How do you construct a conical frustum?

A conical frustum can be constructed from a sector of a circle by cutting along the slanted height and determining the inner radius x, outer radius y, and central angle theta.

How is the slanted height H of a frustum calculated?

The slanted height H is calculated using Pythagorean's theorem: H = sqrt(h^2 + (R - r)^2), where h is the height, R is the lower base radius, and r is the upper base radius.

How do you find the central angle theta for constructing a frustum?

The central angle theta in radians is given by the formula theta = (2 * pi * (R - r)) / H.

How to Construct a Conical Frustum | Step-by-Step Geometry Guide

Understanding the Conical Frustum

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. The upper base of the frustum has radius \(r\) and the lower base has radius \(R\). The height of the frustum is \(h\).

Cutting the frustum along the slanted height \(H\) yields a shape that can be used to construct a conical frustum, which forms part of a circular sector. We begin by expressing \(H\) in terms of \(r\), \(R\), and \(h\) using Pythagoras' theorem in the right triangle:

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

Deriving Construction Parameters

Prolonging the sides of the sector figure yields the complete sector shown below:

To construct the frustum, you need to find \(x\), \(y\), and the central angle \(\theta\). Using the arc length formula:

\[ 2 \pi r = x \theta \quad \text{and} \quad 2 \pi R = y \theta \]

Dividing these equations gives:

\[ \frac{2 \pi R}{2 \pi r} = \frac{y \theta}{x \theta} \implies \frac{R}{r} = \frac{y}{x} \]

Since \(y = x + H\), substituting into the proportion and solving for \(x\):

\[ \frac{R}{r} = \frac{x + H}{x} \implies Rx = rx + rH \implies x = \frac{r H}{R - r} \]

Then, solving for \(y\):

\[ y = x + H = \frac{r H}{R - r} + H = \frac{R H}{R - r} \]

Finally, using the arc length formula to find the central angle \(\theta\) (in radians):

\[ \theta = \frac{2 \pi r}{x} = \frac{2 \pi r}{\frac{r H}{R - r}} = \frac{2 \pi (R - r)}{H} \]

Worked Example

Example: Frustum with \(r = 10\) cm, \(R = 20\) cm, and \(h = 25\) cm

Find \(H\), \(x\), \(y\), and \(\theta\) for a frustum with \(r = 10\) cm, \(R = 20\) cm, and \(h = 25\) cm.

1. Find \(H\):

\[ H^2 = 25^2 + (20 - 10)^2 = 625 + 100 = 725 \] \[ H = \sqrt{725} \, \text{cm} \]

2. Find \(x\) and \(y\):

\[ x = \frac{r H}{R - r} = \frac{10 \sqrt{725}}{20 - 10} = \sqrt{725} \, \text{cm} \] \[ y = x + H = 2 \sqrt{725} \, \text{cm} \]

3. Find \(\theta\):

\[ \theta = \frac{2 \pi r}{x} = \frac{2 \pi (10)}{\sqrt{725}} \, \text{radians} \]

In degrees, angle \(\theta\) is:

\[ \theta = 180 \left( \frac{2 \pi (10)}{\sqrt{725}} \right) / \pi = 133.7^\circ \, \text{(rounded to 1 decimal place)}. \]

Understanding the Conical Frustum

Deriving Construction Parameters

Worked Example

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