Area Calculator

The area of a rectangle is length × width. If a room is 5 meters long and 4 meters wide, the area is 5 × 4 = 20 m². For a square (a special rectangle where all sides are equal), the formula simplifies to side². Always use the same units for both dimensions — if length is in feet, width must also be in feet, giving area in square feet.

The area of a circle is π × r², where r is the radius (distance from center to edge) and π ≈ 3.14159. If the diameter is given, the radius is half the diameter. Example: a circle with radius 7 cm has area = π × 7² = π × 49 ≈ 153.94 cm². To find the radius from the circumference: r = C / (2π). The circumference (perimeter of a circle) is 2πr.

The most common formula is ½ × base × height, where height is the perpendicular distance from the base to the opposite vertex. Example: base = 10 cm, height = 6 cm → area = ½ × 10 × 6 = 30 cm². If you only know the three sides (a, b, c), use Heron's Formula: s = (a+b+c)/2, Area = √(s(s−a)(s−b)(s−c)). For a right triangle, the two legs serve as base and height directly.

The area of a trapezoid (or trapezium) is ½ × (a + b) × h, where a and b are the lengths of the two parallel sides and h is the perpendicular height between them. Example: parallel sides of 8 cm and 12 cm, height 5 cm → area = ½ × (8 + 12) × 5 = ½ × 20 × 5 = 50 cm². This formula works because a trapezoid is essentially an average of two rectangles with widths a and b, times the height.

Area conversions use the square of the linear conversion factor. Since 1 m = 100 cm, then 1 m² = 100² cm² = 10,000 cm². Key conversions: 1 ft² = 144 in², 1 yd² = 9 ft², 1 m² = 10.764 ft², 1 acre = 43,560 ft² = 4,047 m², 1 hectare = 10,000 m² = 2.471 acres. A common mistake is using the linear factor instead of its square — always square it for area conversions.

Calculate the room's area (length × width) then add 10–15% for waste from cuts, pattern matching, and future repairs. For irregular rooms, divide into rectangles, calculate each area, then sum them. For hardwood, add 15% for diagonal installation. Example: a 12 × 15 ft room = 180 ft². Adding 10% waste = 198 ft² → order 200 ft². Always buy slightly more than calculated — matching batches exactly later can be difficult if the product is discontinued.

The area of a parallelogram is base × height, where height is the perpendicular distance between the two parallel sides (not the slant side). This is the same formula as a rectangle because a parallelogram can be rearranged into a rectangle by moving a triangular piece from one end to the other. Example: base = 10 cm, perpendicular height = 6 cm → area = 60 cm². Note: using the slant side instead of perpendicular height gives an incorrect, larger result.

The area of an ellipse is π × a × b, where a is the semi-major axis (half the longer diameter) and b is the semi-minor axis (half the shorter diameter). An ellipse is like a "stretched circle" — if a = b = r, it becomes a circle with area πr². Example: an ellipse with semi-major axis 6 cm and semi-minor axis 4 cm has area = π × 6 × 4 ≈ 75.40 cm². The perimeter of an ellipse has no simple exact formula — it requires an approximation.

Area refers to the 2D space of a flat shape. Surface area refers to the total area of all faces of a 3D object. A rectangle has an area (length × width). A rectangular box has a surface area equal to the sum of the areas of all six faces. This calculator computes 2D area for flat shapes. For 3D objects like spheres (4πr²), cylinders (2πrh + 2πr²), or cubes (6s²), you need a surface area calculator.

For irregular shapes, use one of these approaches: (1) Decomposition — divide the shape into simpler shapes (rectangles, triangles, circles), calculate each area, then add them; (2) Grid counting — draw the shape on graph paper, count full squares, estimate partial squares; (3) Coordinate formula — if you have the x,y coordinates of all vertices, use the Shoelace Formula: Area = ½|Σ(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)|; (4) Digital tools — GIS software, CAD programs, or Google Maps can measure irregular areas directly from satellite images.