A parallelogram is a four-sided shape with two pairs of parallel sides, and its area tells how much flat space it covers. The key formula is area = base × perpendicular height, written A = bh. This matters because many shapes in geometry can be understood by cutting, sliding, or rearranging them into simpler shapes.
For a parallelogram, the simpler shape is a rectangle with the same base and height.
The formula works because a slanted triangular piece on one side of the parallelogram can be moved to the other side to form a rectangle. This transformation changes the shape but does not change the area, as long as no pieces are stretched or lost. The height must be measured at a right angle to the base, not along the slanted side.
In applications such as flooring, design, and coordinate geometry, using the perpendicular height gives the correct area even when the shape is tilted.
Key Facts
- Area of a parallelogram: A = bh
- b is the length of the chosen base.
- h is the perpendicular height, measured at a 90 degree angle to the base.
- A parallelogram can be rearranged into a rectangle with the same base and height.
- Slanted side length is not the height unless it is perpendicular to the base.
- If b = 12 cm and h = 5 cm, then A = 12 × 5 = 60 cm².
Vocabulary
- Parallelogram
- A quadrilateral with two pairs of opposite sides that are parallel.
- Base
- The side of a parallelogram chosen as the reference side for measuring area.
- Perpendicular height
- The shortest distance between the base and the opposite side, measured at a right angle to the base.
- Area
- The amount of flat surface inside a two-dimensional shape.
- Shear
- A transformation that slants a shape while keeping parallel lines parallel and preserving area.
Common Mistakes to Avoid
- Using the slanted side as the height is wrong because height must be perpendicular to the base.
- Multiplying all side lengths together is wrong because area of a parallelogram uses only base times perpendicular height.
- Forgetting square units is wrong because area is measured in units such as cm², m², or in².
- Changing the base without changing the matching height is wrong because each chosen base has its own perpendicular height.
Practice Questions
- 1 A parallelogram has a base of 14 cm and a perpendicular height of 6 cm. Find its area.
- 2 A parallelogram has area 96 m² and base 12 m. Find its perpendicular height.
- 3 Two parallelograms have the same base and the same perpendicular height, but one is much more slanted than the other. Explain why their areas are the same.