Exploring Area of Polygons

Exploring the Area of a Parallelogram

Objective: Students will derive the formula for the area of a parallelogram.

Materials:•Index Cards•Ruler•Scissors•Tape

Step 1: Find the Area of your index card.

Step 2: Use a straightedge to draw a line through one of the vertices of your index card.

Step 3: Cut out the triangle. Tape the triangle to the opposite side to form a parallelogram.

1. How does the area of the parallelogram compare to the area of the rectangular index card?

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2. How do their bases compare?

3. How do their heights compare?

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4. Write a conjecture about the formula for the area of a parallelogram.

Area of parallelogram bh

Think About It

Exploring the Area of a Triangle

Objective: Students will derive the formula for the area of a triangle.

Materials:•Grid paper•Colored pencils or markers•Scissors

Step 1: Draw a triangle on grid paper. Then draw a rectangle that encloses the triangle. Write down the dimensions of the rectangle.

Step 2: Cut out the rectangle. Then cut the triangles out of the rectangle.

Step

Step 3: Arrange the two smaller triangles to cover the area of the large triangle .

1. Do the two smaller triangles cover the same area as the large triangle?

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2. How is the area of the large triangle related to the area of the original rectangle?

Area of 1

2Area of rectangle

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3. Use the dimensions of the rectangle in step 1 to find the area of the rectangle, then use your answer to find the area of the large triangle.

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4. Use the diagram below to write a conjecture about the area of a triangle given its base b and its height h.

Area of 1

2Area of rectangle

bh

1

2

Exploring the Area of a Trapezoid

Materials:•Grid Paper•Ruler•Scissors•Tape

Objective: Students will derive the formula for the area of a trapezoid.

Step 1: On grid paper, cut out two identical trapezoids. Label the bases b1 and b2, respectively, and label the heights h.

What is the shape formed by the twoidentical trapezoids?

Step 2: Then turn one trapezoid upside down and tape it to the other trapezoid as shown

1. Write an expression to represent the base of the parallelogram you created.

b1 b2

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2. Write an expression to represent the area of the parallelogram you created.

Total Area (b1 b2)h

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3. How does the area of each trapezoid compare to the area of the parallelogram you created?

Area of trapezoid 1

2Total Area

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Total Area (b1 b2)h

4. Write a conjecture about the formula for thearea of the trapezoid.

1

2

b1 b2 h

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Area of trapezoid 1

2Total Area

Exploring the Area of a Circle

Materials:•Paper•Compass•Ruler•Scissors

Objective: Students will derive the formula for the area of a circle.

Step 1: Use a compass to draw a circle on a piece of paper. Cut the circle out. Fold the circle in half, four times.

Step 2: Cut the circle along the fold lines to divide the circle into 16 equal wedges.

Step 3: Arrange the wedges to form a shape resembling a parallelogram. The base and height of the parallelogram are labeled.

1. How does the area of the original circle compare to the area of the parallelogram you created?

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2. Write an expression for the height of the parallelogram you created.

height r

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3. Write an expression for the base of the parallelogram you created.

base 1

2C

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4. Write an expression for the area of the parallelogram you created.

r

1

2C

Area 1

2C r

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5. Use the fact that to rewrite the area.

Area 1

2C r

C 2r

1

22r r

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6. Use the fact that to rewrite the area.

Area 1

2C r

C 2r

1

22r r

1

2 2

r r

r2

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7. Write a conjecture about the formula for the area of a circle.

Area of circle r2

Think About It

Circle Applet

NCTM-Circle Lesson