Find the volume of the described solid S?

Question

S is a frustum of a right circular cone with height h, base radius R, and top radius r. ( A frustum (plural: frusta) is the portion of a solid - normally a cone or pyramid - which lies between two parallel planes cutting the solid. the planes are perpendicular to the conter axis of the solid.) Please find a volume of this. (in a basic formula)

Answer 1

Imagine the volume of two cones:
1) The cone formed by the "tip" cut off to make the frustum
2) The cone formed by the frustum AND the tip.

The volume of the frustum can be derived by subtracting the volume of the small cone from the big cone to get the frustum.

Let H be the height of the smaller cone (the "tip")

The volume of the tip cone:
Vsmall = 1/3 π r² H

The volume of the big cone:
Vbig = 1/3 π R² (H + h)

The volume of the frustum:
Vfrustum = Vbig - Vsmall
V = 1/3 π R² (H + h) - 1/3 π r² H

Pull out the common 1/3 π
V = 1/3 π [ R² (H + h) - r² H ]

From similar triangles of the big cone and little cone we have:
R / r = (H + h) / H

H + h = RH / r

Substituting in the formula above gives:
V = 1/3 π [ R² (RH/r) - r² H ]

Pull out the common H:
V = 1/3 π H [ R^3 / r - r² ]

Get a common denominator of r inside:
V = 1/3 π H [ R^3 / r - r^3 / r ]
V = 1/3 π H [ (R^3 - r^3) / r ]

Now use the similar triangles of the small cone and the angled edge of the frustum. Here the difference is R - r for the frustum.
H / h = r / (R - r)

Solve for H:
H = hr / (R - r)

Substitute back in for H:
V = 1/3 π H [ (R^3 - r^3) / r ]
V = 1/3 π (hr / (R - r) [ (R^3 - r^3) / r ]

Cancel out r:
V = 1/3 π h [ (R^3 - r^3) / (R - r) ]

Factor the difference of cubes:
R^3 - r^3 = (R-r)(R² + Rr + r²)

So the item in brackets is just R² + Rr + r²
V = 1/3 π h (R² + Rr + r²)

So the final formula is:

...... π h (R² + Rr + r²)
V = -----------------------
................... 3

Answer 2

we are in a position to describe the backside through the equation x^2 + y^2 = 25, or y = +-sqrt(25 - x^2) The squares have area length the area the better and decrease branches of the circle, it extremely is s = 2sqrt(25 - x^2) so their area is dA = 4*(25 - x^2) and quantity dV = (one hundred - 4x^2) dx Now we merely might desire to combine between x=5 and x=-5: V = int(one hundred - 4x^2)dx with limits [-5,5] V = (100x - 4/3 x^3) with limits [-5,5] V = 2000/3

Answer 3

Wesley S is wrong, Puzzling is right. See the reference Puzzling put under sources.

Answer 4

1/3 * pi * (R^3 - r^3)