Area of Rectangle, Square, Parallelogram

Area measures how much flat surface a shape covers — the number of unit squares that fit inside it. The rectangle, square, and parallelogram share the simplest area rule in geometry: area equals base times height.

These three shapes matter because their area formula is the seed from which almost every other area formula grows. Triangles, trapeziums, and rhombi are all derived by cutting or doubling parallelograms, so mastering this one idea unlocks the rest.

Key Concepts

• Area: the amount of surface enclosed, measured in square units: cm², m², km².
• Base and height: the height must be perpendicular to the chosen base. In a rectangle the adjacent side serves as the height; in a slanted parallelogram the height is shorter than the slant side.
• Why base × height works: cutting a triangle off one end of a parallelogram and moving it to the other end forms a rectangle with the same base and height — so the parallelogram has the rectangle's area.
• Square: a rectangle with all sides equal; its area is side × side = side².
• Same base, same height, same area: all parallelograms (and rectangles) on the same base and between the same parallels have equal areas, however slanted.

Formulas

• Rectangle: A = length × width. Assumes the sides are perpendicular (true for every rectangle).
• Square: A = side². A direct special case of the rectangle formula.
• Parallelogram: A = base × height, where height is the perpendicular distance between the base and the opposite side. Units: square units.
• Rearranged forms: base = A / height; height = A / base. Useful when the area and one dimension are known.

Worked Example

A parallelogram has a base of 8 cm and a perpendicular height of 5 cm. Its slant side is 6 cm. Find its area.

1. Use A = base × height with the perpendicular height: A = 8 × 5 = 40 cm².
2. Ignore the slant side 6 cm — it is a distractor and is not used in the area formula.
3. Double-check with the rearranged form: height = 40 / 8 = 5 cm. Matches the given height. Correct.

Common Misconceptions

• Never use the slant side as the height. The height is the perpendicular distance; the slant side of a parallelogram is always longer than the height, so using it inflates the area.
• Area units are squared. Writing 40 cm instead of 40 cm² loses marks and confuses area with perimeter.
• Doubling the sides quadruples the area. If a rectangle's length and width both double, the area becomes 4 times larger, not 2 times — area scales with the square of the length scale.

Real-World Applications

• Flooring and painting: tiles, carpet, and paint are all estimated from rectangular areas — floor area divided by tile area gives the tile count.
• Agriculture: field areas in acres or hectares come from length × width measurements of rectangular plots.
• Screens and paper: display sizes, paper sizes (A4 = 21 cm × 29.7 cm ≈ 623.7 cm²), and solar panels are all rectangles.
• Construction: parallelogram-shaped plots and ramps use base × height to compute land area and material needs.

The rectangle, square, and parallelogram all share one area rule: base times perpendicular height, giving side² for the square. The cut-and-paste argument shows why slant does not matter — only the perpendicular height counts. Remember the square units and ignore the slant side, and these area problems are foolproof.