Finding the perimeter and area of parallelograms and trapezoids

Area of parallelograms and trapezoids

We learnt how to find the area of rectangles (and squares) in the previous section. We can use this knowledge to help us to find the area of parallelograms and trapezoids.

For this activity, you are going to need some blank paper, a ruler, a pencil and a pair of scissors.

Part 1:

1. Draw the following rectangle on your piece of paper. Be as accurate as you can.

2. What is the area of this rectangle?

3. Cut your rectangle out and then cut a triangle shape off the one end as shown.

4. Move this triangle over to the other side of the shape. What kind of shape do you create?

5. Is the area of this parallelogram the same as the area of the original rectangle? How do you know?

7. Will we alway be able to make a parallelogram out of a rectangle like this?

8. What can you say about how to find the area of a parallelogram?

Answer 1:

We can find the area of the original rectangle as A = b x h = 9 cm x 5 cm = 45 cm².

The area of the parallelogram we created must be the same as the area of the original rectangle because it is made up of the same amount of paper. Therefore, the area of the new parallelogram is also 45 cm².

We can calculate the area of any parallelogram as A = b x h.

BUT, h must be the perpendicular height. In other words, it is the distance between the two base sides NOT the length of the other sides.

Part 2:

1. Draw the following trapezoid on your piece of paper. Be as accurate as you can.

2. Now cut this shape out. Use it as a stencil to cut another copy of this shape out so that you have two identical trapezoids.

3. Arrange these trapezoids into a parallelogram.

4. What is the area of this parallelogram?

5. How does the area of this parallelogram compare to the area of your original trapezoid?

6. What can you say about how to find the area of a trapezoid?

Answer 2:

The length of the base of the parallelogram you created is 10 cm + 6 cm = 16 cm. Therefore, the area of the parallelogram is A = b x h = 16 cm x 7 cm = 112 cm².

The area of the original trapezoid is half the area of the parallelogram. But to find the area of the parallelogram we had to add the lengths of the two parallel sides of the original trapezoid.

This means that the area of a trapezoid is

\( \text{Area of trapezoid}=\frac{1}{2}\text{area of parallelogram} \)

\( \text{Area}=\frac{1}{2}\left(\text{base}_1 + \text{base}_2\right)\times \text{perpendicular height} \)

or \( A=\frac{1}{2}\left(b_1 + b_2 \right)\times h \)