If the volume of a cone is 1/3 pi r^2 h, then how can it also be 1/3 pi r^3?
That would only be true if the height is equal to the radius
The first formula is correct.
The volume of a cone is 1/3(Area of Base)(height) = 1/3 π r^2 h
The formula for the volume of a cone is given by V = (1/3)πr^2h, where V represents the volume, r is the radius of the base, and h is the height of the cone.
To understand why the volume can also be expressed as (1/3)πr^3, it's crucial to understand that the radius (r) in this alternate equation represents the slant height of the cone, not the radius of the base.
The slant height (l) is the distance from the tip of the cone to any point on its base along a straight line that connects them. It is calculated by applying the Pythagorean theorem, using the radius (r) and the height (h) of the cone. The Pythagorean theorem states that the square of the hypotenuse (l) is equal to the sum of the squares of the other two sides.
By applying the Pythagorean theorem to a cone, we can express the slant height (l) as follows: l = √(r^2 + h^2)
Now, let's substitute this value of the slant height into the volume formula:
V = (1/3)πr^2h
V = (1/3)π(r^2)h
Since the slant height (l) can be substituted for r, we get:
V = (1/3)π(l^2)h
Now, using the formula for the slant height (l), we can simplify further:
V = (1/3)π((√(r^2 + h^2))^2)h
V = (1/3)π(r^2 + h^2)h
V = (1/3)πr^2h + (1/3)πh^3
The second term, (1/3)πh^3, represents a different volume calculation based on the height of the cone. However, the first term, (1/3)πr^2h, represents the volume calculation based on the radius of the base and the height of the cone, which is the commonly used formula.
So, in conclusion, the volume of a cone can be expressed as both (1/3)πr^2h and (1/3)πr^3, but the latter representation incorporates the slant height rather than the radius of the base.