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?
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.
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?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
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?
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 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?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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?
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
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 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
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
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?
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?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
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?
Can you find the area of a parallelogram defined by two vectors?
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.
Can you describe this route to infinity? Where will the arrows take you next?
What angle is needed for a ball to do a circuit of the billiard
table and then pass through its original position?
Can you work out the dimensions of the three cubes?
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?
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?
Different combinations of the weights available allow you to make different totals. Which totals can you make?
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?
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?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
If it takes four men one day to build a wall, how long does it take
60,000 men to build a similar wall?
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
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?
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. . . .
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?
Can you find six numbers to go in the Daisy from which you can make all the numbers from 1 to a number bigger than 25?
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.
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
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
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.
How many winning lines can you make in a three-dimensional version of noughts and crosses?
Here's a chance to work with large numbers...