If you cut off the top part of a cone with a plane perpendicular to the height of the cone, you obtain a conical frustum. See figure on the left: the upper base of the frustum has radius r and the lower base has radius R. The height of the frustum is h.
![]() 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
ExampleFind 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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