Applications

In water science, we find the area of watersheds, reservoirs, and lakes. This information is vital for water management, irrigation planning, and assessing potential flood risks. These calculations are generally completed in geographic information system (GIS) software. So you’ll jump into that soon in Introduction to GIS!

The calculations we complete by hand or in a spreadsheet will more often involve a building project. A guzzler is an inexpensive way to increase water in arid habitats and potentially support more game species. The amount of water a guzzler provides will depend on the area of its roof. The roof of a guzzler funnels rainwater and snowmelt into a water trough for wildlife.

If one inch of rain falls on the guzzler shown below, how much water (gallons) will fall into the trough?

We have seen that the perimeter of a polygon is the distance around the outside. Perimeter is a length, which is one-dimensional, and so it is measured in linear units (feet, centimeters, miles, etc.). The area of a polygon is the amount of two-dimensional space inside the polygon, and it is measured in square units: square feet, square centimeters, square miles, etc.

You can always think of area as the number of squares required to completely fill in the shape.

Area: Rectangles and Squares

There are of course formulas for finding the areas of rectangles and squares; we don’t have to count little squares.

Area: Parallelograms

Another common polygon is the parallelogram, which looks like a tilted rectangle. As the name implies, the pairs of opposite sides are parallel and have the same length. Notice that, if we label one side as a base of the parallelogram, we have a perpendicular height which is not the length of the other sides.

The following set of diagrams shows that we can cut off part of a parallelogram and rearrange the pieces into a rectangle with the same base and height as the original parallelogram. A parallelogram with a base of [latex]7[/latex] units and a vertical height of [latex]6[/latex] units is transformed into a [latex]7[/latex] by [latex]6[/latex] rectangle, with an area of [latex]42[/latex] square units.

Therefore, the formula for the area of a parallelogram is identical to the formula for the area of a rectangle, provided that we are careful to use the base and the height, which must be perpendicular.

Area: Triangles

When we are finding the area of a triangle, we need to identify a base and a height that is perpendicular to that base. If the triangle is obtuse, you may have to imagine the height outside of the triangle and extend the base line to meet it.

As shown below, any triangle can be doubled to form a parallelogram. Therefore, the area of a triangle is one half the area of a parallelogram with the same base and height.

As with a parallelogram, remember that the height must be perpendicular to the base.

Area: Trapezoids

A somewhat less common quadrilateral is the trapezoid, which has exactly one pair of parallel sides, which we call the bases. The first example shown below is called an isosceles trapezoid because, like an isosceles triangle, its two nonparallel sides have equal lengths.

There are a number of ways to show where the area formula comes from, but the explanations are better in video because they can be animated.^[2][3][4]

Don’t be intimidated by the subscripts on [latex]b_1[/latex] and [latex]b_2[/latex]; it’s just a way to name two different measurements using the same letter for the variable. (Many people call the bases [latex]a[/latex] and [latex]b[/latex] instead; feel free to write it whichever way you prefer.) Whatever you call them, you just add the two bases, multiply by the height, and take half of that.

Area: Circles

The area of a circle is [latex]\pi[/latex] times the square of the radius: [latex]A=\pi{r^2}[/latex]. The units are still square units, even though a circle is round. (Think of the squares on a round waffle.) Because we can’t fit a whole number of squares—or an exact fraction of squares—inside the circle, the area of a circle will be an approximation.

If your calculator doesn’t have a [latex]\pi[/latex] key, use the approximation [latex]\pi\approx3.1416[/latex].

Exercise Answers