What Is Area?
Area is the amount of two-dimensional space enclosed within a shape's boundary. It is measured in square units: square meters (m²), square centimeters (cm²), square feet (ft²), square inches (in²), etc.
Understanding area is essential for real-world tasks like calculating floor space, painting walls, tiling rooms, designing gardens, and countless engineering applications.
Area Formulas for Common Shapes
Rectangle
Where l = length and w = width.
Example: A room is 6 m long and 4 m wide. Area = 6 × 4 = 24 m²
Square
Where s = side length.
Example: A square tile with side 0.3 m. Area = 0.3² = 0.09 m²
Triangle
Where b = base length and h = perpendicular height.
Example: A triangle with base 8 cm and height 5 cm. Area = (1/2) × 8 × 5 = 20 cm²
Tip: The height must be perpendicular (at a right angle) to the base — not the slant side.
Circle
Where r = radius (half the diameter) and π ≈ 3.14159.
Example: A circle with radius 7 cm. Area = π × 7² = π × 49 ≈ 153.94 cm²
Trapezoid
Where a and b are the lengths of the two parallel sides and h is the perpendicular height between them.
Example: A trapezoid with parallel sides of 6 cm and 10 cm, and a height of 4 cm.
Area = (1/2) × (6 + 10) × 4 = (1/2) × 16 × 4 = 32 cm²
Parallelogram
Where b = base and h = perpendicular height.
Note: This is the same as a rectangle formula! A parallelogram can be thought of as a "slanted rectangle."
Ellipse
Where a = semi-major axis and b = semi-minor axis (half of each diameter).
Regular Hexagon
Where s = side length.
Composite Shapes
Many real-world shapes are combinations of simpler shapes. To find the area of a composite shape:
- Break the shape into simpler parts (rectangles, triangles, circles).
- Calculate the area of each part separately.
- Add (or subtract) the areas as needed.
Example: L-Shaped Room
An L-shaped room can be split into two rectangles. If one rectangle is 5 m × 3 m and the other is 4 m × 2 m:
Total area = (5 × 3) + (4 × 2) = 15 + 8 = 23 m²
Tips for Remembering Area Formulas
- Triangle = half a parallelogram: Any triangle can be doubled to form a parallelogram, which is why the formula has a factor of 1/2.
- Circle vs. Circumference: Area uses r² (squared), while circumference uses just r: C = 2πr. Don't mix them up.
- Units matter: If you mix cm and m in the same problem, convert everything to one unit first.
- Height is perpendicular: For triangles, trapezoids, and parallelograms, the height is always the perpendicular distance — never the slant side.