# Area Calculator - Square, Rectangle, Triangle, Circle, Trapezoid Find plane area for seven common shapes. Each result shows the formula that ran. Default is an 8 × 5 rectangle (area 40). ### Shape and Dimensions Shape Length l _8_ Width w _5_ Clear All Default is a rectangle with l = 8 and w = 5 (area 40). All lengths must be positive. Circle and ellipse use Math.PI. ### Area Result Area and the formula used will appear here ### What Is an Area Calculator and Why Do You Need One? An **area calculator** answers how much flat space a region covers. That is a two-dimensional question: length times length, or an equivalent formula that still produces **square units**. A garden bed, a rectangular room, a circular patio, a triangular sail, and an elliptical running track all have area. They do not have volume until you add thickness, and they do not have surface area in the solid-geometry sense until you wrap a 3-D object. This page handles seven plane figures: square (A = s²), rectangle (A = l × w), triangle (A = ½bh or Heron from three sides), parallelogram (A = bh), trapezoid (A = (a+b)/2 × h), circle (A = πr²), and ellipse (A = πab). The default load is a rectangle 8 by 5, which is exactly 40 square units - useful as a sanity check that the tool is multiplying, not adding (8+5=13) and not doubling a perimeter (26). Arithmetic stays in the browser. If you need the full set of circle quantities (diameter, circumference, arc, sector), open the [circle calculator](https://www.toolcomb.com/circle-calculator). If the “area” you actually wanted is the skin of a box or tank, that is [surface area](https://www.toolcomb.com/surface-area-calculator), not this page. If you needed how much the tank holds, that is the [volume calculator](https://www.toolcomb.com/volume-calculator). ### How to Use This Free Online Area Calculator Using this **square rectangle triangle circle area tool** takes four steps: 1. **Pick the shape** that matches the region. A square is not “almost a rectangle” for the formula: you still use s², which is the special case l = w. A rhombus that is not a square belongs under parallelogram with perpendicular height, not under square. 2. **Enter every required length** as a positive number in the same unit. Height for a triangle, parallelogram, or trapezoid is the perpendicular distance, not a slanted side. Circle wants radius, not diameter. 3. Read the **formula, the plugged-in substitution, and the numeric area**. The number is square in whatever unit you used for the lengths. 4. **Copy** the result or clear the fields. Shape and dimensions are stored locally for up to 30 days. **Note:** there is no unit dropdown. 3 cm and 3 m are both “3” to the formula, but 3² cm² is not 3² m². Convert first. Irregular polygons, composite L-shapes, and GIS polygons are not implemented. ### How to Calculate the Area of a Rectangle A = l × w Rectangle. Length times width, both positive. A rectangle is a four-sided figure with right angles. Opposite sides are equal. Area is the product of two adjacent sides. The default on this page is **l = 8, w = 5**, so A = 40. If you swap length and width you still get 40; order does not matter. If a problem gives perimeter and one side, you do not yet have area: you must recover the other side first. Perimeter of the default rectangle is 2(8+5) = 26, which is not 40 and must not be reported as area. #### Worked example - 8 × 5: - Identify length 8 and width 5 in the same unit. - Multiply: 8 × 5 = 40. - Attach square units: 40 m² if the sides were metres, 40 ft² if they were feet. #### Worked example - square of side 6: - A square is the special rectangle with l = w = s. - A = 6² = 36, not 24 (that would be the perimeter 4s). Scaling: if you double both sides of an 8 × 5 rectangle you get 16 × 10 = 160, which is four times 40, not twice. Area scales with the square of a linear scale factor. That is why a drawing at 1:100 is not “one percent of the real area”; it is one ten-thousandth. ### Area of a Triangle - Base Times Height and Heron's Formula The school formula **A = (1/2) b h** needs a chosen base b and the perpendicular height to that base. A slanted side is not h. If you only have the three side lengths, switch this tool to Heron: let s = (a+b+c)/2, then A = √[s(s−a)(s−b)(s−c)]. The sides must form a genuine triangle (strict triangle inequality). Degenerate sticks laid end to end have area 0 and are rejected. #### Worked example - base 6, height 4: A = (1/2) × 6 × 4 = 12. If someone forgets the 1/2 they report 24, which is the parallelogram with the same base and height, not the triangle. #### Worked example - sides 3, 4, 5 (Heron): s = 6. A = √[6(6−3)(6−4)(6−5)] = √[6 × 3 × 2 × 1] = √36 = 6. That matches (1/2)×3×4 for a right triangle. For a general SSS case that is not right, use the [triangle calculator](https://www.toolcomb.com/triangle-calculator) if you also need angles; this area page only returns the number of square units. Heron can be numerically unhappy on extremely skinny triangles (one tiny height, sides almost collinear). IEEE-754 square roots will still run; the result may lose digits. For ordinary homework sides it is exact enough. ### Area of a Circle - Why the Formula Is πr² A disk of radius r has **A = π r²**. This calculator uses JavaScript **Math.PI** (about 3.141592653589793), not 3.14 and not 22/7. Diameter is 2r. Typing diameter into the radius field overstates area by a factor of four because (2r)² = 4r². Circumference 2πr is a length around the rim; it is not area. If a problem gives circumference C, recover r = C/(2π) first, or use the circle calculator which does that inversion for you. Example: r = 7 gives A = 49π ≈ 153.93804002589985. Example: r = 1 gives π itself. An ellipse with semi-axes a and b has A = πab; when a = b = r the ellipse formula collapses to the circle. Semi-axes are half the full width and full height of the bounding rectangle, not the full diameters. ### Square vs Rectangle vs Parallelogram vs Trapezoid vs Ellipse | Shape | Inputs | Formula | Notes | | --- | --- | --- | --- | | Square | s | s² | All sides equal, right angles | | Rectangle | l, w | l × w | Default 8 × 5 = 40 | | Triangle | b, h or a, b, c | ½bh or Heron | h is perpendicular | | Parallelogram | b, h | b × h | Not base × slanted side | | Trapezoid | a, b, h | (a+b)/2 × h | a, b are the parallel sides | | Circle | r | π r² | Math.PI; not diameter | | Ellipse | a, b | π a b | Semi-axes, not full diameters | A parallelogram with base 10 and perpendicular height 3 has area 30. The same parallelogram with a slanted side of 5 is still 30 if the height is 3; multiplying 10 × 5 would be wrong unless that 5 happens to be the perpendicular height. A trapezoid with parallel sides 3 and 5 and height 4 is (3+5)/2 × 4 = 16. The non-parallel legs are not entered. ### Where Area Calculations Show Up in Real Work #### 1. Flooring, paint coverage, and rooms Flooring is sold by square metre or square foot. A 8 × 5 m room is 40 m² of floor. Walls are rectangles too, but painting a room is closer to surface area of a box minus doors; do not use this plane tool for the ceiling-plus-walls total without adding the faces yourself. Waste factors (5–10%) are a purchasing choice, not part of the geometry. #### 2. Land, gardens, and simple plots A rectangular lot is l × w. A circular bed is πr². A triangular corner leftover is ½bh. Surveyors use more sophisticated polygons and geodesic area on a curved Earth; this page is Euclidean plane geometry only. #### 3. Physics and engineering cross-sections Stress = force / area. Pipe flow uses interior cross-section πr², which is a circle, not the outer wall. If you needed the curved wall of the pipe as a wrap, that is cylinder lateral area on the surface-area page. #### 4. Design, printing, and ellipses An elliptical table top with semi-axes 0.9 m and 0.6 m has area π × 0.9 × 0.6 ≈ 1.69646 m². Using full diameters 1.8 and 1.2 by accident would multiply by four. Print shops quote coverage in square inches of paper, still A = l × w for a rectangular sheet. ### Common Mistakes When Calculating Area - **Reporting perimeter as area.** 8 × 5 is 40 square units, not 26 linear units. The words “around” and “covers” are different questions. - **Using a slanted side as triangle or parallelogram height.** Height is the perpendicular drop to the base line. - **Forgetting 1/2 on a triangle** and getting the parallelogram instead. - **Putting diameter in the circle radius box** and getting 4× too much area. - **Mixing cm and m.** Convert to one unit. Area conversion is the square of the length conversion: 1 m² = 10,000 cm². - **Calling volume “area.”** A tank’s capacity is cubic. A label’s printed face is square. ### Why Area Units Are Squared Two lengths multiplied produce a square unit: m², cm², in², ft². One metre by one metre is a square metre, regardless of whether the region is a circle or a triangle - the unit does not change with the shape, only the numeric factor does. This tool never converts among those names. If l is in inches and you wanted square feet, divide the inch-area by 144 yourself. Changing all linear dimensions by a factor k changes area by k². Triple every side of a square and the area becomes nine times larger. That scaling is why model drawings and full-size floors do not share a 1:1 area ratio with their scale factor. ### Area vs Surface Area vs Volume **Area** (this page) is 2-D measure of a region in a plane. **Surface area** is the total of the faces of a solid - still square units, but you are wrapping a 3-D object. **Volume** is cubic. A cube of side 4 has face area 16 per square, six faces 96 (surface area), and interior 64 (volume). People search “area of a cube” when they mean surface area; this calculator will not invent a sixth dimension for a square. Use the matching tool. A circle is not a sphere. πr² is the disk. 4πr² is the sphere’s skin. (4/3)πr³ is the ball’s volume. Mixing those three is the most common exam error when a word problem mentions “round.” ### Honest Limits of This Area Calculator - **Seven plane shapes only.** No regular pentagon, no sector (use the circle calculator’s central angle), no annulus, no composite L-shape. - **Perpendicular heights.** Triangle, parallelogram, and trapezoid h is altitude, not a side. - **No unit conversion** and no paint-coverage litres. The output is a pure number in square input-units. - **IEEE-754 π** for circle and ellipse. Very large r² can overflow to Infinity. - **Positive lengths only.** Empty, zero, and negative dimensions error. Coordinates of polygons are not accepted. ### Frequently Asked Questions (FAQ) - Area Calculator #### How do I calculate area? Area is the measure of a flat region. Pick the shape, then use its formula: a rectangle is length × width, a square is side², a triangle is (1/2)×base×height (or Heron's formula from three sides), a parallelogram is base × height, a trapezoid is (a+b)/2 × height, a circle is πr², and an ellipse is πab. This calculator rejects zero or negative dimensions and shows the formula it used. Default example: an 8 × 5 rectangle has area 40 square units. #### What is the area of a triangle? The usual school formula is A = (1/2)bh, where b is a chosen base and h is the perpendicular height to that base. If you know all three side lengths instead, Heron's formula uses the semi-perimeter s = (a+b+c)/2 and A = √[s(s−a)(s−b)(s−c)]. The sides must satisfy the triangle inequality or the figure is not a triangle. #### What is the area of a circle? Is it πr²? Yes. A disk of radius r has area A = πr². This page uses JavaScript Math.PI (about 3.141592653589793). Diameter is 2r, so typing the diameter into the radius box overstates area by a factor of four. Circumference is 2πr, which is a length, not an area. #### What is the difference between the area of a rectangle and its perimeter? Area is length × width (square units). Perimeter is 2(length + width) (linear units). An 8 × 5 rectangle has area 40 and perimeter 26. Doubling both sides multiplies area by four but only doubles the perimeter. They answer different questions and must not be mixed. #### Why are area units squared? Area multiplies two lengths, so the unit is a square: m², cm², in², ft². If you measure in centimetres, the result is cm², not cm and not cm³. This tool does not convert units. Mixing metres with centimetres without converting first will be wrong by a factor of 10,000 for area. #### Does this area calculator upload my numbers? No. The arithmetic runs in your browser with JavaScript. Nothing is sent to a server. After the page loads, the tool still works if the network drops. The last shape and dimensions are stored only in localStorage on this device for up to 30 days. ### Why Choose Our Area Calculator? - **Free, no account.** Square, rectangle, triangle, parallelogram, trapezoid, circle, ellipse. - **Private.** Dimensions never leave this browser. - **Formula shown** next to the number so you can quote A = l × w, not only a decimal. - **Heron option** when you know three sides instead of height. - **Math.PI** for circles and ellipses instead of a silent 3.14. - **Rejects non-positive lengths** instead of printing a negative area. - **Works offline** after the first page load. ## Related Tools Discover more free tools that might help with your next task. 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