A frustum of a right circular cone with height h, lower base radius R, and top radius r.
-what's the volume using h, R, & r in the answer?
Figure it as the difference of two right cones, 1/3 base area*height.
To find the volume of a frustum of a right circular cone, you can use the formula:
V = (1/3) * π * h * (R^2 + Rr + r^2)
where:
V is the volume of the frustum,
h is the height of the frustum,
R is the radius of the lower base of the frustum, and
r is the radius of the upper base of the frustum.
Now, let's understand how this formula is derived by considering the frustum as the difference of two right cones.
Let's consider the larger cone (with radius R and height h) and the smaller cone (with radius r and height h). The volume of each cone can be calculated using the formula for the volume of a cone:
V1 = (1/3) * π * h * R^2
V2 = (1/3) * π * h * r^2
The volume of the frustum is the difference between the volumes of these two cones:
V = V1 - V2 = (1/3) * π * h * R^2 - (1/3) * π * h * r^2
Now, let's factor out π * h / 3 from both terms:
V = (1/3) * π * h * (R^2 - r^2)
We can further factorize the expression R^2 - r^2 as (R^2 + Rr + r^2) since it represents the difference of squares. Substituting this back into the equation, we get:
V = (1/3) * π * h * (R^2 + Rr + r^2)
So, the volume of the frustum of a right circular cone can be calculated using the formula mentioned above, which is derived by considering it as the difference of two right cones.