Square Prism Problems, 3

Category: Solid Geometry

"Published in Newark, California, USA"

A right prism of altitude 7 in. and square base 6 in. on an edge is cut by a plane forming section ABCD as shown. (a) Find the length of the diagonal AC. (b) Find angle ABC. (c) Find the area of section ABCD. (d) Find the angle which the plane of the section ABCD makes with the plane of the base.

Photo by Math Principles in Everyday Life

Solution:

To understand more the problem, it is better to label further the figure as follows

Photo by Math Principles in Everyday Life

From point A, draw four line segments which are perpendicular to the lateral edges of a prism so that plane AEFG is parallel to the base of a prism.

(a) The length of AC which is the diagonal of a plane section is 

(b) By Pythagorean Theorem, the length of AB is

By Pythagorean Theorem, the length of BC is

Therefore, by Cosine Law, ∠ABC is

                         or

(c) Since plane ABCD is a parallelogram as you can see from the figure, therefore, the area of plane ABCD is
 

 

 

 

 

(d) The angle which the plane of the section ABCD makes with the plane of the base is
 

 

                            or

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