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.
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.