Geometry

    

Geometry is a continuation of the knowledge acquired within Shapes
We have already learned about several two-dimensional shapes.  

We now have the skills to calculate the perimeter of any polygon.  We should also have a firm understanding of the different formulas needed to calculate the area of any shape and the circumference of a circle.  
(Remember that Pi = 3.14159...)

Please note that  within every triangle:  all angles of any triangle add up to 180°



Now the time has come to add a new dimension.  The third dimension, depth.  Three dimensional objects are often referred to as solids.

There are two common calculations with 3D objects, surface area and volume (capacity)

              

    a cube is the most common term for a where all sides are equal
 
    a cube's surface area is the area of one side of the cube (H x W) multiplied by 6

    the volume of a square can be found by multiplying the height, width, and depth
                                                            or   V = H x W x D





a rectangular solid or box or both terms common to this 3D shape 

the surface area can be found by adding twice the area of each non-opposing sides
    or to use the example given:  2(H x W) + 2(H x D) + 2(D x W)
                                                        2(2 x 4) + 2(2 x 3) + 2(3 x 4)

the volume can be found much like that of a cube   V = H x W x D




 a sphere or ball is a three-dimensional circle 

 the surface area of a sphere can be found by multiplying 4 x ( π x r2 ) 
      or to use the example given:  4 (π x 142)

 the volume of a sphere can be found by using the following formula
                          V =  4  x π x r3
                                  3



            a cylinder is the most common term for this 'can-like' shape

            to find the surface area of a cylinder, add both circle areas to the radius times the height
                                     or  surface area = 2(π x r2) x r x h

            the volume of a cylinder can be found using the following formula
                                               V = h x π x r2
            (think of this as one circle stacked atop another for the height of the cylinder)




                 a cone is the term used for this shape regardless of it's orientation (facing up or down)
 
                 the surface area of a cone can be found by adding ( π x r2 ) with (pi x height x length of side)
            
                 the volume of a cone can be found using the following formula
                                                   V = ⅓ x h x π x r2




      a pyramid is the name used for this three-dimensional object

      the surface area of a pyramid can be found by adding the area of the square base to
                                                                                                   the areas of each triangle side
                   or      surface area = b2 + 2 x base x length of side
      
      the volume of a pyramid can be found using the following formula
                                       V = ⅓ x b x h




These are the most basic shapes you will encounter.  However, three-dimensional shapes can be found in many other different forms.
                         
                                                                    Such as :


We will now take a closer look at angles and their attributes.




As shown in this image, 

complementary angles are two angles who's sum add up to make a right angle (90°) 






    Complementary Angles




As shown in this example, 

supplementary angles are two angles who's sum add up to make a straight line (180°) 
                         ∠ DEF + ∠ ABC = 180°







Use this figure to refer to similarities between angles,

and note that when two parallel lines are intersected by a straight line the following is true:
      opposite angles are equal      
   b = c = f = g              and        a = d = e = h


more coming soon...