Area

Area is the measure of a region's size on a surface. The area of a plane region or plane area refers to the area of a shape or planar lamina, while surface area refers to the area of an open surface or the boundary of a three-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analogue of the length of a curve (a one-dimensional concept) or the volume of a solid (a three-dimensional concept).

The area of a shape can be measured by comparing the shape to squares of a fixed size. In the International System of Units (SI), the standard unit of area is the square metre (written as m²), which is the area of a square whose sides are one metre long. A shape with an area of three square metres would have the same area as three such squares. In mathematics, the unit square is defined to have area one, and the area of any other shape or surface is a dimensionless real number.

There are several well-known formulas for the areas of simple shapes such as triangles, rectangles, and circles. Using these formulas, the area of any polygon can be found by dividing the polygon into triangles. For shapes with curved boundaries, calculus is usually required to compute the area. In all, the problem of determining the area of plane figures was a major motivation for the historical development of calculus.

Formal definition

Area is a way to measure how much space a flat shape takes up. We can think of it like this: imagine you have a special rule book that tells you how to measure different flat shapes. This rule book, or "area function," has a few important rules. For example, the area of any shape is always zero or more. If you put two shapes together, their total area is the sum of their individual areas, minus the area where they overlap. Also, if one shape fits perfectly inside another, the area of the space between them can be found by subtracting the smaller shape’s area from the larger one’s.

One basic shape we know how to measure is a rectangle. If a rectangle is h long and k wide, its area is simply h times k. Using these rules, we can also figure out the area of more complicated shapes by breaking them down into simpler parts.

Units

Every unit of length has a matching unit of area, like the area of a square with sides of that length. We measure areas in square metres (m²), square centimetres (cm²), square millimetres (mm²), and many others such as square feet (ft²) and square miles (mi²). The main unit for area in the SI system is the square metre, which is made from the basic unit of length, the metre.

To change from one area unit to another, you square the change in length units. For example, because 1 foot equals 12 inches, 1 square foot equals 144 square inches. Other examples include 1 square yard equalling many square feet, and 1 square mile equalling even more square yards and feet. There are also special units like the hectare for land and the barn for very tiny areas used in nuclear physics. Some places still use older units like the acre or traditional South Asian units, but these can vary in size.

Main article: Category:Units of area

History

Circle area

Long ago, people discovered ways to find the area of shapes. In the 5th century BCE, Hippocrates of Chios showed that the area of a circle is connected to the square of its diameter. Later, Archimedes used geometry to prove that the area inside a circle is the same as the area of a special triangle. He also came close to finding the exact value of π, which helps us calculate the area of circles.

Triangle area

Quadrilateral area

In the 7th century CE, a mathematician named Brahmagupta found a way to calculate the area of certain four-sided shapes inscribed in circles. Much later, in the 1800s, two German mathematicians found a more general way to calculate the area of any four-sided shape.

General polygon area

In the 1600s, René Descartes introduced a way to place points on a grid, which helped mathematicians find the area of any polygon by knowing the positions of its corners.

Areas determined using calculus

The invention of calculus in the late 1600s gave mathematicians powerful new tools. With these tools, they could find the areas of more complex shapes, like ellipses, and even figure out the surface areas of curved three-dimensional objects.

Area formulas

The area of a shape tells us how much space it covers. For simple shapes like rectangles and squares, we can easily calculate the area using basic formulas. For more complex shapes, there are special methods and formulas that help us find the area.

Polygon formulas

For shapes with straight sides, such as rectangles and triangles, we have straightforward ways to find the area. A rectangle’s area is found by multiplying its length by its width. For a square, since all sides are equal, we square the length of one side to get the area. Triangles have their area calculated by taking half the product of their base and height.

Area of curved shapes

For shapes with curved edges, like circles and ellipses, we use different formulas. The area of a circle is found using the formula π times the radius squared. An ellipse’s area depends on the lengths of its wide and narrow axes, using a formula similar to that of a circle.

Non-planar surface area

Some shapes, like spheres and cylinders, have surfaces that are not flat. To find the area of these surfaces, we use special formulas. For example, the surface area of a sphere is four times π times the radius squared.

General formulas

There are general ways to find the area of many different shapes, whether they are simple or complex. For triangles, there are multiple formulas depending on what information we have about the triangle’s sides or angles. For three-dimensional shapes like cones, cubes, and cylinders, we also have specific formulas to find their surface areas.

List of formulas

Here are some common formulas for finding the area of different shapes:

• Triangle: Half the product of its base and height.
• Rectangle: Product of its length and width.
• Square: Square of its side length.
• Circle: π times the radius squared.
• Ellipse: π times the product of its semi-major and semi-minor axes.
• Sphere: Four times π times the radius squared.
• Cone: π times the radius times the sum of the radius and the slant height.
• Cube: Six times the side length squared.
• Cylinder: Two π times the radius times the sum of the radius and the height.
• Prism: Two times the base area plus the perimeter of the base times the height.

Relation of area to perimeter

For any shape with a closed border, there is a relationship between its area and the length of its border, called the perimeter. A circle has the largest area for a given perimeter. Other shapes can have smaller areas for the same perimeter length.

Fractals

Some special kinds of shapes, called fractals, have areas that change in unusual ways when their sizes change. These shapes are studied in a branch of mathematics called fractal geometry.

Additional common formulas for area:

Shape	Formula	Variables
Square	A = s 2 {\displaystyle A=s^{2}}
Rectangle	A = a b {\displaystyle A=ab}
Triangle	A = 1 2 b h {\displaystyle A={\frac {1}{2}}bh\,\!}
Triangle	A = 1 2 a b sin ⁡ ( γ ) {\displaystyle A={\frac {1}{2}}ab\sin(\gamma )\,\!}
Triangle (Heron's formula)	A = s ( s − a ) ( s − b ) ( s − c ) {\displaystyle A={\sqrt {s(s-a)(s-b)(s-c)}}\,\!}	s = 1 2 ( a + b + c ) {\displaystyle s={\tfrac {1}{2}}(a+b+c)}
Isosceles triangle	A = c 4 4 a 2 − c 2 {\displaystyle A={\frac {c}{4}}{\sqrt {4a^{2}-c^{2}}}}
Regular triangle (equilateral triangle)	A = 3 4 a 2 {\displaystyle A={\frac {\sqrt {3}}{4}}a^{2}\,\!}
Rhombus/Kite	A = 1 2 d e {\displaystyle A={\frac {1}{2}}de}
Parallelogram	A = a h a {\displaystyle A=ah_{a}\,\!}
Trapezoid	A = ( a + c ) h 2 {\displaystyle A={\frac {(a+c)h}{2}}\,\!}
Regular hexagon	A = 3 2 3 a 2 {\displaystyle A={\frac {3}{2}}{\sqrt {3}}a^{2}\,\!}
Regular octagon	A = 2 ( 1 + 2 ) a 2 {\displaystyle A=2(1+{\sqrt {2}})a^{2}\,\!}
Regular polygon ( n {\displaystyle n} sides)	A = n a r 2 = p r 2 {\displaystyle A=n{\frac {ar}{2}}={\frac {pr}{2}}} = 1 4 n a 2 cot ⁡ ( π n ) {\displaystyle \quad ={\tfrac {1}{4}}na^{2}\cot({\tfrac {\pi }{n}})} = n r 2 tan ⁡ ( π n ) {\displaystyle \quad =nr^{2}\tan({\tfrac {\pi }{n}})} = 1 4 n p 2 cot ⁡ ( π n ) {\displaystyle \quad ={\tfrac {1}{4n}}p^{2}\cot({\tfrac {\pi }{n}})} = 1 2 n R 2 sin ⁡ ( 2 π n ) {\displaystyle \quad ={\tfrac {1}{2}}nR^{2}\sin({\tfrac {2\pi }{n}})\,\!}	p = n a   {\displaystyle p=na\ } (perimeter) r = a 2 cot ⁡ ( π n ) , {\displaystyle r={\tfrac {a}{2}}\cot({\tfrac {\pi }{n}}),} a 2 = r tan ⁡ ( π n ) = R sin ⁡ ( π n ) {\displaystyle {\tfrac {a}{2}}=r\tan({\tfrac {\pi }{n}})=R\sin({\tfrac {\pi }{n}})} r : {\displaystyle r:} incircle radius R : {\displaystyle R:} circumcircle radius
Circle	A = π r 2 = π d 2 4 {\displaystyle A=\pi r^{2}={\frac {\pi d^{2}}{4}}} ( d = 2 r : {\displaystyle d=2r:} diameter)
Circular sector	A = θ 2 r 2 = L ⋅ r 2 {\displaystyle A={\frac {\theta }{2}}r^{2}={\frac {L\cdot r}{2}}\,\!}
Ellipse	A = π a b {\displaystyle A=\pi ab\,\!}
Integral	A = ∫ a b f ( x ) d x ,   f ( x ) ≥ 0 {\displaystyle A=\int _{a}^{b}f(x)\mathrm {d} x,\ f(x)\geq 0}
	Surface area
Sphere	A = 4 π r 2 = π d 2 {\displaystyle A=4\pi r^{2}=\pi d^{2}}
Cuboid	A = 2 ( a b + a c + b c ) {\displaystyle A=2(ab+ac+bc)}
Cylinder (incl. bottom and top)	A = 2 π r ( r + h ) {\displaystyle A=2\pi r(r+h)}
Cone (incl. bottom)	A = π r ( r + r 2 + h 2 ) {\displaystyle A=\pi r(r+{\sqrt {r^{2}+h^{2}}})}
Torus	A = 4 π 2 ⋅ R ⋅ r {\displaystyle A=4\pi ^{2}\cdot R\cdot r}
Surface of revolution	A = 2 π ∫ a b f ( x ) 1 + [ f ′ ( x ) ] 2 d x {\displaystyle A=2\pi \int _{a}^{b}\!f(x){\sqrt {1+\left[f'(x)\right]^{2}}}\mathrm {d} x} (rotation around the x-axis)

Area bisectors

Main article: Bisection § Area bisectors and perimeter bisectors

Many lines can split the area of a triangle in half. Three special lines, called medians, connect the middle of each side to the opposite corner and all meet at a point called the centroid. For a parallelogram, any line that goes through the middle point splits its area in half. In circles and ellipses, lines that go through the center, known as diameters or chords, always divide the area equally.

Optimization

When we talk about optimizing area, we look for shapes that cover the most space with the least amount of material. For example, soap bubbles naturally form shapes called minimal surfaces, which have the smallest possible area for their outline. Among all two-dimensional shapes with the same outer edge, a circle has the largest area. Similarly, for polygons with the same number of sides and side lengths, a cyclic polygon—one that fits perfectly inside a circle—will always have the largest area.

Triangles also follow special rules. The triangle with the greatest area for a given perimeter is always an equilateral triangle, meaning all its sides and angles are equal. This same type of triangle also has the largest area when inscribed inside a circle, and the smallest area when its circle is inside the triangle. These patterns show how shapes can be optimized to use space most efficiently.

Main article: Isoperimetric inequality