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Figuring Out Formulas for Area


Date: 08/24/99 at 19:39:10
From: Tarah Theoret
Subject: Area

I'm having trouble remembering formulas for squares, triangles, 
cylinders, etc. and I don't know how I can remember them. Please help!


Date: 08/27/99 at 11:13:24
From: Doctor Jeremiah
Subject: Re: Area

Hi there - thank you for writing to Dr. Math.

I don't think you need to remember them. Well, all except the circle. 
The area of a circle just has to be remembered.

A square or rectangle is easy to figure out. You don't need to 
remember it. Imagine if you could cut it up into small pieces. If each 
of them had an area of one you could do two things to find the area of 
the whole object. You could count up all the small pieces, but that 
would take too long, or you could notice that there are 10 squares 
along each row and that there are 5 rows. If we add up the areas of 
the rows we get 10+10+10+10+10 = 5*10. Adding the same number more 
than once is the same as multiplication, so the area of a square or 
rectangle is just the multiplication of the length of the bottom by 
the total height of the object: 10*5 = 50

     +---+---+---+---+---+---+---+---+---+---+
     |   |   |   |   |   |   |   |   |   |   |
     +---+---+---+---+---+---+---+---+---+---+
     |   |   |   |   |   |   |   |   |   |   |
     +---+---+---+---+---+---+---+---+---+---+
     |   |   |   |   |   |   |   |   |   |   |
     +---+---+---+---+---+---+---+---+---+---+
     |   |   |   |   |   |   |   |   |   |   |
     +---+---+---+---+---+---+---+---+---+---+
     |   |   |   |   |   |   |   |   |   |   |
     +---+---+---+---+---+---+---+---+---+---+

Knowing how to get the area of a rectangle helps us figure out the 
area of a triangle. Notice that a triangle is just half of a 
rectangle. In fact even if it doesn't look like half a rectangle, it 
is, so when you want to figure out the area of a triangle think of one 
that does look like half a rectangle. 

The area of a rectangle is the length of the bottom times the total 
height, so the area of a triangle is the length of the bottom times 
the total height divided by two. In the example below, the bottom is 6 
and the total height is 6 so the area is 6*6/2 = 36/2 = 18

     +-----------------------+
     | \                     |
     +---+                   |
     |   | \                 |
     +---+---+               |
     |   |   | \             |
     +---+---+---+           |
     |   |   |   | \         |
     +---+---+---+---+       |
     |   |   |   |   | \     |
     +---+---+---+---+---+   |
     |   |   |   |   |   | \ |
     +---+---+---+---+---+---+

Remember that this works even if the triangle doesn't look like half a 
rectangle. In the example below, the bottom is 8 and the total height 
is 4 so the area of the triangle is 8*4/2 = 32/2 = 16

                  +
                / | \
              +---+---+
            / |   |   | \
          +---+---+---+---+
        / |   |   |   |   | \
      +---+---+---+---+---+---+
    / |   |   |   |   |   |   | \
  +---+---+---+---+---+---+---+---+


A cylinder is a little harder. The top and bottom are circles, so we 
have to remember that the area of a cylinder includes the areas of two 
circles. What about the sides? Take a piece of paper and stick the 
left edge next to the right edge. Now you have a cylinder without a 
top or bottom. That means the sides of a cylinder are just a rectangle 
in disguise, so the area of a cylinder is 2 circles and one rectangle.

The circles at the top and bottom are the same size, so instead of 
figuring out both and adding them up we should figure out the bottom 
one and multiply by two. The length of the rectangle is the same as 
the length around the circle (the circumference). We know this because 
the edge of the circle and the length of the side have to touch all 
the way around.

The circumference of a circle is 2*Pi*r. This is the length of the 
rectangle, so the area of the rectangle is 2*Pi*r times the total 
height of the cylinder.

The area of a circle is Pi*r^2 so the area of a cylinder is
2*Pi*r^2 + 2*Pi*r*height. The first term is the two circles and the 
last one is the rectangle that makes up the sides.

When you want to figure out the area of a cylinder just imagine making 
one out of paper: two circles and a rectangle.


A cube is just the area of six rectangles, because there are six 
sides. If the sides are all squares, we can multiply the area of the 
square by 6 to get the area of the cube. Even if the sides are 
rectangles and not squares, all we have to do is add up the areas of 
all six sides.

I hope that helps. I don't think memorizing the formulae of areas for 
all the shapes is worth doing. I think you are better off figuring it 
out, imagining how to calculate it.

Parallelograms are a perfect example. Who wants to remember the area 
formula for that?

                 +---+---+---+---+---+---+---+---+---+---+---+
               / |   |   |   |   |   |   |   |   |   |   | /
             +---+---+---+---+---+---+---+---+---+---+---+
           / |   |   |   |   |   |   |   |   |   |   | /
         +---+---+---+---+---+---+---+---+---+---+---+
       / |   |   |   |   |   |   |   |   |   |   | /
     +---+---+---+---+---+---+---+---+---+---+---+

Just imagine taking the triangle off the right side and sticking it 
on the left. You would get a rectangle with the same length along the 
bottom and the same total height. So the formula for the area of a 
parallelogram is the same as for the area of a rectangle: the length 
of the bottom times the total height.

                 +-------------------------------+-----------+
               /                                 |         /
             /                                   |       /
           /                                     |     /
         /                                       |   /
       /                                         | /
     +-------------------------------------------+


     +-----------+-------------------------------+
     |         /                                 |
     |       /                                   |
     |     /                                     |
     |   /                                       |
     | /                                         |
     +-------------------------------------------+

See what I mean? Figuring it out is easier than memorizing all those 
equations. 

I hope that helps. If you need more examples, email me back.

- Doctor Jeremiah, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Euclidean/Plane Geometry
High School Geometry
High School Higher-Dimensional Geometry
Middle School Geometry
Middle School Higher-Dimensional Geometry
Middle School Two-Dimensional Geometry

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