings.

Perimeter and Area

Is it possible for two shapes to have the same area but different perimeters? Explain your answer by using words and drawings.

Perimeter and Area

Shapes with the same area can have different perimeters. The perimeter is smallest when the length and width are the same. This would be a square. The perimeter is largest when it is long and narrow.

We think the greatest possible perimeter can be determined by doubling the number of tiles and adding two. 2n+2=largest possible perimeter when arranging tiles in a rectangle with a width of 1 tile.

What are the areas and perimeters of these parallelograms?

What are the areas and perimeters of these parallelograms?

This is a family of parallelograms.

1. Find the area of each parallelogram

2. What pattern do you see?

3. Why do you think these parallelograms are called a family?

Make a rectangle out of Geo-Strips.

Tilt the rectangle out of shape until it makes a different parallelogram.

1. How will the sides, angles, area and perimeter of the new parallelogram compare to the original rectangle?

2. What relations among the sides and angles of rectangles are also true of parallelograms?

Make a rectangle out of Geo-Strips.

Tilt the rectangle out of shape until it makes a different parallelogram.

1. How will the sides, angles, area and perimeter of the new parallelogram compare to the original rectangle?

The sides will remain the same, as well as the perimeter. The angles are not right angles anymore. They are acute and obtuse, but the sum is still 360 degrees. The area gets smaller because the height decreases.

2. What relations among the sides and angles of rectangles are also true of parallelograms?

The opposite angles are equal and the sum of all the angles is 360°. Opposite sides are also equal and parallel.

Area of Parallelogram

• What is the formula for finding the area of a parallelogram?

Multiply the base times the height

Challenge

• What is the area of this parallelogram? 7 square units

• How do you know?Make a rectangle around the parallelogram. The area of that rectangle is 15. Find the area of the triangles outside the parallelogram, but inside the rectangle. The combined area of those triangles is 8. Subtract 8 from 15 to get the area of the parallelogram.

Use the geoboards and worksheets to find a formula for areas of triangles and trapezoids.

How many ways can squares be joined together?

Rules:

1.Squares must touch along one entire edge.

2. If a pattern can fit on top of another using a flip or turn, it is considered the same.

How many ways can squares be joined together?

Number of Squares

Number of Patterns

Drawings of Patterns

1 1

2

3

4

How many ways can 5 squares be joined together?

There are 12 pentomino nets.

Which of the pentomino nets can make an open box?