Area of a parallelogram
A parallelogram is a quadrilateral with two pairs of opposite sides parallel. A rectangle is a special type of parallelogram but parallelograms do not have to have right angles. Both shapes below are examples of parallelograms.
You may recall that we can find the area of a rectangle using the formula \text{Area}=\text{length}\times\text{width}, and we will see that finding the area of a parallelogram is very similar. We will make use of the base and perpendicular height of the parallelogram to find its area.
Notice that a rectangle is a type of parallelogram, but not all parallelograms are rectangles. Why might this be? Think of what each shape has in common and how they differ.
Exploration
Parallelograms can be easily rearranged into rectangles. Explore this using the applet below.
- Click and drag the blue circle on the \text{Slide} slider. This will rearrange the parallelogram into a rectangle.
- Click and drag the blue circle on the \text{Slant} slider. Does this change the area of the shape?
- Click the button \text{Change dimensions}. You can now adjust the base and height to make a new parallelogram. This can be done with the b slider and h slider or by dragging the vertices of the parallelogram.
- The area of the parallelogram is being calculated with a formula as you change its dimensions. What part of the calculation changes when you change a dimension? Can you work out the formula?
- Click the button \text{Decompose }to see if the new parallelogram can also rearrange into a rectangle.
- Click the button \text{Show grid }for a grid. Assume that this is a square centimetres grid. To remove it, click the button \text{Hide grid}.
After using the applet above, we can make the following observations:
- Changing the slant of the parallelogram without changing the base and height did not affect its area. This means that the area of a parallelogram depends only upon its base and its perpendicular height, not the slanted height.
- The base of the parallelogram is the same as the length of the rectangle it creates.
- The perpendicular height of the parallelogram is the same as the width of the rectangle it creates.
- As the area of a rectangle can be found with \text{Area}=\text{length}\times\text{width}, then the area of a parallelogram can be found in a similar way.
The area of a parallelogram is given by
\begin{aligned} \text{Area}&=\text{base}\times\text{height}\\ A&=b\times h \end{aligned}
Unlike a rectangle, there are generally no right angles in a parallelogram. But we should remember that the height and base are at right angles to each other when we work out the area of a parallelogram.
Examples
Example 1
Consider the following parallelogram.
If the parallelogram is formed into a rectangle, what would the length and width of the rectangle be?
Find the area of the parallelogram.
Example 2
Find the area of a parallelogram whose base is 15\text{ cm} and height is 7\text{ cm}.
The area of a parallelogram is given by: