Hexagon, Cylinder Volume – Geometry Problems

GEOMETRY:
QUESTION:11
Find the volume of hexagon and cylinder. The cylinder has 7cm as height and radius is 4cm.
The hexagon exactly fits inside the cylinder.

 
Solution:
The volume of cylinder is = pie r^2 h
Here, r = 4cm
And h = 7cm
So, the volume of cylinder is = pie * 4*4*7 cm^3
= 112 pie cm^3

 
Now,

The volume of hexagon exactly fits inside the cylinder:
The volume of hexagon is base area * height.
Base area:
using sine law we can find base area. The regular hexagon
has 120 degree as interior angle. So, the triangle we drawn
through the center of the circle is equilateral triangle. The
radius is 4cm.
using. Sine law


 
sin 90 sin 30
------ = -------
4 x

 
1 0.5
-- = ---
4 x

 
X = 2cm
the length of the side of the hexagon is 2+2cm = 4cm.
the base area of the hexagon is:

 
3 root 3
--------- side^2
2

 
3 root 3
---------- * 4*4cm^2
2

 
=24 root 3.

 
The volume of the hexagon is:

 
= base area * height

 
= 24 root 3 cm^2 * 7 cm

 
= 168 root 3 cm^3

 

 

 

 

 

 
GEOMETRY:
QUESTION:12
Find the sides of a square if AM = 8cm

 
Solution:

 
According to the properties of a square, the diagonals bisect each
other. So,
AM = MC

 
AC = AM + MC

 
AC = 8cm + 8cm

 
AC = 16cm.

 
Using pythagoras theorem,

 
Take AB = x

 
x^2 + x^2 = 16^2

 
2x^2 = 256

 
x^2 = 128

 
x = 11.314.

 
Side of the square is 11.314cm