118 Fun Facts About Geometry

The interior angles of any triangle add up to how many degrees?

A triangle's interior angles sum to 180 degrees, which is why it can have at most one right or obtuse angle; the other two share the rest.

A triangle with all three sides the same length is called what?

A triangle with three equal sides is equilateral; each of its angles is 60 degrees, and the inverted triangle of a yield sign is one.

A triangle with all sides of different lengths is called what?

A triangle with no two sides equal is scalene, from the Greek for 'uneven'; one with exactly two equal sides is isosceles.

What is the name for the longest side of a right-angled triangle?

The hypotenuse, opposite the right angle, takes its name from the Greek for 'stretching under'; its square equals the sum of the squares of the other two sides.

How many degrees are in a right angle?

A right angle is 90 degrees, a quarter turn; the name comes from the Latin angulus rectus, meaning an upright angle.

A triangle in which one angle is larger than 90 degrees is called what?

A triangle with one angle over 90 degrees is obtuse; if every angle is under 90 degrees it is acute, and 'obtuse' means blunt.

How many sides does a hexagon have?

The name comes from Greek hex, meaning six, and each interior angle of a regular one is 120 degrees.

What is the sum of the interior angles of a pentagon?

A pentagon's interior angles sum to 540°, so each corner of a regular pentagon is 108°, which is why regular pentagons cannot tile a floor.

Which four-sided shape has all four sides the same length but does not need right angles?

A rhombus has four equal sides and equal opposite angles; the playing-card diamond suit is a rhombus, and the shape is also called a lozenge.

What do Americans call a four-sided shape with exactly one pair of parallel sides?

In American usage a trapezoid is a quadrilateral with one pair of parallel sides, which the British call a trapezium; the parallel sides are its bases.

A quadrilateral with two pairs of adjacent equal-length sides is named after which toy?

A kite has two pairs of equal adjacent sides, like the toy; the non-convex version of the shape is called a dart.

What is the sum of the exterior angles of any convex polygon?

The exterior angles of any convex polygon sum to 360°: walk once around it, turning at each corner, and you make exactly one full turn.

How many sides does a chiliagon have?

A chiliagon has 1,000 sides, from the Greek chilioi; Descartes used it in his Meditations as a shape one can reason about but not picture.

How many sides does a hendecagon have?

A hendecagon has 11 sides, from the Greek hendeka, 'eleven'; Canada's one-dollar coin, the loonie, is an 11-sided shape.

Which regular polygon is the simplest one that cannot be constructed with compass and straightedge?

The regular heptagon is the first that cannot be drawn with compass and straightedge alone; it can be built with a marked ruler by neusis.

Gauss proved a regular polygon with how many sides can be built with compass and straightedge?

He did it in 1796 at age 19, and it was the first such discovery since the ancient Greeks.

How many books make up Euclid's Elements?

Euclid's Elements runs to 13 books, covering plane and solid geometry, number theory and incommensurable magnitudes, and dominated teaching for two thousand years.

Which of Euclid's five postulates is known as the parallel postulate?

The fifth postulate is the parallel postulate; centuries of failed attempts to prove it from the other four led to hyperbolic and elliptic geometry.

Euclid, the 'father of geometry', spent his career in which ancient city?

Euclid worked in Alexandria around 300 BC under Ptolemy I; almost nothing else is known of his life, most of it from Proclus and Pappus centuries later.

The Pythagorean theorem is Proposition 47 in which part of Euclid's Elements?

Proposition 47 of Book I is the Pythagorean theorem; Proposition 48, right after it, proves the converse: if the squares add up, the angle is right.

Which civilisation used the Pythagorean rule over a thousand years before Pythagoras?

The Babylonians knew the rule more than a millennium before Pythagoras: Plimpton 322, from about 1800 BC, lists fifteen rows of Pythagorean triples.

The book The Pythagorean Proposition collects how many different proofs of the theorem?

Elisha Loomis's The Pythagorean Proposition (second edition, 1940) collects 370 proofs; only the law of quadratic reciprocity rivals that count.

Which future US president published his own proof of the Pythagorean theorem in 1876?

James A. Garfield, then an Ohio congressman, published a trapezoid-based proof in the New-England Journal of Education in 1876, five years before becoming president.

According to legend, Hippasus of Metapontum was drowned at sea for revealing the existence of what?

Legend says the Pythagoreans drowned Hippasus for revealing irrational numbers, probably that a square's diagonal cannot be measured exactly against its side (√2 is irrational).

How many Platonic solids are there?

Theaetetus, a contemporary of Plato, may have written the first proof that no others exist.

Which Platonic solid has twelve pentagonal faces?

The dodecahedron has twelve pentagonal faces; in the Timaeus, Plato says the god used this fifth solid for 'embroidering the constellations on the whole heaven'.

In Plato's Timaeus, which classical element was matched with the cube?

Plato matched the cube with earth in the Timaeus because of its stability; fire got the tetrahedron, air the octahedron and water the icosahedron.

The twenty-sided die used in tabletop role-playing games is which solid?

The twenty-sided die is a regular icosahedron, twenty equilateral triangles meeting five at each corner; the adenovirus has the same shape.

How many edges does a cube have?

A cube has 12 edges, along with eight vertices and six faces; Euler's formula V − E + F = 2 checks: 8 − 12 + 6 = 2.

Who first used the Greek letter π for the ratio of a circle's circumference to its diameter, in 1706?

William Jones introduced π for the ratio in his 1706 Synopsis Palmariorum Matheseos; Euler's adoption of the symbol from the 1730s made it universal.

Around 250 BC, Archimedes bounded pi using polygons with how many sides?

Archimedes started with hexagons inside and outside a circle and doubled to 96 sides, proving pi lies between 223/71 and 22/7.

William Shanks calculated pi to 707 digits, but his figures were wrong from which digit onward?

Shanks's 1873 value went wrong at the 528th decimal place; the faulty digits were even painted in the pi room of the Palais de la Découverte in Paris until 1949.

Pi Day is celebrated on which date in the United States?

Pi Day is 14 March, written 3/14 in US date order; in 2015 the moment 3/14/15 9:26:53 matched pi's first ten digits, while Tau Day fans prefer 28 June.

Squaring the circle was proved impossible in 1882 once pi was shown to be what?

Ferdinand von Lindemann proved in 1882 that pi is transcendental, not the root of any polynomial with rational coefficients, so no compass-and-straightedge construction can square the circle.

Who proved in 1837 that an arbitrary angle cannot be trisected with compass and straightedge?

Pierre Wantzel's 1837 paper showed angle trisection and cube doubling are impossible with compass and straightedge, nine years before Galois's work was published by Liouville.

The ancient problem of doubling the cube is named after the citizens of which Greek island?

The Delian problem is named for Delos, whose citizens were told by an oracle to double Apollo's cubic altar to end a plague; Plato read it as a lesson in geometry.

Which Hungarian published non-Euclidean geometry in the early 1830s, independently of Lobachevsky?

János Bolyai's 1832 appendix to his father Farkas's textbook laid out a non-Euclidean geometry; Gauss told Farkas he had reached the same ideas years earlier without publishing.

In which geometry are there infinitely many lines through a point that never meet a given line?

In hyperbolic geometry infinitely many lines through a point miss a given line; in elliptic geometry none do, and in Euclidean geometry exactly one.

Who proved the Poincaré conjecture in 2002–03, then turned down a $1 million prize?

Grigori Perelman posted his proof on arXiv in 2002-03 and refused the 2010 Millennium Prize, saying Richard Hamilton, whose Ricci flow programme he completed, deserved equal credit.

The four colour theorem, proved in 1976, was the first major theorem proved with the help of what?

Appel and Haken's 1976 proof checked nearly 2,000 map configurations by computer, the first major theorem to rely on one; many mathematicians balked because no human could verify the case analysis.

Francis Guthrie first raised the four colour problem in 1852 while trying to colour a map of what?

Francis Guthrie noticed that the counties of England could be coloured with four colours; his brother Frederick took the puzzle to Augustus De Morgan, who could not settle it.

Which three regular polygons are the only ones that can tile a flat surface by themselves?

Only the equilateral triangle, square and regular hexagon tile the plane alone, because their corner angles (60°, 90°, 120°) divide 360° evenly.

Which Dutch artist, inspired by the Alhambra's tiles, became famous for tessellations of interlocking animals?

M. C. Escher copied the Alhambra's Moorish tilings on a 1936 visit and went on to draw interlocking-animal tessellations, later hyperbolic ones in his Circle Limit prints.

Which regular grid divides a plane into equal areas with the least total perimeter?

A hexagonal grid divides the plane into equal cells with the least total perimeter; Varro stated this honeycomb conjecture around 36 BC and Thomas Hales proved it in 1999.

Who coined the word 'fractal' in 1975?

Benoît Mandelbrot coined 'fractal' in 1975 from the Latin fractus, 'broken', in his book Les Objets fractals.

The Koch snowflake curve has a finite area but what kind of perimeter?

The Koch snowflake's perimeter is infinite, growing by a factor of 4/3 at every step, although its area stays finite; Helge von Koch described it in 1904.

Colouring the odd numbers black in Pascal's famous number pyramid approximates which fractal?

Colouring Pascal's triangle's odd entries gives the Sierpiński triangle; Wacław Sierpiński described the fractal in 1915, but the pattern appears in medieval Cosmati mosaics centuries earlier.

The golden ratio, roughly 1.618, is the ratio of a regular pentagon's diagonal to what?

In a regular pentagon each diagonal is φ ≈ 1.618 times its side, the link that puts the golden ratio into the dodecahedron and icosahedron.

Luca Pacioli's 1509 book about the golden ratio, illustrated by Leonardo da Vinci, was called what?

Pacioli's Divina Proportione, written in Milan in 1498 and printed in Venice in 1509, carried Leonardo's drawings of polyhedra; the term 'golden section' came only in the 1800s.

The Möbius strip was discovered as a mathematical object in 1858 by August Möbius and which other German?

Johann Benedict Listing found the one-sided band independently of Möbius in 1858; a third-century Roman mosaic shows the same twist, as does the recycling symbol.

Which surface, first described in 1882, has no distinct inside or outside and self-intersects in 3D space?

The Klein bottle, described in 1882, is a closed one-sided surface with no boundary edge, unlike the Möbius strip; in three dimensions it must pass through itself.

Topologically, a coffee mug with a handle is the same as which shape?

A coffee mug with a handle is topologically a doughnut: both have genus one, a single hole, so one can be stretched into the other without cutting.

In Thales's theorem, joining both ends of a diameter to any third point on the circle makes what kind of angle?

Thales's theorem says the angle in a semicircle is a right angle; Euclid recorded it as Proposition 31 of Book III of the Elements.

Archimedes asked for which two shapes to be carved on his tomb, marking his proudest discovery?

He had shown that a sphere has two-thirds the volume of the cylinder that just encloses it; Cicero later found the neglected tomb.

The Reuleaux triangle, a shape of constant width, is used in drill bits that bore holes of what shape?

A Reuleaux triangle turning inside a square stays in contact with all four sides, so Harry Watts's 1910s drill bores an almost square hole.

Which shape has the smallest surface area of any surface enclosing a given volume?

The sphere encloses a given volume with the least surface area, which is why surface tension pulls a soap bubble into a sphere.

James Thomson, who first printed the word 'radian' in 1873, was the brother of which physicist?

James Thomson's brother was William Thomson, Lord Kelvin; 'radian' first appeared in print in James's 1873 examination questions, and one radian is about 57.3 degrees.

The first geodesic dome was built after World War I in Jena, Germany, to house what?

Walther Bauersfeld built the first geodesic dome on the Zeiss factory roof in Jena in 1922-23 to test a planetarium projector; Buckminster Fuller patented his version in 1954.

According to one theory, why does a full circle have 360 degrees?

One theory is that 360 is roughly the days in a year: Babylonian astronomers used a 360-day ideal year of twelve 30-day months, which also suited their base-60 counting.

The orbit of each planet is approximately an ellipse with the Sun located where?

Kepler's first law puts the Sun at one focus of each planet's elliptical orbit; strictly the focus is the Sun-planet barycentre, which lies inside the Sun for most planets.

Who created analytic geometry, describing curves with coordinates and equations, in the 17th century?

René Descartes (La Géométrie, 1637) and Pierre de Fermat independently created analytic geometry, turning curves into equations so algebra and geometry stopped being separate.

The Mandelbrot set was first defined and drawn in 1978 by Robert Brooks and which collaborator?

Robert Brooks and Peter Matelski drew the first crude plot of the set in 1978; Mandelbrot's high-quality images followed in 1980 at IBM.

What name is given to a polygon whose vertices all lie on a single circle?

A polygon whose vertices all lie on one circle is cyclic; every triangle is cyclic, and a quadrilateral is cyclic exactly when its opposite angles sum to 180°.

Thales of Miletus used geometry in the 6th century BC to calculate the height of what?

Thales measured the height of the pyramids from their shadows, comparing them with his own shadow; he is also said to have found the distance of ships from shore.

What value of pi does the Rhind Papyrus's rule for a circle's area imply?

The Rhind Papyrus (c. 1550 BC) takes a circle's area as that of a square on 8/9 of its diameter, implying pi ≈ 256/81, about 3.16.

Which Greek mathematician's Conics gave the ellipse, parabola and hyperbola their names?

Apollonius of Perga's Conics (3rd century BC) studied the curves cut from a cone and gave the ellipse, parabola and hyperbola their names.

Which Persian mathematician found geometric solutions to cubic equations using intersecting conics?

Omar Khayyam solved cubic equations geometrically by intersecting conic sections in his 11th-century Treatise on Algebra; his work on quadrilaterals fed into later studies of the parallel postulate.

Which mathematician's Erlangen programme made symmetry the central idea uniting geometries?

Felix Klein's 1872 Erlangen programme classified geometries by their symmetry groups, covering Euclidean and non-Euclidean geometry alike.

Which country's flag, two stacked pennants, is the world's only national flag that is not four-sided?

Nepal joined the pennons of two rival branches of its ruling dynasty into a single flag. It is also the only national flag that is taller than it is wide.

Which everyday name for a rhombus is also given to a baseball infield, really a 90-foot square?

Diamond is the name the playing-card suit takes from the same shape. A baseball diamond's four bases mark the corners of a square with 90-foot sides.

Which Chinese puzzle cuts a square into seven flat pieces, one of them a parallelogram?

Tangram pieces are called tans, and the parallelogram is the only one that must be flipped over to make its mirror image. Trading ships carried the puzzle from China to Europe and America.

The word trapezoid traces back to the Greek trapeza, the name of which piece of furniture?

Table in Greek was trapeza, built from words for 'four' and 'foot', and the shape was a trapezion, a 'little table'. Classical Greek tables are said to have had one side shorter than the one opposite.

Which other word for a quadrilateral also names the enclosed courtyard of an Oxford college?

Quadrangle literally means 'four-angle', and the courtyard is often shortened to 'quad'. At Cambridge the same enclosed spaces are always called courts instead.

A parallelogram whose two diagonals are exactly the same length as each other must be which shape?

A rectangle is the only kind of parallelogram with equal diagonals. Its name comes from the Latin rectangulus, built from rectus, 'right', and angulus, 'angle'.

Which word describes a four-right-angled shape, such as a banknote, that is longer than it is wide?

Oblong comes from the Latin oblongus, 'more long than broad', so a perfect square can never be one. Euclid's Elements listed the two as separate kinds of quadrilateral.

Copies of any uncrossed quadrilateral, however lopsided, can always be arranged to do what?

Every such quadrilateral can tile an endless floor with no gaps, however irregular its sides and angles. The trick is to keep rotating copies about the midpoints of their edges.

A quadrilateral whose sides cross each other is nicknamed after a butterfly or which item of neckwear?

A bow tie quadrilateral is also called crossed or self-intersecting. Traced around its outline, its four 'interior' angles add up to 720° rather than the usual 360°.

The two sides of a trapezoid that are not its bases share their name with which parts of the body?

Legs is also the name for the two shorter sides of a right-angled triangle. On a trapezoid they are sometimes called the lateral sides instead.

Roger Penrose's non-repeating tiling uses two quadrilaterals: a kite and which partner, named after a missile?

A dart is the arrowhead-shaped partner of the kite, and the two tiles can be joined to make a rhombus. Laid by Penrose's rules, they cover a surface without the pattern ever repeating.

Which word describes a quadrilateral with one angle over 180°, so that a diagonal lies outside it?

Concave quadrilaterals look dented, like an arrowhead, yet their four interior angles still add up to 360° just as in any other simple quadrilateral.

In every parallelogram, from a leaning one to a perfect square, what do the two diagonals always do?

The diagonals bisect each other in every parallelogram, so their crossing point is the midpoint of both. Only special cases such as the rhombus add a right-angled crossing.

A trapezoid's midsegment runs midway between its two bases; its length is what of the two base lengths?

Their average is the answer, so the midsegment sits between the two bases in length as well as in position. It is also known as the median of the trapezoid.

Which quadrilateral has equal diagonals and one line of symmetry that cuts through two opposite sides?

Isosceles trapezoid and kite are the only two uncrossed quadrilaterals with exactly one line of symmetry. The kite's mirror line runs through two corners instead of two sides.

Joining the midpoints of a convex quadrilateral's sides makes a parallelogram with what share of its area?

One half of the original area always ends up inside. The new shape is called the Varignon parallelogram after Pierre Varignon, whose proof was only published after his death, in 1731.

Which rarely used Greek-derived word is a counterpart to the Latin 'quadrilateral' for any four-sided flat shape?

Tetragon joins the Greek tetra, 'four', with gon, 'corner' or 'angle'. Quadrilateral, the usual name, is built from Latin instead and counts sides rather than corners.

A square is the only quadrilateral with all its sides and all its angles equal, earning it which label?

Regular polygons have every side and every angle equal, and the square is the only four-sided one. That makes it the quadrilateral with the least perimeter for a given area.

A leaning parallelogram's area is not base times slanted side but base times which other measurement?

Its height, measured at right angles to the base, is what counts. Slice a corner off one end and slide it to the other, and the shape squares up with the same base and area.

A square has more lines of symmetry than any other quadrilateral; counting its diagonals, how many has it?

Four lines work: the two diagonals plus the two lines through the midpoints of opposite sides. Add its rotations and the square has eight symmetries in all.

Which carmaker's logo of three rhombuses is a picture of what its own Japanese name means?

Mitsubishi joins mitsu, 'three', with hishi, the Japanese word for a water caltrop and hence for a rhombus. Hishi turns into 'bishi' when the two words are run together.

Which circus apparatus takes its name, by way of French, from the Late Latin word for a four-sided shape?

Trapeze probably got the name because its bar, its ropes and the ceiling outline a trapezium. Frenchman Jules Léotard, who performed the first flying trapeze routine in 1859, also gave his name to the leotard.

A classic quilt pattern made only of 60° rhombuses tricks the eye into seeing a stack of which solids?

Cubes appear because the pattern matches a corner-on view of stacked boxes. The eye can read the stack in two different ways, so it is also called the reversible cubes illusion.

Which South American civilisation built doorways, windows and wall niches in a distinctive trapezoid shape?

The Inca gave most wall openings sloping sides, and some were dressed up with double or triple jambs. A doorway of this kind is one of the familiar sights of Machu Picchu.

The four corners of a quadrilateral are properly called what, from a Latin word for a summit or whirlpool?

Vertices is the plural of vertex, Latin for 'highest point' and originally a whirling column, from vertere, 'to turn'. Each diagonal of a quadrilateral joins an opposite pair of them.

Which muscles between the spine and shoulder blades are named after a slanted, unequal-sided parallelogram?

Rhomboids keep a name mathematicians have dropped: a rhomboid was a parallelogram with unequal neighbouring sides and no right angles. The muscles pull the shoulder blades together.

The name rhombus comes from the Greek rhombos, used for objects such as a bullroarer that do what?

Spin is the root idea: rhombos could mean a spinning top, a bullroarer or a sorcerer's magic wheel. The Greeks also gave the same name to a kind of flatfish.

Which term describes any quadrilateral, such as a kite or a rhombus, whose diagonals cross at right angles?

Orthodiagonal quadrilaterals share a handy area rule: multiply the two diagonals together and divide by two. No other convex quadrilateral with the same diagonals encloses more area.

Which Greek astronomer's theorem on the diagonals of cyclic quadrilaterals helped him build a table of chords?

Ptolemy needed the chord table for the astronomy in his Almagest, and it became the earliest trigonometric table extensive enough for practical use.

Which 7th-century Indian mathematician's formula gives a cyclic quadrilateral's area from its side lengths?

Brahmagupta's formula extends Heron's rule for a triangle: shrink one side of the quadrilateral to nothing and the two formulas become the same.

In the Elements, Euclid defined a point, the simplest object in geometry, as 'that which has' what?

No part means a point cannot be cut any smaller: it marks an exact position but has no length, width or thickness of its own.

The first postulate in Euclid's Elements says a straight line can be drawn between any how many points?

Two points are enough, and Euclid's wording was 'to draw a straight line from any point to any point'. His second postulate lets that line be extended as far as you like.

Which geometric figure starts at a fixed point and runs on forever in one direction, like a beam of light?

A ray has one fixed end, known as its initial point. Any point on a line splits that line into two of them, heading in opposite directions.

Three points always fix exactly one plane unless all three sit in one straight row, a condition called what?

Collinear points turn up in surprising places: in any triangle the centroid, circumcentre and orthocentre always sit on one straight line, known as the Euler line.

Which word describes points or lines that all lie in one flat surface, like dots on a single sheet of paper?

Coplanar lines must either cross or run parallel. Only out in space, where two lines need not share a plane, can a pair manage to do neither.

Which word names the point, line or curve that two geometric figures have in common, as where two roads cross?

Intersection can also come up empty: two parallel lines share no points at all. Two different straight lines that do cross can only ever share a single point.

In geometry notation, the letters AB under a plain bar with no arrowheads stand for which figure?

A segment has two endpoints and a length that can be measured. The bar drawn above the two letters is known as an overline or vinculum.

Euclid's Elements defines a line as a length that has none of which other measurement?

Breadth is what a line lacks: Euclid called it 'breadthless length'. For him a line could curve, and a straight one was a line that 'lies evenly with the points on itself'.

Which word describes two lines that never meet yet are not parallel, like opposite edges of a tetrahedron?

Skew lines cannot exist on a flat sheet, where two lines that never meet are always parallel. Out in space they can miss each other while pointing different ways.

Which Latin word for 'place' names the set of all points that obey one rule, as a circle's points do?

Locus is how older textbooks defined shapes: a circle was the locus of a point moving at a fixed distance from a centre. Modern mathematics prefers to speak of sets of points.

In a perspective drawing, parallel lines such as straight railway tracks appear to meet at what?

The vanishing point is where the drawn images of parallel lines come together, although the real lines never meet. Lines at right angles to the picture give one-point perspective.

On a coordinate grid, what name is given to the point (0, 0), where the x-axis and y-axis cross?

Origin is the fixed reference for every other position on the grid and is usually labelled with the letter O. It splits each axis into a positive side and a negative side.

A point has none, a line has one and a plane has two of which basic geometric property?

Dimensions count the coordinates needed to pin down a position: one number is enough along a line, but a surface such as a globe needs two, like latitude and longitude.

Which word names a starting statement, such as 'all right angles are equal', that is accepted without proof?

Axiom and postulate now mean the same thing: a statement taken to be true as a starting point for reasoning. Euclid believed his were self-evident facts about physical reality.

Which 1884 novella by Edwin Abbott is narrated by a square whose entire world is a single plane?

Flatland's men are polygons with various numbers of sides, and its narrator signs himself 'A Square'. Abbott, a schoolmaster, used the flat world to satirise Victorian class and gender hierarchies.

Which word describes three or more lines that all pass through one single point, as a triangle's medians do?

Concurrent lines are everywhere in a triangle: its three medians meet at the centroid, the spot on which a cut-out triangle of even thickness would balance.

On a globe, which curves, such as the equator, play the part that straight lines play on a flat plane?

Great circles contain the shortest path between any two places on a globe, which is why they are used to plan routes for air and sea navigation.

In modern geometry, point, line and plane are deliberately left undefined, earning them which label?

Primitive notions are the handful of starting words used without explaining their meanings, because every definition has to rest on earlier terms and reasoning must begin somewhere.

Which German mathematician said points, lines and planes might as well be called chairs and beer mugs?

David Hilbert meant that only the stated relationships between the objects matter, not what they are. His 1899 book rebuilt Euclid's geometry on a short list of carefully chosen assumptions.

Which branch of geometry, born from perspective art, adds 'points at infinity' where parallel lines meet?

Projective geometry lets parallel lines meet, like railway tracks at the horizon, and has no use for measuring angles. Kepler and Desargues each came up with the point at infinity independently.