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
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?
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
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?
Can you maximise the area available to a grazing goat?
What is the same and what is different about these circle
questions? What connections can you make?
Explore when it is possible to construct a circle which just
touches all four sides of a quadrilateral.
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
A napkin is folded so that a corner coincides with the midpoint of
an opposite edge . Investigate the three triangles formed .
Can you find rectangles where the value of the area is the same as the value of the perimeter?
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.
What is the smallest number with exactly 14 divisors?
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?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
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?
How many pairs of numbers can you find that add up to a multiple of
11? Do you notice anything interesting about your results?
Triangle ABC is isosceles while triangle DEF is equilateral. Find
one angle in terms of the other two.
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
Which set of numbers that add to 10 have the largest product?
Think of two whole numbers under 10. Take one of them and add 1.
Multiply by 5. Add 1 again. Double your answer. Subract 1. Add your
second number. Add 2. Double again. Subtract 8. Halve this. . . .
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?
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?
Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten
numbers from the bags above so that their total is 37.
A circle is inscribed in a triangle which has side lengths of 8, 15
and 17 cm. What is the radius of the circle?
Do you know a quick way to check if a number is a multiple of two?
How about three, four or six?
Here are four tiles. They can be arranged in a 2 by 2 square so
that this large square has a green edge. If the tiles are moved
around, we can make a 2 by 2 square with a blue edge... Now try. . . .
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?
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
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?
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
The number 2.525252525252.... can be written as a fraction. What is
the sum of the denominator and numerator?
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?
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?
Can you describe this route to infinity? Where will the arrows take you next?
Explore the effect of combining enlargements.
Explore the effect of reflecting in two parallel mirror lines.
If: A + C = A; F x D = F; B - G = G; A + H = E; B / H = G; E - G =
F and A-H represent the numbers from 0 to 7 Find the values of A,
B, C, D, E, F and H.
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. . . .
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. . . .
Show that is it impossible to have a tetrahedron whose six edges
have lengths 10, 20, 30, 40, 50 and 60 units...
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
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. . . .
Can you find the area of a parallelogram defined by two vectors?
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
Different combinations of the weights available allow you to make different totals. Which totals can you make?
A hexagon, with sides alternately a and b units in length, is
inscribed in a circle. How big is the radius of the circle?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
How many different symmetrical shapes can you make by shading triangles or squares?
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. . . .
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?