# cone

## 22 - Smallest cone that may circumscribe a sphere

**Problem 22**

A sphere of radius *a* is dropped into a conical vessel full of water. Find the altitude of the smallest cone that will permit the sphere to be entirely submerged.

## 13 - Sphere cut into a circular cone

**Problem 13**

A sphere is cut in the shape of a circular cone. How much of the material can be saved?

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## 12 - Cone of maximum convex area inscribed in a sphere

**Problem 12**

Find the altitude of the circular cone of maximum convex surface inscribed a sphere of radius a.

## Solids for which Volume = 1/3 Area of Base times Altitude

This is a group of solids in which the volume is equal to one-third of the product of base area and altitude. There are two solids that belong to this group; the **pyramid** and the **cone**.

## Derivation of Formula for Lateral Area of Frustum of a Right Circular Cone

The lateral area of frustum of a right circular cone is given by the formula

where

R = radius of the lower base

r = radius of the upper base

L = length of lateral side

## 66 - 68 Maxima and minima: Pyramid inscribed in a sphere and Indian tepee

**Problem 66**

Find the largest right pyramid with a square base that can be inscribed in a sphere of radius a.

## 64 - 65 Maxima and minima: cone inscribed in a sphere and cone circumscribed about a sphere

**Problem 64**

A sphere is cut to the shape of a circular cone. How much of the material can be saved? (See Problem 63).

## 62 - 63 Maxima and minima: cylinder inscribed in a cone and cone inscribed in a sphere

**Problem 62**

Inscribe a circular cylinder of maximum convex surface area in a given circular cone.

## 58 - 59 Maxima and minima: cylinder surmounted by hemisphere and cylinder surmounted by cone

**Problem 58**

For the silo of Problem 57, find the most economical proportions, if the floor is twice as expensive as the walls, per unit area, and the roof is three times as expensive as the walls, per unit area.