Area

The area of a figure refers to how much space you would need to fill it completely. It is typically recorded in square units, though terminology other than units may be used. For example, you might be asked to calculate the area of a backyard in ${\mathrm{ft}}^2$ or ${\mathrm{cm}}^2$ . You might also see "square" abbreviated to "sq." in some answers.

The simplest area problems are graphed for you, allowing you to count the squares to find the area. For instance, consider the following diagram:

This funky shape has an area of 9.5 square units: 8 whole squares colored red above and three half-squares in blue. Adding 8 and 1.5 gives us a final answer of 9.5 square units.

Most area problems are formula-driven though, and you'll need to memorize each formula and which shapes they apply to. The rest of this article will focus on how to find the area of common plane figures.

The area of a square

Squares are among the easiest shapes to work with because you know that all four sides are exactly equal. The formula for finding the area of a square is $A=s^2$ where s equals the length of one of the four sides. For instance, let's say that you have a square where each side measures 5 inches. The area of the square would be calculated as $5^2$ , or 25 sq. in.

The area of a rectangle

The area of a rectangle is obtained by the formula $A=LW$ where L and W correspond to the length and width of the rectangle, respectively. If you have a rectangle that's 7 inches long and 4 inches wide, you would calculate its area by multiplying 7 and 4 to get 28 square inches. Notably, it doesn't matter which number is the length and which is the width so long as you're working with the correct pair of numbers.

The area of a triangle

The basic formula for calculating the area of a triangle is $A=\frac{1}{2}bh$ where b is the triangle's base and h is its height. The reason this formula works is that any rectangle can be split with a diagonal line across to create two congruent triangles.

We know that the area for calculating the area of a rectangle is $A=LW$ and that each of the triangles above must be half of that figure. Therefore, the area of each triangle is half that of the total rectangle, or $A=\frac{1}{2}bh$ .

For example, let's say that you have a triangle with a base of 3 centimeters and a height of five centimeters. The area is equivalent to $5\left(3\right)\left(\frac{1}{2}\right)$ , or $7.5{\mathrm{cm}}^2$ .

If you don't know the height of a triangle, you can still calculate its area. However, the process becomes considerably more challenging. For instance, you could use Heron's Formula if you know the length of each of the triangle's three sides:

$A=\sqrt{s(s-a)(s-b)(s-c)}$

The a, b, and c above refer to the length of the triangle's three sides in any order while the s stands for semiperimeter. The semiperimeter is simply half of the triangle's perimeter, and you can calculate it using the following formula:

$\frac{a+b+c}{2}$

Let's say there's a triangle with sides measuring 3, 4, and 5 units. The first step is getting a value for s, so we add all three numbers and divide by 2 to get 6. Next, we plug all four numbers into Heron's Formula:

First, we need to calculate the semiperimeter of the triangle, which is half the perimeter:

$s=\frac{a+b+c}{2}$

$s=\frac{3+4+5}{2}$

$s=6$

Next, we can plug the values of a, b, c, and s into Heron's formula:

$A=\sqrt{s(s-a)(s-b)(s-c)}$

$A=\sqrt{6(6-3)(6-4)(6-5)}$

$A=\sqrt{6\times 3\times 2\times 1}$

$A=\sqrt{36}$

$A=6\mathrm{ sq.\; units}$

You can also use trigonometric functions to calculate the area of a triangle if you know the length of two sides and the measurement of one of its angles, but we'll save that for another day.

The area of a parallelogram

The area of a parallelogram is calculated using the formula $A=bh$ where b represents the length of the base and h is the height of the figure. It's very similar to calculating the area of a rectangle, and again, you don't have to worry about which number is b and which is h as long as you're working with the correct two numbers.

For instance, let's say that you have a parallelogram with a base of 10 kilometers and a height of 5 kilometers. Finding the area is as easy as multiplying $10\times 5=50{\mathrm{km}}^2$ .

The area of a trapezoid

You can determine the area of a trapezoid with the formula $A=\frac{b_1+b_2}{2}\times h$ , where b1 and b2 are the lengths of the parallel sides and h is the height, or the distance between the parallel sides. Consider the diagram below as an example:

The two parallel sides measure 3 and 11, so we have values for b1 and b2. Likewise, we can see that 7 units separate the parallel sides, giving us a value for h. Now, all we have to do is plug these numbers into the formula above:

$A=\frac{3+11}{2}\times 7$

$A=\frac{14}{2}\times 7$

$A=7\times 7$

$A=49\mathrm{ sq.\; units}$

The area of a circle

The formula for the area of a circle is $A=\pi r^2$ where r is the radius of the circle. Problems involving the area of a circle may try to trick you by giving the diameter instead of the radius, so watch out for that as you work out your answer.

For example, what is the area of a circle with a diameter of 10 feet? A circle's radius is always equal to half of its diameter, meaning that 5 (not 10) is the number we want to plug into our formula. Using 3.14 as the value of pi gives us:

$A=\pi r^2$

$A=\pi {\left(5\right)}^2$

$A=\pi \left(25\right)$

$A=78.5\mathrm{ square\; feet}$ (rounded to one decimal place)

Area practice questions

a. What is the area of a square with a side measuring 7 inches?

$A=s^2$

$A={\left(7\mathrm{in}\right)}^2$

$A=49\mathrm{ \; sq\; in}$

b. What is the area of a rectangle with a length of 4 centimeters and a width of 6 centimeters?

$A=l\times w$

$A=4\mathrm{cm}\times 6\mathrm{cm}$

$A=24\mathrm{ sq\; cm}$

c. What is the area of a triangle with a base of 10 feet and a height of 4 feet?

$A=\frac{1}{2}bh$

$A=\frac{1}{2}\left(10\mathrm{ft}\right)\left(4\mathrm{ft}\right)$

$A=20\mathrm{ \; sq\; ft}$

d. What is the area of a triangle with sides measuring 5 yards, 7 yards, and 11 yards?

$s=\frac{a+b+c}{2}$

$s=\frac{5\mathrm{ yards}+7\mathrm{ yards}+11\mathrm{yards}}{2}$

$s=\frac{23\mathrm{ yards}}{2}$

$s=11.5\mathrm{ yards}$

$A=\sqrt{s(s-a)(s-b)(s-c)}$

$A=\sqrt{11.5(11.5-5)(11.5-7)(11.5-11)}$

$A=\sqrt{11.5\times 6.5\times 4.5\times 0.5}$

$A=12.97$

About 13 sq. yds.

e. What is the area of a parallelogram with a base of 8 inches and a height of 5 inches?

$A=b\times h$

$A=8\mathrm{ in}\times 5\mathrm{in}$

$A=40\mathrm{ sq\; in}$

f. What is the area of a trapezoid with parallel sides measuring 7 and 5 millimeters and a height of 6 millimeters?

$A=\frac{\mathrm{b1}+\mathrm{b2}}{2}\times h$

$A=\frac{7\mathrm{mm}+5\mathrm{mm}}{2}\times 6\mathrm{mm}$

$A=6\mathrm{ mm}\times 6\mathrm{mm}$

$A=36{\mathrm{mm}}^2$

g. What is the area of a circle with a diameter of 8 feet? Use 3.14 as the value of pi to determine your answer.

$A=\pi r^2$

$A=3.14\times {\left(4\mathrm{ft}\right)}^2$

$A=3.14\times 16{\mathrm{ft}}^2$

$A=50.24{\mathrm{ft}}^2$

Topics related to the Area

Area of a Circle

Word Problems Involving Width, Length and Area

Surface Area

Flashcards covering the Area

3rd Grade Math Flashcards

Basic Geometry Flashcards

Practice tests covering the Area

Basic Geometry Diagnostic Tests

Basic Geometry Diagnostic Tests

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Area is both a practical concept used to plan construction and renovation projects and a building block for more advanced mathematical concepts, so your student needs to understand how it works. Whether your young learner is struggling to memorize all of the formulas or figure out which formula applies to each figure, a professional 1-on-1 math tutor can provide the personalized support they need to pursue educational success. Contact the Educational Directors at Varsity Tutors today for more information.