Area of Shapes Calculator

Rectangle: base × height. This is the definition every other area formula is measured against — how many unit squares fit inside. Square: side², which is just the rectangle with both sides equal. Triangle: base × height ÷ 2. Take any triangle, copy it, rotate the copy half a turn and join the two along a matching side: you get a parallelogram with the same base and height. The triangle is exactly half of it. That is the whole derivation. Trapezoid: (long base + short base) × height ÷ 2. Read it as the average of the two parallel sides, multiplied by the height — a rectangle whose width is the mean of the top and the bottom. When both bases are equal it collapses to the rectangle formula, as it should. Circle: π × radius². If you know the diameter instead, use π × d² ÷ 4, which is about 0.7854 × d² — the version pipe and duct sizing usually wants. A parallelogram, though not in the selector, is base × height as well: shearing a rectangle sideways moves area around without creating or destroying any.

This is the mistake that ruins more area calculations than every other combined. In a triangle or a trapezoid, the height is the perpendicular distance from the base to the opposite vertex or side. It is not the slanted edge. On a leaning triangle the slant is visibly longer than the true height, so using it inflates the area — sometimes badly. The test is a right angle. Drop a straight line from the apex to the base line and make sure it meets at 90°. That length is your height, and on an obtuse triangle it may land outside the triangle, on an extension of the base. That is normal and the formula still works. The same applies to the trapezoid: the height is the gap between the two parallel sides, measured perpendicular to both — never along a sloping leg. Every triangle has three base-height pairs, and all three give the identical area. If you know one side and the perpendicular distance to it, you are done; there is no need to hunt for a particular orientation. When you know all three sides but no height at all, Heron's formula is the tool for the job.

Doubling a linear dimension does not double the area — it quadruples it. Triple it and area goes up ninefold. Area lives in two dimensions, so it follows the square of the scale factor, and this catches people out constantly. Pizza is the standard demonstration. A 16-inch pizza is not a third bigger than a 12-inch one; the ratio is (16 ÷ 12)² ≈ 1.78, so it is nearly 78% more food. Two 12-inch pizzas give less area than one 16-inch, which is worth checking against the prices. The same reasoning drives real budgets. A room 20% larger in each direction needs about 44% more flooring, paint or tile. A photo enlarged from 4×6 to 8×12 covers four times the paper. Scaling a garden bed from 2 m square to 3 m square nearly doubles the soil required. It runs the other way too: to double an area you multiply each dimension by only √2 ≈ 1.41, not by 2. And if you are comparing shapes rather than sizes, note that a fixed perimeter does not fix the area. Of all shapes with the same perimeter, the circle encloses the most, and long thin rectangles enclose the least — a 1 × 9 rectangle and a 5 × 5 square both have a perimeter of 20 but areas of 9 and 25.

Real surfaces are rarely one clean shape, and the method is always the same: decompose, compute, combine. An L-shaped room is two rectangles. Draw the dividing line, measure each piece, add the areas. There is usually more than one way to split it, and both give the same answer — a useful way to check your work. Subtraction handles cut-outs. A wall to be painted is the wall rectangle minus the window and door rectangles. A washer is the outer circle minus the inner one. A patio with a circular fountain in it is the patio minus that circle. A third technique is the enclosing rectangle: draw the smallest rectangle around the whole shape, then subtract the triangular or circular corners that are not part of it. For shapes with one awkward diagonal edge, this is often faster than splitting. For genuinely irregular outlines — a lake, a plot with a curved boundary — you can grid the area and count squares, taking part-filled cells as roughly a half. It is approximate by nature, but the error shrinks as the grid gets finer, and it is how estimates were made long before software. One practical warning: for material estimates, add a waste allowance on top of the geometric area. Tile, flooring and wallpaper all need cutting, and the calculator only reports the true surface.

Area is always in the square of whatever unit you measured in. Sides in metres give square metres; sides in feet give square feet. This calculator does not convert, so all inputs for one shape must use the same unit. The conversions between square units are where people slip, because they are the square of the linear factors. A metre is 100 centimetres, but a square metre is 10,000 square centimetres, not 100. A foot is 12 inches, so a square foot is 144 square inches. A yard is 3 feet, so a square yard is 9 square feet — the number that matters when carpet is priced by the yard and measured in feet. Mixing units mid-calculation is the other common failure. Measuring a room as 3.5 m by 240 cm and multiplying directly gives 840 of nothing. Convert first, then compute. For land, the customary units are worth knowing: an acre is 43,560 square feet, and a hectare is 10,000 square metres, which is one square hectometre by construction. If you need to convert an area from one system to another afterwards, remember to square the linear factor: one inch is 2.54 cm exactly, so one square inch is 2.54² = 6.4516 square centimetres.