Select a shape and enter dimensions to calculate area and perimeter.
Area Formulas for Every Shape
Area is the space inside a flat shape, measured in square units. For a rectangle it is length × width, so a 12 ft by 15 ft room is 180 square feet, and for a circle it is π × r². Pick a shape above and enter its dimensions to get the area, the perimeter where one applies, and the result in other units.
Area measures the 2D space inside a shape, expressed in square units (cm², m², ft², etc.). Every shape has its own formula derived from geometric properties. This calculator supports 8 of the most common shapes used in everyday measurement, construction, and design.
Rectangle: A = l × w
Circle: A = π × r²
Triangle: A = ½ × b × h
Trapezoid: A = ½ × (a + b) × h
Parallelogram: A = b × h
Ellipse: A = π × a × b
Rhombus: A = ½ × d&sub1; × d&sub2;
Area vs Perimeter
Unit Conversion
Real-World Uses
Heron's Formula for Triangles
Area Formulas with Worked Examples
Every example below was run through the same formulas the calculator uses. Height always means the perpendicular height, not a slanted side.
| Shape | Formula | Example | Area |
|---|---|---|---|
| Square | s² | side 12 ft | 144 ft² |
| Rectangle | l × w | 12 ft by 15 ft | 180 ft² |
| Circle | π × r² | radius 5 in | 78.54 in² |
| Semicircle | ½ × π × r² | radius 10 ft | 157.08 ft² |
| Triangle | ½ × b × h | base 10 cm, height 6 cm | 30 cm² |
| Trapezoid | ½ × (a + b) × h | parallel sides 8 and 12 m, height 5 m | 50 m² |
| Parallelogram | b × h | base 10 in, height 6 in | 60 in² |
| Ellipse | π × a × b | semi-axes 6 and 4 cm | 75.40 cm² |
| Rhombus | ½ × d₁ × d₂ | diagonals 10 and 8 ft | 40 ft² |
An ellipse has no exact perimeter formula in elementary terms, so the calculator uses Ramanujan's second approximation. For semi-axes 6 and 4 it gives 31.7309, which matches the exact value to more than six decimal places.
Square Units: Conversion Table
An area factor is the length factor squared. One yard is 3 feet, so one square yard is 3 × 3 = 9 square feet, not 3.
| Unit | Equals | Metric or US equivalent |
|---|---|---|
| 1 square inch | 1/144 ft² | 6.4516 cm² (exact) |
| 1 square foot | 144 in² | 0.0929 m² |
| 1 square yard | 9 ft² | 0.8361 m² |
| 1 square meter | 10,000 cm² | 10.7639 ft² or 1.1960 yd² |
| 1 acre | 43,560 ft² | 4,046.86 m² |
| 1 hectare | 10,000 m² | 2.4711 acres |
| 1 square mile | 640 acres | 2.59 km² |
Carpet, sod and some flooring are sold by the square yard, so divide square feet by 9. For other unit pairs, the area converter handles the arithmetic.
Three Worked Examples
An L-shaped room
Split the room into two rectangles: 12 ft × 15 ft = 180 ft² and 6 ft × 8 ft = 48 ft². Total area is 228 ft². With 10% extra for cuts and waste you need 250.8 ft², so buy at least 251 ft² of flooring, rounded up to whole boxes.
A round patio
The patio is 12 ft across, so the radius is 6 ft. Area = π × 6² = 113.10 ft². Using the diameter by mistake (π × 12²) gives 452.39 ft², four times too much.
A gable end wall
The triangle above the eaves has a base of 24 ft and a height of 6 ft: ½ × 24 × 6 = 72 ft². Add the rectangular wall below it to get the full siding area, then subtract windows and doors.
Common Mistakes
- Radius and diameter mixed up. The circle formula needs the radius. Halve the diameter first.
- Slanted side used as height. Triangles, parallelograms and trapezoids need the perpendicular height. A slanted side always gives too large an answer.
- Mixed units. A room 8 ft 6 in wide is 8.5 ft, not 8.6 ft. Convert every dimension to one unit before multiplying.
- Linear factor used for area. 1 m² is 10,000 cm², not 100 cm².
- No allowance for waste. Tile, flooring and sod need extra for cuts, usually 10% for straight layouts and 15% for diagonal ones.