Lesson 61: Area of a Parallelogram, Angles of a Parallelogram

New Concept

To find a parallelogram's area, multiply its base by its height. The base is a side, and the height is the perpendicular distance to the opposite side.
> Area of a parallelogram = base · height
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What’s next

Next, you will apply these concepts in worked examples. We will tackle problems on finding both the area and the missing angles of various parallelograms.

Property

The base is one of the parallel sides of the parallelogram. The height is the perpendicular distance between the base and the opposite side.

Examples

In a parallelogram with base 12 cm and a slanted side of 10 cm, the height is a shorter 8 cm.
If a rectangular frame 80 inches tall is bumped, its side length stays 80 inches, but its new height becomes shorter, like 75 inches.

Explanation

Think of the base as the floor and the height as how tall the parallelogram stands, measured straight up. Don't be fooled by the slanted side; the height always forms a perfect right angle with the base, showing its true 'altitude'!

Property

Area of a parallelogram = base · height

Examples

A park with a base of 80 yards and a height of 40 yards has an area of $80 \times 40 = 3200$ square yards.
A piece of paper with a base of 10 cm and a height of 9 cm has an area of $10 \times 9 = 90$ square cm.

Explanation

Imagine you snip a triangle off one side and slide it to the other. Presto! You've magically transformed the parallelogram into a rectangle. That's why the area is simply its base multiplied by its straight-up height, not the slanted side length.

Property

1. Nonadjacent angles (in opposite corners) have equal measures.
2. Adjacent angles (sharing a side) are supplementary, meaning their sum is $180^\circ$.

Examples

If one angle is $110^\circ$, its opposite is also $110^\circ$. The other two angles are each $180^\circ - 110^\circ = 70^\circ$.
In a parallelogram with a $75^\circ$ angle, the adjacent angles are $105^\circ$, and the opposite angle is $75^\circ$.

Explanation

When you squish a rectangle, angles in opposite corners stay twins, always having the same measure. Any two neighbors, however, become partners that always add up to a straight line, or 180 degrees. It's all about balance in the parallelogram family!