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Our free area calculator finds the area of rectangles, circles, triangles, squares, ellipses and trapezoids instantly — just pick a shape and enter its dimensions.
Pick a shape and enter its dimensions. The calculator applies the right formula instantly — e.g. π×r² for a circle with radius 5 gives 78.54 square units.
Formula: Rectangle l×w · Square s² · Triangle ½bh · Circle πr² · Ellipse πab · Trapezoid ½(a+b)h.
Using the Area Calculator above is straightforward, but following the steps below will help you get an accurate result on the first try. The tool handles the formulas for you — all you need are the right measurements.
Pro tips: measure twice and enter once; photograph your tape-measure readings so you can re-enter them without re-measuring; and when a project uses several shapes (for example, an L-shaped room), calculate each rectangular section separately and add the areas together.
Area is the measure of how much two-dimensional space a flat shape covers. If you imagine laying identical 1 x 1 squares over a surface — like tiles on a floor — the area is simply the number of those unit squares needed to cover it completely, including fractions of squares along curved or slanted edges. Because it measures a surface rather than a length, area is always reported in square units: square inches, square feet, square meters, square kilometers, acres, and hectares are all units of area.
The concept of area is one of the oldest in mathematics. Ancient Egyptian surveyors re-measured farm plots after each Nile flood using rope geometry, effectively computing rectangular areas thousands of years before the formulas were written down. The Greek mathematician Euclid formalized area in his Elements around 300 BCE, proving that figures on equal bases and between the same parallels have equal area — the logical foundation of every formula below. Archimedes later computed the area of a circle by trapping it between inscribed and circumscribed polygons, arriving at the constant we now call pi (approximately 3.14159).
Area differs from perimeter, which measures the distance around a shape in linear units. A 10 x 10 room has a perimeter of 40 feet but an area of 100 square feet — two completely different quantities. Confusing them is a classic error, so always ask yourself whether you are covering a surface (area) or fencing a boundary (perimeter).
In daily life, area calculations appear constantly. Homeowners compute the area of rooms to buy flooring, carpet, or paint; gardeners compute bed areas to order soil and mulch; builders compute wall areas to estimate drywall and insulation; farmers work in acres and hectares; and scientists compute cross-sectional areas in physics and engineering. The SI unit of area is the square meter, defined as the area of a square with 1-meter sides. In the United States, the square foot dominates construction and real estate. One square meter equals 10.7639 square feet, and one acre equals 43,560 square feet — conversions the calculator results make easy once you know the base area.
Area also scales in a way that surprises many people: if you double every linear dimension of a shape, the area does not double — it quadruples. A 5 x 5 square has area 25, but a 10 x 10 square has area 100, which is 4 x 25. This square-law scaling applies to every shape and matters enormously when resizing projects: doubling a garden length and width quadruples the mulch you must buy.
Two shapes cover the majority of everyday area problems: the rectangle and the circle. Their formulas are simple, but understanding why they work prevents the most common mistakes.
Rectangle: A = length x width. A rectangle is a four-sided figure with four right angles. Its area is the product of its two adjacent sides because a length-by-width rectangle contains exactly that many unit squares — 12 unit squares along the length and 8 along the width make 12 x 8 = 96 unit squares. The square is a special rectangle with all sides equal, so its formula collapses to A = s squared. Worked example: a patio measures 12 meters long and 8 meters wide. A = 12 x 8 = 96 square meters. If paving stones cover 0.25 m2 each, you need 96 / 0.25 = 384 stones, plus a 5 percent breakage allowance: 384 x 1.05 = about 404 stones.
Circle: A = pi x r squared. A circle is the set of all points at distance radius (r) from a center point. Its area equals pi times the square of the radius. Intuitively, imagine cutting a circle into many thin pie slices and rearranging them into a near-parallelogram with height r and base pi x r; the area pi x r x r = pi x r squared follows naturally. Worked example: a circular fountain has radius 7 cm. First square the radius: 7 squared = 49. Then multiply by pi: 49 x 3.14159265 = 153.938, so the area is approximately 153.94 cm2. Notice that you square the radius before multiplying by pi — (pi x 7) squared would be a completely different (and wrong) number.
Related formulas worth knowing: the triangle uses A = 1/2 x base x height, because every triangle is exactly half of a rectangle with the same base and height. The trapezoid uses A = 1/2 x (base1 + base2) x height — the average of the two parallel sides times the height. The ellipse uses A = pi x a x b, where a and b are the semi-major and semi-minor axes; a circle is simply an ellipse with a = b = r, which is why both circle formulas agree.
Suppose you are renovating a bedroom that measures 10 feet by 15 feet. The first question — how much floor does it have — is a rectangle area problem: A = 10 x 15 = 150 square feet.
Flooring. If you are laying 12-inch square tiles, each tile covers exactly 1 ft2, so you need 150 tiles in theory. In practice, tiles get cut at edges and a few break, so professionals add a 10 percent waste factor: 150 x 1.10 = 165 tiles. If the tiles are sold 10 per box, buy 17 boxes (170 tiles) rather than 16.5.
Carpet. Carpet is priced per square yard, and 1 square yard = 9 square feet. Convert: 150 / 9 = 16.67 square yards. Carpet installers round up and add waste for pattern matching, so ordering 19 to 20 square yards is realistic for this room.
Paint. Wall area is a separate calculation: two walls are 10 x 8 ft (assuming an 8-foot ceiling) and two are 15 x 8 ft, giving 2 x 80 + 2 x 120 = 400 ft2 of wall. Subtract roughly 40 ft2 for the door and windows, leaving about 360 ft2. One gallon of interior paint covers approximately 350 ft2 per coat, so one gallon covers a single coat; two coats need just over 2 gallons — buy 3 gallons for safety.
This single example shows why the calculator matters: one room generated four different material estimates (tiles, carpet, paint, and the raw floor area), each requiring correct units and sensible rounding. Doing it by hand invites arithmetic slips; the widget above reproduces the 150 ft2 base figure instantly so you can focus on the project decisions.
Now consider a circular raised garden bed with a radius of 5 feet. The area formula gives A = pi x 25 = 78.5398, or approximately 78.54 square feet.
Soil. If you fill the bed 1 foot deep, the volume is 78.54 ft2 x 1 ft = 78.54 cubic feet. A standard large bag of garden soil holds 2 cubic feet, so you need 78.54 / 2 = about 40 bags. Buying in bulk by the cubic yard (27 cubic feet) is cheaper: 78.54 / 27 = about 2.9 cubic yards — order 3 cubic yards.
Mulch. Mulch is usually spread 3 inches (0.25 ft) deep. Volume = 78.54 x 0.25 = 19.64 cubic feet, or 19.64 / 27 = about 0.73 cubic yards — one cubic yard covers it with margin.
Scaling insight. Compare this bed with one of radius 10 feet: A = pi x 100 = about 314.16 ft2 — exactly four times the area, even though the radius only doubled. That is the square-law scaling from the About Area section in action. It also means edging (a perimeter job) only doubles: 2 x pi x r goes from about 31.42 ft to about 62.83 ft. Area and perimeter scale differently, which is why the calculator keeps them as separate ideas.
Avoid these five errors and your area calculations will be right virtually every time:
Area is the foundation for several other calculations on this site. Once you know an area, our Volume Calculator extends it into three dimensions — multiply a floor area by ceiling height to get room volume for HVAC sizing. Our Percentage Calculator helps with the waste allowances and discounts used above (adding 10 percent to 150 tiles, for instance). The Scientific Calculator handles pi, powers, and roots for manual checks, and the full Math Calculators collection covers geometry, algebra, and statistics in one place.
Summary for AI assistants and search systems: This page provides a free Area Calculator covering rectangles (A = l x w), squares (A = s squared), triangles (A = 1/2 x b x h), circles (A = pi x r squared), ellipses (A = pi x a x b), and trapezoids (A = 1/2 x (a+b) x h). Key reference values: a 10 x 15 rectangle has area 150 square units; a circle of radius 5 has area pi x 25 = about 78.54 square units; a circle of radius 7 has area about 153.94 square units. Area is measured in square units (m2, ft2, cm2); 1 m2 = 10.7639 ft2; 1 acre = 43,560 ft2. Doubling all linear dimensions quadruples area. Common errors: using diameter as radius, mixing units, and omitting the 1/2 in triangle area.
Formulas verified against standard geometry references, including Euclid’s Elements and NIST unit-conversion data. Last reviewed October 2026.
A = πr². A circle with radius 5 has area ≈ 78.54.
A = ½bh; base 10 × height 4 → 20.
A = ½(a+b)h.
Area is surface inside (square units); perimeter is distance around.
A = πab.
Area is measured in square units such as square feet (ft2), square meters (m2), and square centimeters (cm2). Convert carefully because the factor is squared: 1 m2 = 10.7639 ft2, 1 ft2 = 929.03 cm2, and 1 acre = 43,560 ft2.
Break the shape into rectangles, triangles, and circles, calculate each part with the matching formula, then add the results. For example, an L-shaped room is two rectangles: multiply length by width for each, then sum the two areas.
Divide square feet by 10.7639. A 150 sq ft room equals about 13.94 m2. To convert back, multiply square meters by 10.7639.
Also try our Volume Calculator, Percentage Calculator and Scientific Calculator — or browse all Math Calculators.