Geometry Trivia
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8correct to beat the crowd
This geometry trivia quiz has 118 free questions with answers and starts with the basics everyone half-remembers from school, the angles in a triangle, the name of the longest side, what a rhombus and a trapezoid are, and then keeps going. There are rounds on the five Platonic solids and the elements Plato assigned to them, on pi and the men who spent their lives computing it (one of them wrongly), and on Euclid's Elements, the textbook that ran for two thousand years. From there it reaches the good stories: the drowned Pythagorean who leaked the irrationals, the president who published his own proof of the Pythagorean theorem, the plague that supposedly started the doubling-the-cube problem, the map-colouring puzzle that needed a computer, and the mathematician who turned down a million dollars. Fractals, the Möbius strip, the Klein bottle and the honeycomb theorem round it out. It suits a maths classroom, a pub quiz science round or anyone who likes shapes. Every answer has been checked against an encyclopaedic or primary source and each question shows its citation once answered.
30 of 118 questions with answers and explanations. Play the quiz
Q 01The interior angles of any triangle add up to how many degrees?
180
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.
Q 02A triangle with all three sides the same length is called what?
Equilateral
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.
Q 03A triangle with all sides of different lengths is called what?
Scalene
A triangle with no two sides equal is scalene, from the Greek for 'uneven'; one with exactly two equal sides is isosceles.
Q 04What is the name for the longest side of a right-angled triangle?
Hypotenuse
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.
Q 05How many degrees are in a right angle?
90
A right angle is 90 degrees, a quarter turn; the name comes from the Latin angulus rectus, meaning an upright angle.
Q 06A triangle in which one angle is larger than 90 degrees is called what?
Obtuse
A triangle with one angle over 90 degrees is obtuse; if every angle is under 90 degrees it is acute, and 'obtuse' means blunt.
Q 07How many sides does a hexagon have?
Six
The name comes from Greek hex, meaning six, and each interior angle of a regular one is 120 degrees.
Q 08What is the sum of the interior angles of a pentagon?
540°
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.
Q 09Which four-sided shape has all four sides the same length but does not need right angles?
Rhombus
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.
Q 10What do Americans call a four-sided shape with exactly one pair of parallel sides?
Trapezoid
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.
Q 11A quadrilateral with two pairs of adjacent equal-length sides is named after which toy?
A kite
A kite has two pairs of equal adjacent sides, like the toy; the non-convex version of the shape is called a dart.
Q 12What is the sum of the exterior angles of any convex polygon?
360°
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.
Q 13How many sides does a chiliagon have?
1,000
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?
Q 21Which civilisation used the Pythagorean rule over a thousand years before Pythagoras?
The Babylonians
The Babylonians knew the rule more than a millennium before Pythagoras: Plimpton 322, from about 1800 BC, lists fifteen rows of Pythagorean triples.
Q 22The book The Pythagorean Proposition collects how many different proofs of the theorem?
370
Elisha Loomis's The Pythagorean Proposition (second edition, 1940) collects 370 proofs; only the law of quadratic reciprocity rivals that count.
Q 23Which future US president published his own proof of the Pythagorean theorem in 1876?
James A. Garfield
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.
11
A hendecagon has 11 sides, from the Greek hendeka, 'eleven'; Canada's one-dollar coin, the loonie, is an 11-sided shape.
Q 15Which regular polygon is the simplest one that cannot be constructed with compass and straightedge?
Heptagon
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.
Q 16Gauss proved a regular polygon with how many sides can be built with compass and straightedge?
17
He did it in 1796 at age 19, and it was the first such discovery since the ancient Greeks.
Q 17How many books make up Euclid's Elements?
13
Euclid's Elements runs to 13 books, covering plane and solid geometry, number theory and incommensurable magnitudes, and dominated teaching for two thousand years.
Q 18Which of Euclid's five postulates is known as the parallel postulate?
The fifth
The fifth postulate is the parallel postulate; centuries of failed attempts to prove it from the other four led to hyperbolic and elliptic geometry.
Q 19Euclid, the 'father of geometry', spent his career in which ancient city?
Alexandria
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.
Q 20The Pythagorean theorem is Proposition 47 in which part of Euclid's Elements?
Book I
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.
Q 24According to legend, Hippasus of Metapontum was drowned at sea for revealing the existence of what?
Irrational numbers
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).
Q 25How many Platonic solids are there?
Five
Theaetetus, a contemporary of Plato, may have written the first proof that no others exist.
Q 26Which Platonic solid has twelve pentagonal faces?
Dodecahedron
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'.
Q 27In Plato's Timaeus, which classical element was matched with the cube?
Earth
Plato matched the cube with earth in the Timaeus because of its stability; fire got the tetrahedron, air the octahedron and water the icosahedron.
Q 28The twenty-sided die used in tabletop role-playing games is which solid?
Regular icosahedron
The twenty-sided die is a regular icosahedron, twenty equilateral triangles meeting five at each corner; the adenovirus has the same shape.
Q 29How many edges does a cube have?
12
A cube has 12 edges, along with eight vertices and six faces; Euler's formula V − E + F = 2 checks: 8 − 12 + 6 = 2.
Q 30Who first used the Greek letter π for the ratio of a circle's circumference to its diameter, in 1706?
William Jones
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.