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?
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.
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?
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
Can you find the area of a parallelogram defined by two vectors?
Triangle ABC is isosceles while triangle DEF is equilateral. Find
one angle in terms of the other two.
Can you find rectangles where the value of the area is the same as the value of the perimeter?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
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
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?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
A 2-Digit number is squared. When this 2-digit number is reversed
and squared, the difference between the squares is also a square.
What is the 2-digit number?
Can you maximise the area available to a grazing goat?
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
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?
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
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
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
Take any prime number greater than 3 , square it and subtract one.
Working on the building blocks will help you to explain what is
special about your results.
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
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 number. . . .
Can you explain the surprising results Jo found when she calculated
the difference between square numbers?
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?
The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.
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?
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 are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
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 ?
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?
Why does this fold create an angle of sixty degrees?
An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?
What angle is needed for a ball to do a circuit of the billiard
table and then pass through its original position?
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.
Take any four digit number. Move the first digit to the 'back of
the queue' and move the rest along. Now add your two numbers. What
properties do your answers always have?
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?
Show that is it impossible to have a tetrahedron whose six edges
have lengths 10, 20, 30, 40, 50 and 60 units...
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?
A napkin is folded so that a corner coincides with the midpoint of
an opposite edge . Investigate the three triangles formed .
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.
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?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?