# Ice cream cone height

(I'm only an Year 7 so please explain clearly how you found the solution) An ice-cream dessert is made out of a cone and a scoop of ice-cream in the shape of a hemisphere placed atop the cone. If the slant edge of the cone is 100 mm and the radius of the hemisphere is 28 mm, what is the height of the ice-cream dessert from the tip of the scoop to the bottom of the cone?

I've tried to use Pythagorean Theorem to work it out but since I've only learnt how to use it to find the hypotenuse that didn't work. Then I opened a book full of formulas that I have, but I didn't find any formula about the height of the ice-cream. I finally found the answer online but it was explained in Year 10 format, so I didn't understand it. Can anyone help me? P.S. The answer is 124mm

• The radius of the base of the cone is same as the radius of the scoop. In the cone, the base radius is 28 mm and slant height is 100 mm. By Pythagoras theorem, the height of the cone is $\sqrt{100^2-28^2} = 96$. The total height is therefore $96+28 = 124$
– user348749
Feb 28, 2017 at 11:38

First, let us draw out a diagram with using the information given. Notice that I've drawn the ice cream cone as if looking at it from the side (which in this case won't change the measurements we're interested in). Notice that I've used the 28cm twice since it is both pertinent to the height of the ice cream and also to one leg of a triangle. Notice that in the diagram, we have formed a triangle with a missing side length which I'll label as $x$. Notice that if we can find $x$, then we can find the height of the entire ice cream which will be $x+28$. Notice that in the diagram, we have formed a triangle with legs $x$ and $28$ and hypotenuse $100$. By the Pythagorean theorem, we have that $$x^2+28^2 = 100^2.$$
Solving for $x$, $$x = \sqrt{100^2 - 28^2} = 96.$$
And thus the height is $x+28 = 96+28 = 124$.