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?
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
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?
A napkin is folded so that a corner coincides with the midpoint of
an opposite edge . Investigate the three triangles formed .
Two ladders are propped up against facing walls. The end of the
first ladder is 10 metres above the foot of the first wall. The end
of the second ladder is 5 metres above the foot of the second. . . .
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?
Explore when it is possible to construct a circle which just
touches all four sides of a quadrilateral.
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?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
What is the same and what is different about these circle
questions? What connections can you make?
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
Here is a chance to create some attractive images by rotating
shapes through multiples of 90 degrees, or 30 degrees, or 72
How many different symmetrical shapes can you make by shading triangles or squares?
A circle of radius r touches two sides of a right angled triangle,
sides x and y, and has its centre on the hypotenuse. Can you prove
the formula linking x, y and r?
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 known that the area of the largest equilateral triangular
section of a cube is 140sq cm. What is the side length of the cube?
The distances between the centres of two adjacent faces of. . . .
What angle is needed for a ball to do a circuit of the billiard
table and then pass through its original position?
Is it always possible to combine two paints made up in the ratios
1:x and 1:y and turn them into paint made up in the ratio a:b ? Can
you find an efficent way of doing this?
A hexagon, with sides alternately a and b units in length, is
inscribed in a circle. How big is the radius of the circle?
Explore the effect of reflecting in two parallel mirror lines.
A decorator can buy pink paint from two manufacturers. What is the
least number he would need of each type in order to produce
different shades of pink.
Each of the following shapes is made from arcs of a circle of
radius r. What is the perimeter of a shape with 3, 4, 5 and n
Can you find rectangles where the value of the area is the same as the value of the perimeter?
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
This shape comprises four semi-circles. What is the relationship
between the area of the shaded region and the area of the circle on
AB as diameter?
Find the decimal equivalents of the fractions one ninth, one ninety
ninth, one nine hundred and ninety ninth etc. Explain the pattern
you get and generalise.
Can you maximise the area available to a grazing goat?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
A game for 2 or more people, based on the traditional card game
Rummy. Players aim to make two `tricks', where each trick has to
consist of a picture of a shape, a name that describes that shape,
and. . . .
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle
contains 20 squares. What size rectangle(s) contain(s) exactly 100
squares? Can you find them all?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
A circle is inscribed in a triangle which has side lengths of 8, 15
and 17 cm. What is the radius of the circle?
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
Can you find the area of a parallelogram defined by two vectors?
Show that is it impossible to have a tetrahedron whose six edges
have lengths 10, 20, 30, 40, 50 and 60 units...
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
Explore the effect of combining enlargements.
What is the greatest volume you can get for a rectangular (cuboid)
parcel if the maximum combined length and girth are 2 metres?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
Imagine a large cube made from small red cubes being dropped into a
pot of yellow paint. How many of the small cubes will have yellow
paint on their faces?
Find a cuboid (with edges of integer values) that has a surface
area of exactly 100 square units. Is there more than one? Can you
find them all?
An aluminium can contains 330 ml of cola. If the can's diameter is
6 cm what is the can's height?
Which of these games would you play to give yourself the best possible chance of winning a prize?
Some people offer advice on how to win at games of chance, or how
to influence probability in your favour. Can you decide whether
advice is good or not?
Chris and Jo put two red and four blue ribbons in a box. They each
pick a ribbon from the box without looking. Jo wins if the two
ribbons are the same colour. Is the game fair?
A plastic funnel is used to pour liquids through narrow apertures.
What shape funnel would use the least amount of plastic to
manufacture for any specific volume ?
Start with two numbers. This is the start of a sequence. The next
number is the average of the last two numbers. Continue the
sequence. What will happen if you carry on for ever?