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?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
A circle is inscribed in a triangle which has side lengths of 8, 15
and 17 cm. What is the radius of the circle?
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
Explore when it is possible to construct a circle which just
touches all four sides of a quadrilateral.
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.
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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.
Triangle ABC is isosceles while triangle DEF is equilateral. Find
one angle in terms of the other two.
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?
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?
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 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?
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
In a three-dimensional version of noughts and crosses, how many winning lines 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 .
Can you see how to build a harmonic triangle? Can you work out the
next two rows?
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?
Each of the following shapes is made from arcs of a circle of
radius r. What is the perimeter of a shape with 3, 4, 5 and n
Explore the effect of reflecting in two parallel mirror lines.
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?
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 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?
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. . . .
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 hexagon, with sides alternately a and b units in length, is
inscribed in a circle. How big is the radius of the circle?
Explore the effect of combining enlargements.
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. . . .
Can you find the area of a parallelogram defined by two vectors?
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. . . .
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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?
If it takes four men one day to build a wall, how long does it take
60,000 men to build a similar wall?
Different combinations of the weights available allow you to make different totals. Which totals can you make?
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?
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?
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
What is the smallest number with exactly 14 divisors?
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?
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?
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?
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 car's milometer reads 4631 miles and the trip meter has 173.3 on
it. How many more miles must the car travel before the two numbers
contain the same digits in the same order?
What is the greatest volume you can get for a rectangular (cuboid)
parcel if the maximum combined length and girth are 2 metres?
The number 2.525252525252.... can be written as a fraction. What is
the sum of the denominator and numerator?
Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...