Areas of Polygons and Circles

Let’s move away from the widely-dreaded
two-column proofs
that we have been doing in the previous sections, and
focus on geometry that
revolves solely around computations. In this section, we will use our knowledge
of different
polygons
and circles to help us measure their areas. In the two-dimensional
plane, we use area to express a defined quantity of a figure’s surface.

We can use various units to describe area. These units of measurement include centimeters,
inches, feet, yards, kilometers, miles, and much more. However, because we will
be determining the measurements of various figures in two dimensions, we will need
to attach a “square”, or exponent of 2, to our units. This will signify
that we are not talking about a measurement in one direction, but that we are covering
a span of squares in two directions. An illustration of the difference between a
unit and a unit squared is shown below.

Notice that “3 units” represents a segment of three units, whereas “3 square units”
represents an area of (one unit by one unit) squares that is three units long.

The concept of area plays a significant role in all disciplines of mathematics,
including algebra and calculus. Therefore, it is one of the most widely-utilized
applications in the real world. For instance, we use area when laying down carpet,
surveying land, and putting up wallpaper, to name a few examples. You can even use
area to help you in an argument with your parents regarding why you shouldn’t have
to share your small, square bedroom with a sibling! In fact, after studying this
section, you will know how to measure the areas of polygons and circles of all sizes.

Areas of Parallelograms and Triangles

Main Lesson:
Areas of Parallelograms and Triangles

Learn how to measure the areas of parallelograms and triangles by using their area
formulas. Also, learn about the relationship these figures have in terms of their
areas.

Areas of Trapezoids

Main Lesson:
Areas of Trapezoids

Find out how to use the formula for areas of trapezoids to in order to find the
areas of all trapezoids. Also, take a look at an alternate solution of an exercise
in which a trapezoid is broken up into smaller shapes.

Areas of Rhombuses and Kites

Main Lesson:
Areas of Rhombuses and Kites

Study the area formula for rhombuses and kites. Although one is a parallelogram
and the other is not, we will see that they share a common area formula.

Areas of Circles and Sectors

Main Lesson:
Areas of Circles and Sectors

Our final stop is circles and sectors. Learn how to find the areas of circles and
sectors, as well as how their area formulas relate to each other.

Areas Reference Page

Main Lesson:
Areas Reference Page

Look at two easy-to-read tables that display the important properties of different
figures as well as their area formulas.

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