Area Calculator
Calculate area and volume for common geometric shapes including rectangles, circles, triangles, spheres, cylinders, and cones.
Key features
- Multiple shapes
- Area and volume
- Formula display
- Metric units
Guide
Calculating the area of geometric shapes is a fundamental skill used in construction, landscaping, interior design, engineering, and everyday problem-solving. Area measures the amount of two-dimensional space enclosed within a boundary, expressed in square units (square meters, square feet, square inches, and so on). This guide covers the area formulas for common shapes, the derivations behind them, practical measurement techniques, and real-world applications. A rectangle has an area equal to its length multiplied by its width. A = l x w. A room that is 12 feet long and 10 feet wide has an area of 120 square feet. This is the simplest and most frequently used area formula. It works for any rectangle regardless of size, from a postage stamp to a football field. When measuring rooms, land parcels, or building footprints, you are almost always starting with rectangular area calculations. A square is a special rectangle where all sides are equal. A = s^2, where s is the side length. A 5-meter square patio has an area of 25 square meters. The diagonal of a square equals s x sqrt(2), and you can calculate the area from the diagonal using A = d^2 / 2. This alternate formula is useful when you can measure the diagonal more easily than the sides, such as when measuring a square tile from corner to corner. A triangle has an area equal to half the base multiplied by the height. A = (1/2) x b x h. The height must be perpendicular to the base, not the length of a slanted side. A triangular garden bed with a 6-foot base and 4-foot height has an area of 12 square feet. If you know all three side lengths but not the height, use Heron's formula: A = sqrt(s(s-a)(s-b)(s-c)), where s = (a + b + c) / 2 is the semi-perimeter and a, b, c are the side lengths. A circle has an area equal to pi multiplied by the radius squared. A = pi x r^2. A circular pool with a 3-meter radius has an area of approximately 28.27 square meters. If you know the diameter rather than the radius, divide by 2 first, or use A = pi x d^2 / 4. For a semicircle, take half the circle area: A = pi x r^2 / 2. Circles appear in plumbing (pipe cross-sections), manufacturing (circular parts), landscaping (round gardens), and architecture (domes, arches, round windows). A parallelogram has an area equal to its base multiplied by its height. A = b x h. The height is the perpendicular distance between the base and its opposite side, not the length of the slanted side. A parallelogram with a 10-inch base and 7-inch height has an area of 70 square inches. This formula works because a parallelogram can be rearranged into a rectangle with the same base and height by cutting a triangle from one end and moving it to the other. A trapezoid (trapezium in British English) has two parallel sides of different lengths. Its area equals the average of the two parallel sides multiplied by the height. A = (a + b) / 2 x h, where a and b are the parallel sides and h is the perpendicular distance between them. A garden bed shaped like a trapezoid with parallel sides of 8 feet and 12 feet and a height of 5 feet has an area of (8 + 12) / 2 x 5 = 50 square feet. Trapezoidal shapes are common in land parcels, architectural features, and retaining wall designs. An ellipse (oval) has an area equal to pi multiplied by the semi-major axis multiplied by the semi-minor axis. A = pi x a x b, where a is the half-length of the longer axis and b is the half-length of the shorter axis. An elliptical flower bed with a major axis of 10 feet and minor axis of 6 feet has semi-axes of 5 and 3, giving an area of pi x 5 x 3 = approximately 47.12 square feet. When a = b, the ellipse becomes a circle and the formula reduces to pi x r^2. A regular polygon (all sides and angles equal) has an area formula based on the number of sides and side length. A = (n x s^2) / (4 x tan(pi/n)), where n is the number of sides and s is the side length. For a regular hexagon with 4-inch sides: A = (6 x 16) / (4 x tan(30 degrees)) = 96 / (4 x 0.5774) = 96 / 2.309 = approximately 41.57 square inches. An alternate formula uses the apothem (distance from center to the middle of a side): A = (1/2) x perimeter x apothem. Composite shapes in real-world applications are rarely simple geometric figures. A room might be L-shaped, a garden might combine rectangles and semicircles, and a plot of land might have irregular boundaries. The strategy for composite shapes is to break them into simpler components, calculate each area separately, and add or subtract as needed. An L-shaped room can be split into two rectangles. A window with a semicircular top is a rectangle plus a semicircle. A room with a bay window is a rectangle plus a trapezoid or triangle. Unit conversion is essential when working with area measurements. Area units are squared, which affects conversion factors. 1 foot = 12 inches, but 1 square foot = 144 square inches (12^2). 1 meter = 100 centimeters, but 1 square meter = 10,000 square centimeters (100^2). Common area conversions: 1 acre = 43,560 square feet = 4,047 square meters. 1 hectare = 10,000 square meters = 2.471 acres. 1 square mile = 640 acres = 2.59 square kilometers. Flooring and tiling require area calculations plus waste factors. If a room is 200 square feet and tiles are 1 square foot each, you need at least 200 tiles. But cutting tiles to fit edges, matching patterns, and accounting for breakage means you should buy 10% to 15% extra. For a 200 square foot room, order 220 to 230 tiles. The exact waste factor depends on tile size (larger tiles relative to room size means more waste at edges), pattern complexity, and room shape irregularity. Paint coverage calculations use area to determine how much paint to buy. A standard gallon of interior paint covers approximately 350 to 400 square feet. To paint a room, calculate the wall area (perimeter x wall height) and subtract the area of doors and windows. A 12 x 10 foot room with 8-foot ceilings has a wall perimeter of 44 feet and wall area of 352 square feet. Subtracting a 20 square foot door and two 12 square foot windows leaves 308 square feet. One gallon covers this for one coat. Two coats need two gallons. Landscaping uses area calculations for sod, mulch, gravel, and seed coverage. Sod is sold by the square foot or square yard. Mulch is sold by volume (cubic yards), so you calculate area and multiply by the desired depth. For 3 inches of mulch over a 500 square foot garden bed: convert depth to feet (3/12 = 0.25 feet), multiply area by depth (500 x 0.25 = 125 cubic feet), convert to cubic yards (125 / 27 = 4.63 cubic yards). Order 5 cubic yards to account for settling and irregular coverage. Roof area calculations determine material quantities for shingles, metal panels, or other coverings. Roof area exceeds the building footprint because of the pitch (slope). A flat roof with no pitch has the same area as the footprint. A pitched roof has more area. The multiplication factor depends on pitch: a 4/12 pitch has a factor of 1.054, a 6/12 pitch has 1.118, an 8/12 pitch has 1.202, and a 12/12 (45-degree) pitch has 1.414. Multiply the footprint area by the pitch factor to get actual roof area. Irregular land parcels require either triangulation (dividing the area into triangles and summing their areas) or the surveyor's formula (shoelace formula) if you have coordinate points. The shoelace formula for a polygon with vertices (x1,y1), (x2,y2), ..., (xn,yn) is: A = (1/2) |sum of (xi x y(i+1) - x(i+1) x yi)|. This method works for any polygon, regular or irregular, as long as you know the coordinates of the vertices. Surface area differs from planar area. While the calculator focuses on two-dimensional shapes, understanding the connection is useful. A cylinder's surface area (2 x pi x r x h + 2 x pi x r^2) combines the area of a rectangle (the side) and two circles (the ends). A sphere's surface area (4 x pi x r^2) is exactly four times the area of its circular cross-section. These formulas build directly on the two-dimensional area calculations. Practical measurement tips improve accuracy. Use a tape measure rather than a ruler for rooms and outdoor spaces. Measure in the same units throughout (do not mix feet and inches mid-calculation). Take multiple measurements and average them for important calculations. For irregular spaces, take the largest rectangle that fits and then measure the remaining sections separately. Round up when ordering materials; running short costs more than having a small surplus. The WebRecast area calculator lets you select a shape, enter the required dimensions, and instantly see the calculated area. Supported shapes include rectangles, squares, triangles, circles, parallelograms, trapezoids, ellipses, and regular polygons. The tool shows the formula used and accepts inputs in any unit. For composite shapes, calculate each component separately and sum the results. All calculations happen in your browser, making the tool available offline and keeping your data private.
Frequently asked questions
Which shapes?
Rectangle, circle, triangle, trapezoid, sphere, cylinder, and cone.
