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
Can you maximise the area available to a grazing goat?
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?
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?
Explore when it is possible to construct a circle which just
touches all four sides of a quadrilateral.
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?
If you move the tiles around, can you make squares with different coloured edges?
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
A napkin is folded so that a corner coincides with the midpoint of
an opposite edge . Investigate the three triangles formed .
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?
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
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. . . .
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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?
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?
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
How many winning lines can you make in a three-dimensional version of noughts and crosses?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
Can you find the area of a parallelogram defined by two vectors?
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
Can you describe this route to infinity? Where will the arrows take you next?
Explore the effect of reflecting in two parallel mirror lines.
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. . . .
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
Explore the effect of combining enlargements.
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 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?
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?
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.
Why does this fold create an angle of sixty degrees?
Can you work out the dimensions of the three cubes?
Show that is it impossible to have a tetrahedron whose six edges
have lengths 10, 20, 30, 40, 50 and 60 units...
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
Can all unit fractions be written as the sum of two unit fractions?
Which set of numbers that add to 10 have the largest product?
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
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.
How many more miles must the car travel before the numbers on the milometer and the trip meter contain the same digits in the same order?
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.
Ben passed a third of his counters to Jack, Jack passed a quarter
of his counters to Emma and Emma passed a fifth of her counters to
Ben. After this they all had the same number of counters.
Can you see how to build a harmonic triangle? Can you work out the next two rows?
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?