How many right-angled triangles are there with sides that are all
integers less than 100 units?
A dot starts at the point (1,0) and turns anticlockwise. Can you
estimate the height of the dot after it has turned through 45
degrees? Can you calculate its height?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
Pythagoras of Samos was a Greek philosopher who lived from about 580 BC to about 500 BC. Find out about the important developments he made in mathematics, astronomy, and the theory of music.
Prove that a triangle with sides of length 5, 5 and 6 has the same area as a triangle with sides of length 5, 5 and 8. Find other pairs of non-congruent isosceles triangles which have equal areas.
Liethagoras, Pythagoras' cousin (!), was jealous of Pythagoras and came up with his own theorem. Read this article to find out why other mathematicians laughed at him.
Semicircles are drawn on the sides of a rectangle ABCD. A circle passing through points ABCD carves out four crescent-shaped regions. Prove that the sum of the areas of the four crescents is equal to. . . .
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
The area of a square inscribed in a circle with a unit radius is,
satisfyingly, 2. What is the area of a regular hexagon inscribed in
a circle with a unit radius?
A tennis ball is served from directly above the baseline (assume
the ball travels in a straight line). What is the minimum height
that the ball can be hit at to ensure it lands in the service area?
A square of area 40 square cms is inscribed in a semicircle. Find
the area of the square that could be inscribed in a circle of the
This article for pupils and teachers looks at a number that even the great mathematician, Pythagoras, found terrifying.
ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP
: PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED.
What is the area of the triangle PQR?
The diagram shows a very heavy kitchen cabinet. It cannot be lifted but it can be pivoted around a corner. The task is to move it, without sliding, in a series of turns about the corners so that it. . . .
A floor is covered by a tessellation of equilateral triangles, each having three equal arcs inside it. What proportion of the area of the tessellation is shaded?
Find the area of the annulus in terms of the length of the chord
which is tangent to the inner circle.
A rectangular field has two posts with a ring on top of each post.
There are two quarrelsome goats and plenty of ropes which you can
tie to their collars. How can you secure them so they can't. . . .
What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?
A circle touches the lines OA, OB and AB where OA and OB are
perpendicular. Show that the diameter of the circle is equal to the
perimeter of the triangle
If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
A half-cube is cut into two pieces by a plane through the long diagonal and at right angles to it. Can you draw a net of these pieces? Are they identical?
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
Three squares are drawn on the sides of a triangle ABC. Their areas
are respectively 18 000, 20 000 and 26 000 square centimetres. If
the outer vertices of the squares are joined, three more. . . .
This article discusses how every Pythagorean triple (a, b, c) can be illustrated by a square and an L shape within another square. You are invited to find some triples for yourself.
Draw two circles, each of radius 1 unit, so that each circle goes
through the centre of the other one. What is the area of the
Explain how the thirteen pieces making up the regular hexagon shown
in the diagram can be re-assembled to form three smaller regular
hexagons congruent to each other.
A spider is sitting in the middle of one of the smallest walls in a
room and a fly is resting beside the window. What is the shortest
distance the spider would have to crawl to catch the fly?
Take any rectangle ABCD such that AB > BC. The point P is on AB
and Q is on CD. Show that there is exactly one position of P and Q
such that APCQ is a rhombus.
This is the second article on right-angled triangles whose edge lengths are whole numbers.
Equal circles can be arranged so that each circle touches four or
six others. What percentage of the plane is covered by circles in
each packing pattern? ...
Read all about Pythagoras' mathematical discoveries in this article written for students.
If the hypotenuse (base) length is 100cm and if an extra line
splits the base into 36cm and 64cm parts, what were the side
lengths for the original right-angled triangle?
What is the same and what is different about these circle
questions? What connections can you make?
The first of two articles on Pythagorean Triples which asks how many right angled triangles can you find with the lengths of each side exactly a whole number measurement. Try it!
ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.
Cut off three right angled isosceles triangles to produce a
pentagon. With two lines, cut the pentagon into three parts which
can be rearranged into another square.
Four identical right angled triangles are drawn on the sides of a
square. Two face out, two face in. Why do the four vertices marked
with dots lie on one line?
Six circular discs are packed in different-shaped boxes so that the
discs touch their neighbours and the sides of the box. Can you put
the boxes in order according to the areas of their bases?
A right-angled isosceles triangle is rotated about the centre point
of a square. What can you say about the area of the part of the
square covered by the triangle as it rotates?
Can you make sense of the three methods to work out the area of the kite in the square?
Can you make sense of these three proofs of Pythagoras' Theorem?
Prove Pythagoras Theorem using enlargements and scale factors.
It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?
I keep three circular medallions in a rectangular box in which they
just fit with each one touching the other two. The smallest one has
radius 4 cm and touches one side of the box, the middle sized. . . .
A 10x10x10 cube is made from 27 2x2 cubes with corridors between
them. Find the shortest route from one corner to the opposite
A ribbon runs around a box so that it makes a complete loop with two parallel pieces of ribbon on the top. How long will the ribbon be?
The sides of a triangle are 25, 39 and 40 units of length. Find the diameter of the circumscribed circle.
There are thirteen axes of rotational symmetry of a unit cube. Describe them all. What is the average length of the parts of the axes of symmetry which lie inside the cube?
Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.
If a ball is rolled into the corner of a room how far is its centre
from the corner?