An illustration of Desargues' theorem, a result in Euclidean and projective geometry
3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)
A triangle immersed in a saddle-shape plane (a hyperbolic paraboloid), as well as two diverging ultraparallel lines.
Archimedes used the method of exhaustion to calculate the area under a parabola.
A European and an Arab practicing geometry in the 15th century
The distribution of prime numbers is a central point of study in number theory. This Ulam spiral serves to illustrate it, hinting, in particular, at the conditional independence between being prime and being a value of certain quadratic polynomials.
An osculating circle
Alhazen, 11th-century Arab mathematician and physicist
Woman teaching geometry. Illustration at the beginning of a medieval translation of Euclid's Elements, (c. 1310).
The quadratic formula expresses concisely the solutions of all quadratic equations
Isaac Newton developed the use of calculus in his laws of motion and gravitation.
An illustration of Euclid's parallel postulate
Rubik's cube: the study of its possible moves is a concrete application of group theory
Gottfried Wilhelm Leibniz was the first to state clearly the rules of calculus.
Acute (a), obtuse (b), and straight (c) angles. The acute and obtuse angles are also known as oblique angles.
The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC.
Maria Gaetana Agnesi
A sphere is a surface that can be defined parametrically (by  or implicitly (by x2 + y2 + z2 − r2 = 0.)
Archimedes used the method of exhaustion, depicted here, to approximate the value of pi.
The logarithmic spiral of the Nautilus shell is a classical image used to depict the growth and change related to calculus.
Visual checking of the Pythagorean theorem for the (3, 4, 5) triangle as in the Zhoubi Suanjing 500–200 BC. The Pythagorean theorem is a consequence of the Euclidean metric.
The numerals used in the Bakhshali manuscript, dated between the 2nd century BC and the 2nd century AD.
The Koch snowflake, with fractal dimension=log4/log3 and topological dimension=1
A page from al-Khwārizmī's Algebra
A tiling of the hyperbolic plane
Leonardo Fibonacci, the Italian mathematician who introduced the Hindu–Arabic numeral system invented between the 1st and 4th centuries by Indian mathematicians, to the Western World.
Differential geometry uses tools from calculus to study problems involving curvature.
Leonhard Euler created and popularized much of the mathematical notation used today.
A thickening of the trefoil knot
Carl Friedrich Gauss, known as the prince of mathematicians
Quintic Calabi–Yau threefold
The front side of the Fields Medal
Discrete geometry includes the study of various sphere packings.
The Cayley graph of the free group on two generators a and b
Bou Inania Madrasa, Fes, Morocco, zellige mosaic tiles forming elaborate geometric tessellations
The Pythagoreans discovered that the sides of a triangle could have incommensurable lengths.
Euler's identity, which American physicist Richard Feynman once called "the most remarkable formula in mathematics".

Geometry is, with arithmetic, one of the oldest branches of mathematics.

- Geometry

Mathematics is an area of knowledge that includes such topics as numbers (arithmetic, number theory), formulas and related structures (algebra), shapes and the spaces in which they are contained (geometry), and quantities and their changes (calculus and analysis).

- Mathematics

Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds.

- Differential geometry

Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.

- Calculus

Since then, the scope of geometry has been greatly expanded, and the field has been split in many subfields that depend on the underlying methods—differential geometry, algebraic geometry, computational geometry, algebraic topology, discrete geometry (also known as combinatorial geometry), etc.—or on the properties of Euclidean spaces that are disregarded—projective geometry that consider only alignment of points but not distance and parallelism, affine geometry that omits the concept of angle and distance, finite geometry that omits continuity, and others.

- Geometry

The first systematic or rigorous treatment of geometry using the theory of infinitesimals and notions from calculus began around the 1600s when calculus was first developed by Gottfried Leibniz and Isaac Newton.

- Differential geometry

This was a necessary precursor to the development of calculus and a precise quantitative science of physics.

- Geometry

Such curves can be defined as graph of functions (whose study led to differential geometry).

- Mathematics

Differential geometry

- Calculus
An illustration of Desargues' theorem, a result in Euclidean and projective geometry

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The combined area of these three shapes is approximately 15.57 squares.

Area

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Quantity that expresses the extent of a region on the plane or on a curved surface.

Quantity that expresses the extent of a region on the plane or on a curved surface.

The combined area of these three shapes is approximately 15.57 squares.
This square and this disk both have the same area (see: squaring the circle).
A square metre quadrat made of PVC pipe.
Although there are 10 mm in 1 cm, there are 100 mm2 in 1 cm2.
The area of this rectangle is lw.
A diagram showing how a parallelogram can be re-arranged into the shape of a rectangle.
A parallelogram split into two equal triangles.
A circle can be divided into sectors which rearrange to form an approximate parallelogram.
Archimedes showed that the surface area of a sphere is exactly four times the area of a flat disk of the same radius, and the volume enclosed by the sphere is exactly 2/3 of the volume of a cylinder of the same height and radius.
Integration can be thought of as measuring the area under a curve, defined by f(x), between two points (here a and b).
The area between two graphs can be evaluated by calculating the difference between the integrals of the two functions

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.

For shapes with curved boundary, calculus is usually required to compute the area.

In addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, and is a basic property of surfaces in differential geometry.