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Pass a plane through a cube of edge 6 in. so that the section formed will be a regular hexagon. Find the volume of a right circular cylinder 8 in. long, (a) whose base circumscribed this hexagon, (b) whose base is inscribed in this hexagon.
Solution:
To illustrate the problem, it is better to draw the figure as follows
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| Photo by Math Principles in Everyday Life |
If a cube is cut by a plane that passes through the midpoints of two adjacent sides from the upper base to the opposite lower base with the midpoints of two adjacent sides, then the intersection is a regular hexagon. By Pythagorean Theorem, the length of the sides of a regular hexagon is
Let's consider the section of a cube which is a regular hexagon as follows
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| Photo by Math Principles in Everyday Life |
There are six equal equilateral triangles in a regular hexagon in which their common vertex is a center of a regular hexagon. All sides of equilateral triangles are all equal which are 3√2 in.
(a) If the base of a right circular cylinder circumscribes the regular hexagon, then the radius is equal to 3√2 in. A circle contains all the vertices of a regular hexagon. Therefore, the volume of a right circular cylinder is
(b) If the base of a right circular cylinder inscribes the regular hexagon, then the radius is tangent to all the sides of a regular hexagon. The radius of a right circular cylinder is also an apothem of a regular hexagon and an altitude of an equilateral triangle. The altitude of an equilateral triangle bisects its base. By Pythagorean Theorem, the radius of a right circular cylinder is
Therefore, the volume of a right circular cylinder is
