Area of a Parallelogram

Statement

The area of a parallelogram is given by:

\[ A = b \times h \]

Here, \(A\) is the area, \(b\) is the base length, and \(h\) is the perpendicular height drawn from the opposite side to the base.

Why it’s true (short reason)

• A parallelogram can be transformed into a rectangle of the same base and height by cutting and rearranging a triangle from one side.
• The area formula for a rectangle is base × height, so the same applies to the parallelogram.

Recipe (how to use it)

1. Identify the base (any chosen side).
2. Draw or identify the perpendicular height from the opposite side to the base.
3. Multiply base × height.
4. Include units squared (cm², m², etc.).

Spotting it

• Questions showing slanted four-sided figures with opposite sides parallel.
• Look for a marked perpendicular height inside or outside the figure.
• Sometimes given as sine rule form: \(A=ab\sin(C)\), but GCSE typically uses \(A=bh\).

Common pairings

• Trapezium area: similar theme, but averages two parallel sides.
• Vector cross product: in advanced maths, area of a parallelogram is magnitude of cross product.
• Volume problems: parallelogram bases often appear in prisms.

Mini examples

1. Given: base=10 cm, height=6 cm. Find: area. Answer: 10×6=60 cm².
2. Given: base=15 cm, height=8 cm. Find: area. Answer: 120 cm².

Pitfalls

• Using slanted side instead of perpendicular height: always ensure height is at 90° to base.
• Mixing with perimeter: area is product, not sum.
• Wrong units: area is always squared units.

Exam strategy

• Check whether height is given inside or outside — sometimes it drops down outside the shape.
• Underline which side is being treated as base.
• Always state formula before substituting.
• Double-check units in final step.

Summary

The area of a parallelogram is base × height. Always ensure the height used is perpendicular to the base. This formula connects to rectangles, trapeziums, and more advanced vector geometry.