Geometry

What geometry studies

This page covers the building blocks of geometry: points, lines and planes, angles, triangles, polygons, circles and solids. Modern-day geometry began in approximately 300 BCE with Euclid’s Elements, where he defined the principles of the line, the point and the plane.^[9] Long before that, the Egyptians and Babylonians knew about area, volume, angles and triangles; the Greeks absorbed this knowledge and set out to prove all of it.^[8]

How it works: definitions, postulates, theorems

Three kinds of statement do different jobs.

• A definition says what a term means. Euclid described the terms he needed and called the descriptions definitions, for example a point as “that which has no part.”^[1] An angle is two rays joined at a common endpoint.^[2] Parallel lines lie in the same plane and never meet; perpendicular lines meet at 90°.^[1]
• A postulate, or axiom, is accepted as true without proof.^{[1, 12]} Euclid’s first says a straight line segment can be drawn joining any two points.^[1]
• A theorem is a statement that has been proved.^[1] A proof is a sequence of logical deductions from axioms and previously proved statements.^[12]

This axiom-and-proof approach, now called the axiomatic method, remains the foundation of mathematics today.^[12]

Key results

• Angles: a right angle is 90° and a straight angle 180°.^[2]
• Triangles: the three interior angles add up to 180°.^[3] If all three sides of one triangle equal those of another, the triangles are congruent (the same shape and size).^[3]
• Polygons, closed flat shapes with straight sides, have interior angles adding up to (n − 2) × 180°, where n is the number of sides.^[4]
• Circles: π ≈ 3.141592654 is the ratio of a circle’s circumference to its diameter.^[4] The circumference is C = 2πr and the area is A = πr², where r is the radius.^{[4, 5]}
• The Pythagorean theorem: in a right triangle, the squares of the two legs add up to the square of the hypotenuse, the side opposite the right angle: a² + b² = c².^{[7, 3]}

In coordinate geometry, Descartes introduced the grid of perpendicular axes now called the Cartesian coordinate system.^[10] The formula for the distance between two points on that grid comes from the Pythagorean theorem.^{[10, 7]} A circle becomes an equation: with centre (h, k) and radius r, it is r² = (x − h)² + (y − k)².^[11]

The math (optional)

• Heptagon (7 sides): the interior angles add up to (7 − 2) × 180° = 900°. In a regular heptagon each angle is 900° ÷ 7 ≈ 128.57°.^[4]
• Circle of radius 3 cm: A = π × 3² = 9π ≈ 28.27 cm².^[5]
• Triangle with base 7 cm and height 3.5 cm: A = ½ × base × height = ½ × 7 × 3.5 = 12.25 cm².^[5]

History

Euclid’s dates are uncertain: OpenStax gives 325–265 BC, while MacTutor says he was born about 325 BC and wrote the Elements in about 300 BC.^{[8, 13, 15]} The Elements has 13 books; books one to six deal with plane geometry.^[13] It begins with definitions, five postulates and axioms Euclid called “common notions”.^[13] More than a thousand editions have been published since it was first printed in 1482.^[13]

The Pythagorean theorem is older than Pythagoras. OpenStax says there is evidence the Babylonians knew it as early as 1900–1100 BC, and MacTutor says it was known to them 1,000 years before Pythagoras, who may have been the first to prove it.^{[7, 14]}

The fifth postulate is different from the other four; the first 28 propositions of the Elements are proved without it.^[15] It concerns parallels. MacTutor gives an equivalent form, known as Playfair’s axiom: through a point not on a line, exactly one parallel can be drawn.^[15] Legendre proved it equivalent to the statement that a triangle’s angles add up to two right angles (180°).^[15] Carl Friedrich Gauss worked on the problem from the 1790s but refrained from publishing for fear of scandal.^[16] In the 1820s Nikolai Lobachevsky and János Bolyai independently tackled the question in a radically new way, and Lobachevsky published first, in 1829.^{[16, 15]} In Lobachevsky’s version, two lines through the point are parallel to the given line.^[15] Riemann discussed a “spherical” geometry with no parallels at all.^[15] In 1868 Beltrami produced a model in which Euclid’s first four postulates held but the fifth did not, and Klein completed it in 1871.^[15] Einstein later placed Riemann’s differential geometry at the core of his general theory of relativity (1915, 1916).^[16]

In 1899 David Hilbert’s Grundlagen der Geometrie revitalised the axiomatic approach.^[17]

Common misconceptions

“Euclid discovered most of geometry.” MacTutor says probably no results in the Elements were first proved by Euclid; the organisation and exposition are his.^[13] Sources differ on the triangle angle sum: OpenStax says the Pythagoreans proposed it and Euclid proved it, while MacTutor doubts that Euclid first proved any result in the Elements.^{[7, 13]}

“A triangle’s angles always add up to 180°.” That holds in Euclidean geometry, where it is equivalent to the fifth postulate.^{[3, 15]} In elliptic geometry they always add up to more.^[16]

“Euclidean geometry is the only possible geometry.” Philosophers including Descartes, Locke, Hume and Kant treated it as a model of certain knowledge.^[16] The 19th-century discoveries turned it into one member of a large family of mathematical theories of space.^[16]

Going further

Next on the map is Calculus. For more worked examples, see chapter 10 of OpenStax’s free Contemporary Mathematics; for the history, the MacTutor and Stanford Encyclopedia articles in the sources below.