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?
A napkin is folded so that a corner coincides with the midpoint of
an opposite edge . Investigate the three triangles formed .
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?
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 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?
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?
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 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?
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 maximise the area available to a grazing goat?
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. . . .
Explore when it is possible to construct a circle which just
touches all four sides of a quadrilateral.
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
If you move the tiles around, can you make squares with different coloured edges?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?
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?
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
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
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 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.
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
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?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
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.
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
What does this number mean ? Which order of 1, 2, 3 and 4 makes the
highest value ? Which makes the lowest ?
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?
Explore the effect of combining enlargements.
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?
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 ?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Imagine you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have?
What is the largest number which, when divided into 1905, 2587,
3951, 7020 and 8725 in turn, leaves the same remainder each time?
How many winning lines can you make in a three-dimensional version of noughts and crosses?
Can you work out the dimensions of the three cubes?
Use the differences to find the solution to this Sudoku.
What angle is needed for a ball to do a circuit of the billiard
table and then pass through its original position?
Can you describe this route to infinity? Where will the arrows take you next?
Explore the effect of reflecting in two parallel mirror lines.
Can you guarantee that, for any three numbers you choose, the
product of their differences will always be an even number?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
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?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?