2D Area Calculator

About 2D Area Calculator

Area Formulas at a Glance

Shape	Area Formula	Perimeter Formula
Rectangle	A = l x w	P = 2(l + w)
Triangle	A = (1/2) x b x h	P = a + b + c
Circle	A = πr²	C = 2πr
Trapezoid	A = (1/2)(a + b) x h	P = a + b + c + d
Parallelogram	A = b x h	P = 2(a + b)
Ellipse	A = πab	P ≈ Ramanujan approximation

Each formula uses the perpendicular height (the shortest distance between base and top), not the slant height. For triangles, this means the height must be measured at a right angle to the base, regardless of the triangle's shape.

How Each Formula Works

The rectangle formula is the simplest: multiply length by width. A 12 m by 8 m room has an area of 96 m² and a perimeter of 2(12 + 8) = 40 m. The diagonal, useful for checking squareness during construction, is sqrt(12² + 8²) = sqrt(208) = 14.42 m.

The triangle formula takes exactly half of the rectangle that would enclose it. A triangle with base 10 and height 6 has area (1/2) x 10 x 6 = 30 square units. If all three sides are known but the height is not, Heron's formula provides an alternative: A = sqrt(s(s-a)(s-b)(s-c)), where s is the semi-perimeter (a+b+c)/2. This formula dates back to Heron of Alexandria (c. 60 AD), though mathematician al-Biruni credited it to Archimedes centuries earlier.

The circle formula uses pi (approximately 3.14159) multiplied by the radius squared. A circle with radius 7 has area π x 49 = 153.94 square units and circumference 2π x 7 = 43.98 units. The relationship between area and circumference means doubling the radius quadruples the area but only doubles the circumference.

The trapezoid formula averages the two parallel sides then multiplies by the height. With parallel sides of 8 and 14 and a height of 5: A = (1/2)(8 + 14) x 5 = (1/2)(22)(5) = 55 square units. The perimeter requires all four side lengths, including the two non-parallel legs.

The parallelogram shares its formula structure with the rectangle: base times height. The difference is that the height must be perpendicular to the base, not the slant side length. A parallelogram with base 10, height 6, and slant side 7 has area 60 square units and perimeter 2(10 + 7) = 34 units.

The ellipse area formula is exact: A = πab, where a is the semi-major axis and b is the semi-minor axis. An ellipse with a = 8 and b = 5 has area π x 8 x 5 = 125.66 square units.

Worked Examples with Full Steps

Rectangle (l = 12, w = 8):

• Area = 12 x 8 = 96 square units
• Perimeter = 2(12 + 8) = 40 units
• Diagonal = sqrt(144 + 64) = sqrt(208) = 14.42 units

Triangle (base = 10, height = 6):

• Area = (1/2) x 10 x 6 = 30 square units

Trapezoid (parallel sides 8 and 14, height 5):

• Area = (1/2)(8 + 14) x 5 = (1/2)(22)(5) = 55 square units

Circle (r = 7):

• Area = π x 49 = 153.94 square units
• Circumference = 2π x 7 = 43.98 units

Ellipse (a = 8, b = 5):

• Area = π x 8 x 5 = 125.66 square units
• h = (8 - 5)² / (8 + 5)² = 9/169 = 0.05325
• Perimeter ≈ π(13)(1 + 3(0.05325)/(10 + sqrt(4 - 3(0.05325)))) ≈ 40.92 units

Why Does the Circle Enclose the Most Area?

Different shapes enclose different areas for the same perimeter. The circle is the most efficient, a fact proven mathematically as the isoperimetric inequality. Jacob Steiner first gave geometric proofs of this in 1838, and Karl Weierstrass provided the first rigorous proof around 1870 using the calculus of variations.

Shape	Perimeter = 40 units	Area	% of Circle
Circle	C = 40, r = 6.366	127.32	100%
Square	s = 10	100.00	78.5%
Equilateral triangle	s = 13.33	77.00	60.5%
Rectangle (2:1 ratio)	l = 13.33, w = 6.67	88.89	69.8%
Rectangle (4:1 ratio)	l = 16, w = 4	64.00	50.3%

The more elongated a rectangle becomes, the less area it encloses for the same perimeter. A square is the most efficient rectangle. This is why circular grain silos use less material than rectangular bins for the same storage capacity, and why efficient buildings tend toward square floor plans.

Common Real-World Areas

Knowing a few reference areas makes it easier to sanity-check results. The table below uses official specifications for each item.

Item	Shape	Dimensions	Area
A4 paper (ISO 216)	Rectangle	210 x 297 mm	623.7 cm²
US Letter (ANSI A)	Rectangle	8.5 x 11 in	93.5 in²
Football pitch (FIFA)	Rectangle	105 x 68 m	7,140 m²
Tennis court (ITF doubles)	Rectangle	23.77 x 10.97 m	260.8 m²
Large pizza (14 in)	Circle	r = 17.78 cm	993 cm²
Basketball court (NBA)	Rectangle	28.65 x 15.24 m	436.6 m²
Basketball court (FIBA)	Rectangle	28 x 15 m	420 m²

FIFA recommends 105 x 68 m for international football, though the Laws of the Game allow pitches to range from 100-110 m long and 64-75 m wide. The ITF tennis court dimensions are fixed at 23.77 x 10.97 m for doubles play. The NBA court (94 x 50 ft) is slightly larger than the FIBA standard (28 x 15 m) used internationally.

Common Mistakes When Calculating Area

Mixing up height and slant side: The most common error with triangles and parallelograms is using a slant side instead of the perpendicular height. The height must be measured at a 90-degree angle to the base. A right triangle makes this easy since one leg is the height, but for obtuse triangles the height may fall outside the triangle itself.

Confusing radius and diameter: Circle area uses the radius (half the diameter), not the full diameter. Using the diameter by mistake gives an area four times too large. A 14-inch pizza has a 7-inch radius, so its area is π x 7² = 153.94 in², not π x 14² = 615.75 in².

Forgetting squared units: Area is always in squared units. If the sides are in metres, the area is in square metres (m²). Converting between area units requires squaring the linear conversion factor: 1 m = 100 cm, but 1 m² = 10,000 cm² (not 100 cm²).

Trapezoid parallel sides: The formula requires the two parallel sides, not any two sides. If the sides are not clearly labelled, check which pair is parallel before calculating.

Unit Conversion for Area

Area conversion factors are the square of the corresponding length conversion factors. Since 1 inch = 2.54 cm, 1 in² = 2.54² = 6.4516 cm². The full reference table:

From	To	Multiply by
cm²	m²	0.0001
m²	ft²	10.764
ft²	m²	0.0929
in²	cm²	6.4516
acres	m²	4,046.86
hectares	m²	10,000
acres	hectares	0.4047
m²	acres	0.000247

An acre is defined as exactly 1/640 of a square mile, equal to 43,560 ft² or 4,046.8564224 m². A hectare is exactly 10,000 m². For quick estimation, 1 hectare is roughly 2.47 acres, and 1 acre is roughly the size of a football pitch without the end zones. For length conversions, the length converter handles the linear equivalents.

The Ellipse Perimeter Problem

Unlike every other shape covered here, there is no exact closed-form formula for the perimeter of an ellipse. This calculator uses Ramanujan's second approximation, published in 1914 by the Indian mathematician Srinivasa Ramanujan:

P ≈ π(a + b)(1 + 3h/(10 + sqrt(4 - 3h)))

where h = (a - b)²/(a + b)² and a, b are the semi-major and semi-minor axes. For nearly circular ellipses, the error is negligible. Even for highly eccentric ellipses, the maximum relative error is bounded by about 0.05%. The formula always slightly underestimates the true perimeter. The area of an ellipse, by contrast, is exact: A = πab, derived by scaling a circle along one axis.

Area in Coordinate Geometry

When working with shapes plotted on a coordinate grid, the Shoelace formula (also called the surveyor's formula) can compute the area of any polygon from its vertex coordinates. For a polygon with vertices (x1, y1), (x2, y2), ... (xn, yn):

A = (1/2)|sum of (xi x y(i+1) - x(i+1) x yi)|

This is useful for irregular plots of land, architectural floor plans, or any polygon that does not fit a standard formula. The midpoint and distance calculator can help find the distances between vertices needed for perimeter calculations.

For circle-specific calculations including arc length, sector area, and chord length, the circumference calculator provides extra detail. For 3D shape volumes including spheres, cylinders, cones, and prisms, the volume calculator extends these area formulas into the third dimension.

All calculations run entirely in the browser. No data is sent to any server.

Sources

• Wolfram MathWorld - Area
• Wolfram MathWorld - Ellipse (Ramanujan Approximation)
• Wolfram MathWorld - Heron's Formula
• Wolfram MathWorld - Isoperimetric Inequality
• ISO 216 - Paper Sizes (A Series)
• IFAB - Laws of the Game (Pitch Dimensions)