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 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 napkin is folded so that a corner coincides with the midpoint of
an opposite edge . Investigate the three triangles formed .
Can you maximise the area available to a grazing goat?
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
Explore when it is possible to construct a circle which just
touches all four sides of a quadrilateral.
What is the same and what is different about these circle
questions? What connections can you make?
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 is inscribed in a triangle which has side lengths of 8, 15
and 17 cm. What is the radius of the circle?
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?
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.
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?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
Can you work out the dimensions of the three cubes?
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 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?
Show that is it impossible to have a tetrahedron whose six edges
have lengths 10, 20, 30, 40, 50 and 60 units...
Different combinations of the weights available allow you to make different totals. Which totals can you make?
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
Can you see how to build a harmonic triangle? Can you work out the next two rows?
If it takes four men one day to build a wall, how long does it take
60,000 men to build a similar wall?
My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?
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?
Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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?
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
Five children went into the sweet shop after school. There were
choco bars, chews, mini eggs and lollypops, all costing under 50p.
Suggest a way in which Nathan could spend all his money.
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.
Explore the effect of combining enlargements.
How many different symmetrical shapes can you make by shading triangles or squares?
Can you describe this route to infinity? Where will the arrows take you next?
Can all unit fractions be written as the sum of two unit fractions?
The clues for this Sudoku are the product of the numbers in adjacent squares.
Can you find the area of a parallelogram defined by two vectors?
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?
Use the differences to find the solution to this Sudoku.
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 ?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
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. . . .
A country has decided to have just two different coins, 3z and 5z
coins. Which totals can be made? Is there a largest total that
cannot be made? How do you know?
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
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?
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?
Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?