What is the method to calculate the height of a cone using Pythagoras' Theorem?

To calculate the height of a cone using Pythagoras' Theorem, use the formula: height = √(slant height² - radius²).

In more detail, Pythagoras' Theorem is a fundamental principle in geometry that relates the lengths of the sides of a right-angled triangle. For a cone, imagine a vertical cross-section that passes through the apex and the centre of the base, forming a right-angled triangle. The slant height of the cone is the hypotenuse of this triangle, the radius of the base is one leg, and the height of the cone is the other leg.

To find the height of the cone, you need to know the slant height (l) and the radius (r) of the base. According to Pythagoras' Theorem, in a right-angled triangle, the square of the hypotenuse (slant height) is equal to the sum of the squares of the other two sides (radius and height). This can be written as:

\[ l^2 = r^2 + h^2 \]

Rearranging this formula to solve for the height (h), we get:

\[ h^2 = l^2 - r^2 \]

Taking the square root of both sides gives us:

\[ h = \sqrt{l^2 - r^2} \]

So, if you know the slant height and the radius of the cone, you can substitute these values into the formula to find the height. For example, if the slant height is 10 cm and the radius is 6 cm, the height would be:

\[ h = \sqrt{10^2 - 6^2} = \sqrt{100 - 36} = \sqrt{64} = 8 \text{ cm} \]

This method allows you to determine the vertical height of the cone using the relationship between the sides of a right-angled triangle.