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A circle is inscribed in a triangle which has side lengths of 8, 15 and 17 cm. What is the radius of the circle?
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?
Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.
Triangle ABC is isosceles while triangle DEF is equilateral. Find one angle in terms of the other two.
Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?
Can you find the area of a parallelogram defined by two vectors?
In a three-dimensional version of noughts and crosses, how many winning lines can you 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?
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 same radius.
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 an efficient method to work out how many handshakes there would be if hundreds of people met?
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?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
The sums of the squares of three related numbers is also a perfect square - can you explain why?
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.
There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?
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?
What is the same and what is different about these circle questions? What connections can you make?
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.
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. . . .
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
Can you arrange these numbers into 7 subsets, each of three numbers, so that when the numbers in each are added together, they make seven consecutive numbers?
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?
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.
Can you find rectangles where the value of the area is the same as the value of the perimeter?
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?
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. . . .
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 out.
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 "nodes".
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?
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?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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?
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?
What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?
Different combinations of the weights available allow you to make different totals. Which totals can you make?
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 ?
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
Can you maximise the area available to a grazing goat?
The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?
An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?
On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight hidden squares?
Show that is it impossible to have a tetrahedron whose six edges have lengths 10, 20, 30, 40, 50 and 60 units...
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 combining enlargements.