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. . . .
The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?
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 white cross is placed symmetrically in a red disc with the central square of side length sqrt 2 and the arms of the cross of length 1 unit. What is the area of the disc still showing?
Find the sides of an equilateral triangle ABC where a trapezium
BCPQ is drawn with BP=CQ=2 , PQ=1 and AP+AQ=sqrt7 . Note: there are
2 possible interpretations.
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 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 napkin is folded so that a corner coincides with the midpoint of
an opposite edge . Investigate the three triangles formed .
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?
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
A ribbon is nailed down with a small amount of slack. What is the
largest cube that can pass under the ribbon ?
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.
What remainders do you get when square numbers are divided by 4?
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?
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?
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?
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.
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.
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
Two semi-circles (each of radius 1/2) touch each other, and a semi-circle of radius 1 touches both of them. Find the radius of the circle which touches all three semi-circles.
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? ...
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. . . .
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?
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.
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.
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
Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .
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.
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?
The sides of a triangle are 25, 39 and 40 units of length. Find the diameter of the circumscribed circle.
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 10x10x10 cube is made from 27 2x2 cubes with corridors between
them. Find the shortest route from one corner to the opposite
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?
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.
How many right-angled triangles are there with sides that are all
integers less than 100 units?
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.
What is the same and what is different about these circle
questions? What connections can you make?
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?
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.
This article for pupils and teachers looks at a number that even the great mathematician, Pythagoras, found terrifying.
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.
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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. . . .
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?
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?
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?