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Venice, 2002

My 40th birthday, and the only time both of us had left the girls alone since they were born.   There was a force 10 gale on the way back and our plane was the only plane to land at Stansted airport that day – all other airports were also closed until later in the day.  It was the scariest landing ever and even once on the tarmac the plane jumped around like a bean.  I thought I was never going to get home to my children.

Problem One

The answer in relation to any sphere is always the same if 1m is added to the string tied around the circumference.  The height of the string above the sphere will always be 15.9cm.  Difficult to believe, but not difficult to prove.  See the calculation here.

Problem Two


In order to calculate the height of the cone you need to find the length of one side of a right-angle triangle.

You already know one side of the triangle – the hypoteneuse or longest side – is equal to 10cm because this was the radius of the original circle used to make the cone.

You can calculate another side of the triangle by finding the radius of the cone.  This may be done in the following way.

Calculate the circumference of the original circle using the formula C=2∏r.

The circumference of the original circle is 62.83cm.

Then find the proportion of the circumference left after removing the 72 degree slice.  This will be equal to the circumference of the cone in cm (C2). 

The proportion of the circumference remaining is equal t0 (360-72)/360 of the original circumference.

C2 = 62.83 x 288/360 = 50.27cm

The radius (r2) of the cone can be calculated by dividing C2 by 2∏.

50.27/2∏ = 8.00cm

Having found the radius of the cone, you now know the length of two sides of the right-angle triangle.  Finding the height of the cone involves calculating the third side of the right-angle triangle using Pythagoras’s theorem – that the square of the hypoteneuse of  of a right-angle triangle will always be equal to the sum of the squares of the other two sides. 

We know that the hypoteneuse of the triangle is equal to the radius of the original circle, that is 10cm.  We also now know that another side of the triangle is equal to r2 or 8cm.

So, the height of the triangle may be calculated by finding the square root of the sum of the square of the hypoteneuse less the square of the other side.

h² = (10×10)-(8×8) = 100-64 = 36

h = √36 = 6cm

The height of the cone is 6cm.