Stick some cubes together to make a cuboid. Find two of the angles
by as many different methods as you can devise.
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 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 napkin is folded so that a corner coincides with the midpoint of
an opposite edge . Investigate the three triangles formed .
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?
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. . . .
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
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 hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?
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?
Which of these games would you play to give yourself the best possible chance of winning a prize?
What is the same and what is different about these circle
questions? What connections can you make?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
Can you work out the equations of the trig graphs I used to make my pattern?
An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?
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?
Can you work out the dimensions of the three cubes?
Explore when it is possible to construct a circle which just
touches all four sides of a quadrilateral.
What is the area of the quadrilateral APOQ? Working on the building
blocks will give you some insights that may help you to work it
Your school has been left a million pounds in the will of an ex-
pupil. What model of investment and spending would you use in order
to ensure the best return on the money?
Can you see how to build a harmonic triangle? Can you work out the next two rows?
What does this number mean ? Which order of 1, 2, 3 and 4 makes the
highest value ? Which makes the lowest ?
There are many different methods to solve this geometrical problem - how many can you find?
What is the largest number which, when divided into 1905, 2587,
3951, 7020 and 8725 in turn, leaves the same remainder each time?
A mother wants to share a sum of money by giving each of her
children in turn a lump sum plus a fraction of the remainder. How
can she do this in order to share the money out equally?
What angle is needed for a ball to do a circuit of the billiard
table and then pass through its original position?
Why does this fold create an angle of sixty degrees?
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?
In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
Two motorboats travelling up and down a lake at constant speeds
leave opposite ends A and B at the same instant, passing each
other, for the first time 600 metres from A, and on their return,
400. . . .
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 ?
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
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?
Use the differences to find the solution to this Sudoku.
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 "nodes".
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.
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 sum of the series.
Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP
have equal areas. Prove X and Y divide the sides of PQRS in the
A 2-Digit number is squared. When this 2-digit number is reversed
and squared, the difference between the squares is also a square.
What is the 2-digit number?
Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?
Can you find the area of a parallelogram defined by two vectors?
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
Substitute -1, -2 or -3, into an algebraic expression and you'll
get three results. Is it possible to tell in advance which of those
three will be the largest ?
There are lots of different methods to find out what the shapes are worth - how many can you find?
All CD Heaven stores were given the same number of a popular CD to
sell for £24. In their two week sale each store reduces the
price of the CD by 25% ... How many CDs did the store sell at. . . .
There is a particular value of x, and a value of y to go with it, which make all five expressions equal in value, can you find that x, y pair ?
Take any prime number greater than 3 , square it and subtract one.
Working on the building blocks will help you to explain what is
special about your results.
The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.