Study Guide

Basic Geometry - Area (Polygon, Triangle, Circle, Square)

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Area (Polygon, Triangle, Circle, Square)

Area is the amount of space inside a two-dimensional shape. If you think about the floor of your bedroom, the area would be the maximum amount of floor you could throw your stuff on before you couldn't see any floor remaining.

Area is always expressed as square units (units2). This is because it is two-dimensional (length and height).

You can find the area of shapes by counting the boxes inside the shapes. In these three figures, each box represents 1 cm^2.

three shapes area

  • Figure A takes up 25 small boxes, so it has an area of 25 cm^2
  • Figure B takes up 36 small boxes, so it has an area of 36 cm^2
  • Figure C takes up 21 full boxes and 7 half boxes, so it has an area of 24.5 cm^2

Area of Rectangle = Base x Height

Here is a 10 cm x 6 cm rectangle.

10 by 6 rectangle drawing

If we break it into section 1 cm wide, it would look like this:

10 cm x 6 cm rectangle drawing

Each row contains 10 squares and there are 6 rows, which gives a total of 10 × 6 square cm. That's the same as multiplying the base by the height: 6 x 10 = 60.

Area of Triangle = ½(Base × Height)

Here is a triangle with a base of 5 cm and a height of 6 cm.

6 by 5 triangle drawing

If we place another triangle with the same height and base on top of this one, we get a 6 cm x 5 cm rectangle.

6 cm x 5 cm divided rectangle

Now, we already know how to compute the area of a rectangle (base × height). So, the area of the  rectangle is 6 x 5 = 30

However, we only want the triangle, which is half of the rectangle, 1/2 x 30 =15.  Essentially we took ½ of the area of the whole rectangle, or ½ (base × height).

Area of Parallelogram = Base × Height

Now let's look at a parallelogram with a base of 6 cm and a height of 3 cm.

3 by 6 parallelogram drawing

By moving the small triangle on the left all the way to the right, this shape becomes a rectangle with a base of 6 and a height of 3 cm.

6 by 3 modified paralleogram rectangle

Since you already know how to find the area of a rectangle (base × height), you have all the tools you need to find the area of this parallelogram.

Area of Trapezoid = ½(Base1 + Base2) x Height

isosceles triangle

Imagine cutting off the triangular lower left corner and fitting it onto the upper right corner like this:

modified trapezoid

Now, we just have another rectangle, but with a new base. The base of this new figure is the average of the original bases,4.5 + 7.5  /2 = 6. The area of this new figure is 6 cm x  3 cm = 18 cm^2.Just be careful, because the base we are using is the mean of the two original bases!

Area of Circle = πr2

Finally we will examine the beautiful circle. Here is one with a radius of 6 cm.

circle r =6

Here is the same circle but with lines drawn in at every cm.

divided circle r=6

First we combined the portions of square to make complete square, then we very carefully and diligently counted each and every one of those squares and found there to be approximately 113 squares. This is nearly equal to the radius squared x pi.

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