On the right is the shape obtained if the frustum on the left is cut along the slanted height H. This shape may be used to construct a conical frustum. It is part of a sector of a circle; see figure below. Let us start by expressing H in terms of r, R and h. We use Pythagora's theorem in the right triangle on the left whose hypotenuse is H, as follows:

H^{ 2} = h^{ 2} + (R - r)^{ 2}

hence

H = sqrt [ h^{ 2} + (R - r)^{ 2} ]

If you now prolong the sides of the figure on the right above, you obtain the sector shown below.

To construct the frustum, you need to find x, y and the central angle t. Using the arc length formula, we can write:

2 Pi r = x t

and

2 Pi R = y t

The above formulas may be used to write:

2 Pi R / 2 Pi r = y t / x t

which can be simplified to give:

R / r = y / x

We can also write:

y = x + H

Substitute y = x + H in R / r = y / x and solve for x:

R / r = (x + H) / x

R x = r x + r H

R x - r x = r H

x = r H / (R - r)

We now use y = x + H to find y:

y = r H / (R - r) + H

Any of the formula 2 Pi r = x t or 2 Pi R = y t may be used to find the central angle t (in radians):

t = 2 Pi r / x

= 2 Pi r / [ r H / (R - r) ]

= 2 Pi (R - r) / H
Example: Find H, x, y and t for a frustum with r = 10 cm, R = 20 cm and h = 25 cm.

First find H.

H^{ 2} = 25^{ 2} + (20 - 10)^{ 2}

H = sqrt (725) cm

x = r H / (R - r)

= 10 sqrt (725) / (20 - 10) = sqrt (725) cm

y = x + H = 2 sqrt (725) cm

t = 2 Pi r / x

= 2 Pi 10 / sqrt (725)

In degrees, angle t is given by

t = 180 [ 2 Pi 10 / sqrt (725) ] / Pi

= 133.7 degrees (rounded to 1 decimal place).

Surface Area and Volume of Frustum - Geometry Calculator. Calculate the surface area, the volume and other parameters of a Frustrum given its radius R at the base, its radius r at the top and its height h.

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