Area

Area is the measure of how much space a flat surface takes up. The area of a flat shape or a thin, flat piece of material is called a shape or planar lamina. Surface area is the area of the outside of a three-dimensional object, like the outside of a ball or a box.

Area can be thought of as how much material you would need to make a model of the shape, or how much paint you would need to cover it with one coat. It is like the length of a line (a one-dimensional thing) or the volume of a solid object (a three-dimensional thing), but in two dimensions.

The area of a shape can be measured by comparing it 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 that is one metre on each side. A shape with an area of three square metres would cover the same space as three such squares. In mathematics, the unit square is defined to have an area of one, and the area of any other shape is a dimensionless real number.

There are easy ways to find the area of simple shapes such as triangles, rectangles, and circles. For more complicated shapes made of straight lines, you can divide the shape into triangles to find the area. For shapes with curved edges, special math called calculus is usually needed. The study of area was very important for the historical development of calculus.

Formal definition

Area is a way to measure how much space a flat shape takes up. Think of it like this: imagine you have a rule book that tells you how to measure 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, you can find the area of the space between them 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. We measure areas in square metres (m²), square centimetres (cm²), square millimetres (mm²), and others like square feet (ft²) and square miles (mi²). The main unit for area is the square metre. It comes 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. 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, but these can vary in size.

Main article: Category:Units of area

History

Circle area

Long ago, people learned how to find the size, or area, of different shapes. In the 5th century BCE, Hippocrates of Chios showed that the area of a circle relates to the square of its width. Later, Archimedes used geometry to prove that the area inside a circle matches 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 discovered a way to find the area of some four-sided shapes drawn inside circles. Much later, in the 1800s, two German mathematicians found a better way to calculate the area of any four-sided shape.

General polygon area

In the 1600s, René Descartes introduced a method to place points on a grid. This helped mathematicians find the area of any polygon by knowing where its corners are located.

Areas determined using calculus

The invention of calculus in the late 1600s gave mathematicians 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 calculate the area using easy formulas.

Polygon formulas

For shapes with straight sides, such as rectangles and triangles, we have simple 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. They 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