Lesson 60: Area of a Parallelogram

New Concept

To find the area of a parallelogram, we multiply the perpendicular base and height. The formula for the area of a parallelogram is:

What’s next

This is just the foundation. Soon, we'll use worked examples to find the area of parallelograms and explore how area is affected by dilations.

Property

The formula for the area of a parallelogram is $A = bh$, in which $A$ represents area, $b$ represents the length of the base, and $h$ represents the perpendicular height.

Examples

• A parallelogram with a base of 10 cm and a height of 5 cm has an area of $A = 10 \operatorname{cm} \cdot 5 \operatorname{cm} = 50 \operatorname{cm}^2$.
• For a base of 7 inches and a height of 3 inches, the area is $A = 7 \operatorname{in} \cdot 3 \operatorname{in} = 21 \operatorname{in}^2$.

Explanation

Forget the slanty sides when calculating area! The trick is to multiply the base by the perpendicular height. Think of it like a magic trick: you can snip a triangle off one side and slide it over to the other side to form a perfect rectangle. This new rectangle has the exact same area, base, and height!

Property

The perimeter of a parallelogram is the total length of its four sides. For a parallelogram with adjacent side lengths of $a$ and $b$, the formula is $P = 2a + 2b$.

Examples

• A parallelogram with side lengths of 8 cm and 6 cm has a perimeter of $P = 8 \operatorname{cm} + 6 \operatorname{cm} + 8 \operatorname{cm} + 6 \operatorname{cm} = 28 \operatorname{cm}$.
• For a rhombus where all four sides are 5 meters long, the perimeter is simply $P = 4 \cdot 5 \operatorname{m} = 20 \operatorname{m}$.

Explanation

Perimeter is all about the distance around the outside, like building a fence. You simply add up the lengths of all four sides. Unlike area, you do not need the height for this calculation! Just remember that opposite sides of a parallelogram are equal in length, which makes the addition a little bit easier for you to work with.

Property

When a parallelogram is dilated by a scale factor of $k$, the area of the new image is $k^2$ times the area of the original parallelogram.

Examples

• A parallelogram with an area of 10 sq. units is dilated by a scale factor of 3. The new area is $10 \cdot 3^2 = 10 \cdot 9 = 90$ sq. units.
• If a parallelogram with an area of 50 cm$^2$ is shrunk by a scale factor of $\frac{1}{2}$, its new area is $50 \cdot (\frac{1}{2})^2 = 50 \cdot \frac{1}{4} = 12.5$ cm$^2$.

Explanation

When you make a parallelogram bigger or smaller using a scale factor, the area changes in a super-sized way! The base gets multiplied by the scale factor, and so does the height. Since area is base times height, the total area gets multiplied by the scale factor twice—that’s the scale factor squared! It's a powerful multiplier effect.