How many right-angled triangles are there with sides that are all
integers less than 100 units?
A ribbon is nailed down with a small amount of slack. What is the
largest cube that can pass under the ribbon ?
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.
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 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
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
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?
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.
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?
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
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?
This is the second article on right-angled triangles whose edge lengths are whole numbers.
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!
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?
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
If the altitude of an isosceles triangle is 8 units and the perimeter of the triangle is 32 units.... What is the area of the triangle?
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?
If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
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? ...
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 Pythagoras Theorem using enlargements and scale factors.
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 1 metre cube has one face on the ground and one face against a
wall. A 4 metre ladder leans against the wall and just touches the
cube. How high is the top of the ladder above the ground?
You are given a circle with centre O. Describe how to construct with a straight edge and a pair of compasses, two other circles centre O so that the three circles have areas in the ratio 1:2:3.
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
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?
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?
Rotate a copy of the trapezium about the centre of the longest side
of the blue triangle to make a square. Find the area of the square
and then derive a formula for the area of the trapezium.
Is the sum of the squares of two opposite sloping edges of a
rectangular based pyramid equal to the sum of the squares of the
other two sloping edges?
This article for pupils and teachers looks at a number that even the great mathematician, Pythagoras, found terrifying.
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. . . .
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?
What is the same and what is different about these circle
questions? What connections can you make?
A fire-fighter needs to fill a bucket of water from the river and
take it to a fire. What is the best point on the river bank for the
fire-fighter to fill the bucket ?.
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. . . .
Chris is enjoying a swim but needs to get back for lunch. If she can swim at 3 m/s and run at 7m/sec, how far along the bank should she land in order to get back as quickly as possible?
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?
A napkin is folded so that a corner coincides with the midpoint of
an opposite edge . Investigate the three triangles formed .
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?
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. . . .
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
The ten arcs forming the edges of the "holly leaf" are all arcs of
circles of radius 1 cm. Find the length of the perimeter of the
holly leaf and the area of its surface.
Can you make sense of these three proofs of Pythagoras' Theorem?
Can you make sense of the three methods to work out the area of the kite in the square?
Read all about Pythagoras' mathematical discoveries in this article written for students.
A ladder 3m long rests against a wall with one end a short distance from its base. Between the wall and the base of a ladder is a garden storage box 1m tall and 1m high. What is the maximum distance. . . .
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?
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?
What remainders do you get when square numbers are divided by 4?