Investigate the number of points with integer coordinates on
circles with centres at the origin for which the square of the
radius is a power of 5.
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?
Four circles all touch each other and a circumscribing circle. Find
the ratios of the radii and prove that joining 3 centres gives a
If a ball is rolled into the corner of a room how far is its centre
from the corner?
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?
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?
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.
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
A napkin is folded so that a corner coincides with the midpoint of
an opposite edge . Investigate the three triangles formed .
Prove that the shaded area of the semicircle is equal to the area of the inner 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
A belt of thin wire, length L, binds together two cylindrical
welding rods, whose radii are R and r, by passing all the way
around them both. Find L in terms of R and r.
A small circle fits between two touching circles so that all three
circles touch each other and have a common tangent? What is the
exact radius of the smallest circle?
The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF.
Similarly the largest. . . .
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. . . .
In a right-angled tetrahedron prove that the sum of the squares of
the areas of the 3 faces in mutually perpendicular planes equals
the square of the area of the sloping face. A generalisation. . . .
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?
A ribbon is nailed down with a small amount of slack. What is the
largest cube that can pass under the ribbon ?
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? ...
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 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 ?.
Ten squares form regular rings either with adjacent or opposite
vertices touching. Calculate the inner and outer radii of the rings
that surround the squares.
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.
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 10x10x10 cube is made from 27 2x2 cubes with corridors between
them. Find the shortest route from one corner to the opposite
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
Shows that Pythagoras for Spherical Triangles reduces to
Pythagoras's Theorem in the plane when the triangles are small
relative to the radius of the sphere.
Given any three non intersecting circles in the plane find another
circle or straight line which cuts all three circles orthogonally.
Three circular medallions fit in a rectangular box. Can you find the radius of the largest one?
P is a point inside a square ABCD such that PA= 1, PB = 2 and PC =
3. How big is angle APB ?
A tetrahedron has two identical equilateral triangles faces, of side length 1 unit. The other two faces are right angled isosceles triangles. Find the exact volume of the tetrahedron.
Can you make sense of these three proofs of Pythagoras' Theorem?
ABCD is a rectangle and P, Q, R and S are moveable points on the
edges dividing the edges in certain ratios. Strangely PQRS is
always a cyclic quadrilateral and you can find the angles.
Stick some cubes together to make a cuboid. Find two of the angles
by as many different methods as you can devise.
Can you make sense of the three methods to work out the area of the kite in the square?
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?
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 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 is the same and what is different about these circle
questions? What connections can you make?
If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
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.
What remainders do you get when square numbers are divided by 4?
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. . . .
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.
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?
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?
What are the shortest distances between the centres of opposite
faces of a regular solid dodecahedron on the surface and through
the middle of the dodecahedron?
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.
The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?
The incircles of 3, 4, 5 and of 5, 12, 13 right angled triangles
have radii 1 and 2 units respectively. What about triangles with an
inradius of 3, 4 or 5 or ...?