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# Area of a Triangle

Calculating the area of shapes such as squares and rectangles is relatively straightforward as they have at least two sides of equal length. However, triangles bring an added layer of complexity as they have three unique sides. As you may recall, a triangle is a polygon with three sides, as shown in the diagram:

The relationship between the lengths of the three sides is known as the triangle inequality theorem, which states that the sum of any two sides must be greater than the third. Despite this added complexity, finding the area of a triangle can be made easier with our knowledge of finding the area of rectangles.

## The basic area of a triangle

The area of a rectangle is calculated by multiplying its base and height $A=bh$ . This concept can be applied to find the area of a triangle. We start by drawing a diagonal line on a rectangle so that it can be divided into two congruent triangles:

The formula for the area of a triangle can be derived from the area of a rectangle. By dividing a rectangle into two congruent triangles, we can see that each triangle represents half of the rectangle's total area. This leads us to the formula for the area of a triangle: $A=\frac{1}{2}bh$ . This formula applies to all triangles, not just those created by cutting a rectangle diagonally.

For example, consider the diagram below.

The base of the triangle measures 14 meters and the height is 10 meters, so we have all of the numbers we need to use our formula:

$A=\frac{1}{2}\left(10\right)\left(14\right)$

Remember that area is always expressed in square units, in this case, meters. You're done!

## Finding the area of a triangle with Heron's Formula

If you don't know the height of a triangle, you can use a variety of other methods to determine its area. One of the most popular is Heron's Formula, which looks intimidating but isn't actually as bad as it seems. It looks like this:

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

It's a lot of letters, but each of them represents something very specific. For example, "A" is exactly what we are interested in, the Area, while a, b, and c are the lengths of each of the triangle's three sides. The Associative Property of Multiplication allows us to assign any of the triangle's sides to any of the three letters, so all you have to do is make sure you're working with the right three numbers.

The s stands for semiperimeter, which is just a fancy way of saying half of the triangle's perimeter. You can calculate it like this:

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

Once you have specific values for s, a, b, and c, all you have to do is plug them into the formula above. For example, let's say that your triangle has sides measuring 6, 8, and 10 meters. The first step is determining what s is:

$s=\frac{24}{2}=12$

Next, plug all four values into Heron's Formula:

$A=\sqrt{12\left(12-6\right)\left(12-8\right)\left(12-10\right)}$

$A=\sqrt{12\left(6\right)\left(4\right)\left(2\right)}$

$A=\sqrt{576}$

There you go! It should be noted that the answer above is a perfect square, but this will not always be the case; you might get decimals depending on the numbers you are working with. Remember to express your answer in units squared as well.

## Calculating the area of specific types of triangles

If you're dealing with specific types of triangles, you may have access to specific formulas that wouldn't apply in other situations. For example, a right triangle has one angle measuring 90 degrees while the remaining two acute angles have a sum of 90 degrees. That means that the triangle's height is the length of its perpendicular side, facilitating the use of the $A=\frac{1}{2}$ bh formula.

Alternatively, you might be working with an equilateral triangle where all three sides are equal. In this scenario, you can calculate the triangle's area using the following formula:

$A=\sqrt{\frac{3}{4}}×{5}^{2}$

As you may have guessed, s is the length of one of the sides. So, let's say you have an equilateral triangle with three sides each measuring 5 meters. You would multiply $\sqrt{\frac{3}{4}}$ by 25 (or ${5}^{2}$ ) and get approximately .

You may also get an isosceles triangle, or a triangle where two of the sides are equal and the angles opposite the equal sides are also equal. You can calculate the area of an isosceles triangle with the following formula:

$A=\frac{\sqrt{{a}^{2}-\frac{{b}^{2}}{4}}×b}{2}$

In this example, b is the base of the triangle and a is the measure of one of the two equal sides. If the base of the triangle measures 8 units and the other sides each measure 10 units, we would get:

$A=\frac{\sqrt{{10}^{2}-\frac{{8}^{2}}{4}}×8}{2}$

$A=\frac{\sqrt{100-16}×8}{2}$

$A=\frac{\sqrt{84}×8}{2}$

$A=\frac{9.165×8}{2}$

$A\approx 36.67sq. units$

## Using trigonometry to calculate the area of a triangle

A more advanced method to calculate the area of a triangle involves special functions from trigonometry. If you have the length of two sides of a triangle and the angle of the third side, you can use a variation of the classic $A=\frac{1}{2}bh$ formula to determine its area. Here is the formula:

$A=\frac{ab\mathrm{sin}\left(c\right)}{2}$

Both a and b are the sides you have a length measurement for. $\mathrm{sin}\left(c\right)$ refers to the sine of the angle between side lengths a and b. You'll need to plug the numbers into a graphing calculator to determine your answer, but otherwise, it's a variation of the triangle area formula you've been using for years. This is sometimes called the SAS (Side-Angle-Side) method.

## Area of a triangle practice questions

a. If a triangle has a base of 5 meters and a height of 4 meters, calculate its area:

$A=\frac{1}{2}\left(5m\right)\left(4m\right)$

b. If the three sides of a triangle measure 3, 4, and 5 inches, respectively, calculate its area:

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:

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

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

$A=\sqrt{6×3×2×1}$

$A=\sqrt{36}$

$A=6{\mathrm{in}}^{2}$

c. Calculate the area of an equilateral triangle where one side measures 3 centimeters.

To calculate the area of an equilateral triangle, we need to know the length of one side (a). We can use the formula for the area of an equilateral triangle:

$A=\frac{\sqrt{3}}{4}×{a}^{2}$

where a is the length of one side of the equilateral triangle.

In this case, we have a triangle with one side measuring 3 centimeters. So, plugging in the value of a, we get:

$A=\frac{\sqrt{3}}{4}×{3}^{2}$

$A=\frac{\sqrt{3}}{4}×9$

$A=\frac{1.732}{4}×9$

$A=0.433×9$

$A=3.897{\mathrm{cm}}^{2}$

d. Calculate the area of an isosceles triangle with a base measuring 16 feet and the other 2 sides measuring 20 feet.

$a=20\mathrm{feet}$

$b=16\mathrm{feet}$

$A=\frac{\sqrt{{20}^{2}-\frac{{16}^{2}}{4}}×16}{2}$

$A=\frac{\sqrt{400-64}×16}{2}$

$A=\frac{\sqrt{336}×16}{2}$

$A=\frac{18.33×16}{2}$

About $146.6{\mathrm{ft}}^{2}$ .

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