Area of a Parallelogram — Formula & Examples

The area of a parallelogram comes from multiplying its base by the perpendicular height. As long as you can read off a base $b$ and the matching perpendicular height $h$, the same short procedure handles every parallelogram:

The key word is perpendicular. The height is the shortest straight-up distance from the base to the opposite side, not the slanted side length.

When to use this method

Use $A = bh$ whenever you know a base and the perpendicular height to that base. This is the standard route in basic geometry, and it also explains why the triangle area is $\frac{1}{2}bh$. If you do not know the height but you do know two adjacent sides $a$ and $b$ and the included angle $\theta$, use

because the height relative to side $b$ is $a \sin \theta$. This only works when $\theta$ is the included angle between those sides.

Step by step

1. Identify the base. Pick one side to call $b$.
2. Find the perpendicular height to that base. Use the straight-up distance $h$, not a slanted edge.
3. Multiply. Compute $A = bh$.
4. Write square units. If lengths are in centimeters, the area is in $\text{cm}^2$.

The reason base times height works: imagine cutting a right triangle from one side of the parallelogram and sliding it to the other side. The shape becomes a rectangle, and no area is lost or added. A rectangle with the same base $b$ and height $h$ has area $bh$, so the parallelogram must also have area $A = bh$.

A full worked example

Suppose a parallelogram has base $8$ cm and perpendicular height $5$ cm. Substitute those values:

So the area is $40\text{ cm}^2$.

Now notice something about step 2. If the slanted side happened to be $6$ cm, the answer would still be $40\text{ cm}^2$. Area depends on the base and the perpendicular height, not on the slanted edge by itself.

Where students get stuck, and how to check each step

• Step 2, the height: Using the slanted side instead of the perpendicular height causes most wrong answers. If a diagram shows both a slanted side and a perpendicular height, use the perpendicular height for $h$. With $b = 8$, slanted side $6$, and perpendicular height $5$, the correct area is $8 \times 5 = 40$, not $8 \times 6$.
• Step 3, what you are computing: Area measures square units inside the shape; perimeter measures the distance around the outside. The diagram numbers may overlap, but the two formulas answer different questions.
• Step 4, units: A common slip is dropping the square units.

To test whether the perpendicular-height idea has clicked, find the area of a parallelogram with base $12$ m and perpendicular height $7$ m, then change only the slanted side length and confirm the area stays the same.