Right Circular Cylinder Problems, 28

Category: Solid Geometry

"Published in Newark, California, USA"

Using the vertices of a 9-in. square as centers, and radii equal to 3 in., four arcs are described outside the square. If the figure thus formed is the uniform cross section of cylinder of element 7 in., find the volume and total area of the cylinder.

Solution:

To illustrate the problem, it is better to draw the figure which is the cross section as follows

Photo by Math Principles in Everyday Life

As you can see from the figure above that there are four three-fourth circles. Hence, the area of the shaded region which is also the base of a right cylinder is 

Therefore, the volume of a right cylinder is

Next, we need to get the perimeter of the base of the shaded region which is equal to the sum of the circumference of four three-fourth circles and the length of the four line segments between the arcs of three-fourth circles as follows 

Hence, the surface or lateral area of a right cylinder is 

Therefore, the total area of a circular cylinder is 

Right Circular Cylinder Problems, 27

Category: Solid Geometry

"Published in Newark, California, USA"

Using the vertices of a 9-in. square as centers, and radii equal to 3 in., four quadrants are described within the square. If the figure thus formed is the uniform cross section of a cylinder of element 7 in., find the volume and total area of the cylinder.

Solution:

To illustrate the problem, it is better to draw the figure which is the cross section as follows

Photo by Math Principles in Everyday Life

As you can see from the figure above that there are four quarter circles which is equal to one circle. Hence, the area of the shaded region which is also the base of a right cylinder is

Therefore, the volume of a right cylinder is

Next, we need to get the perimeter of the base of the shaded region which is equal to the sum of the circumference of a whole circle and the length of the four line segments between the arcs of the quarter circles as follows

Hence, the surface or lateral area of a right cylinder is

Therefore, the total area of a circular cylinder is

Right Circular Cylinder Problems, 26

Category: Solid Geometry

"Published in Newark, California, USA"

The base of a right cylinder is shown in the figure. It is formed by describing semicircular arcs within the square upon the four sides as diameters. If the altitude of the cylinder is 12 in., find the volume and total area.
 

Photo by Math Principles in Everyday LIfe

Solution:

Consider the given figure above

Photo by Math Principles in Everyday Life

If you think that the given figure is difficult to get its length and area, then you are right. In this kind of figure, you need to use the principles of integral calculus in order to solve for the length of a curve as well as its area. Well, in this problem, we don't have to use the principles of integral calculus. The given figure consists of a square and the arcs of semicircles whose diameters are the sides of a square. Let's consider the semicircles at the left and right side as follows

Photo by Math Principles in Everyday Life

The area of the unshaded region which is A is
 

 

 

 

 

 

Hence, the area of the base which is the four shaded area is
 

 

 

 

Therefore, the volume of a right cylinder whose base is the four shaded area is
 

 

 

If you look again the given figure, there are four semicircles enclosed the shaded region. The length or circumference of four semicircles is equivalent to the circumference of two circles. Hence, the circumference of two circles is
 

 

 

 

Therefore, the total area of a right cylinder whose base is the four shaded area is