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How to Find the Area of a Triangle: Formula and Examples

Posted by Hannah Muniz | Dec 2, 2019 2:00:00 PM

General Education

 

feature_four_colorful_triangles

Geometry can be fun, but it can also give you a massive headache if you’re not sure what formulas to use or how to approach a problem.

Triangles especially have a lot of unique qualities and formulas you need to know, including the area of triangle formula. How can you figure out the area of a triangle? It’s not as simple as it is for rectanglesbut it’s also not as difficult as you might think.

In this guide, we'll go over how to find the area of a triangle and give you sample problems and tips you can use to further sharpen your skills.

 

Quick Review: What Is Area?

Area is the total amount of space a 2-D shape (or flat surface) takes up. Every shape in maththat is, every square, rectangle, triangle, parallelogram, trapezoid, etc.has an area, or a certain amount of space it occupies.

Area is determined by the lengths of particular sides of a shape and is always given in square units, which could be general units or things like feet, inches, meters, or miles.

The area of a rectangle, for example, is equal to the length multiplied by the width, or, as some might say, the base times the height:

Body_rectangle

You can also simply count the number of units in the rectangle (if supplied):

body_rectangle_area_units

So in this example, if you counted each unit (i.e., each square) in the rectangle, you'd get 10 square units for the area of the rectangle. The other (far quicker) option for finding the area of a rectangle, however, is to multiply the length (5 units) by the width (2 units); this will also get you 10.

This is how to find the area of a rectangle—pretty simple, really. But finding the area of a triangle is a bit trickier. For this, you’ll need to know the area of triangle formula.

 

Area of Triangle Formula

To find the area of a triangle, you’ll need to use the following formula:

$A=1/2bh$

A is the area, b is the base of the triangle (usually the bottom side), and h is the height (a straight perpendicular line drawn from the base to the highest point of the triangle). This formula may also be written like this:

$A={bh}/2$

Here’s a diagram to help you envision this formula at play:

Body_triangle_non-special

This formula works for all types of triangles:

  • Equilateral: A triangle for which all three sides are equal in length

body_equilateral_triangle

  • Isosceles: A triangle for which two sides are equal in length

body_isosceles_triangle

  • Scalene: A triangle for which all three sides differ in length

body_scalene_triangle

  • Right: An isosceles or scalene triangle with one right (90°) angle

body_right_triangle-3

With right triangles, the base and height are simply the two sides that form the right angle

Now, you’re probably wondering how exactly the area of triangle formula works. It’s pretty simple actually: all triangles can be inscribed in a rectangle. This rectangle will always have double the area than that of the triangle inscribed in it.

In short, to find the area of a triangle, all you need to do is take the area of a rectangle formula ($A=bh$) and divide it by 2.

Khan Academy has a nifty drag tool that lets you see how the area of a triangle is found using the rectangle/parallelogram it's inscribed in.

Let’s look at an example. Say that you’ve been asked to find the area of the following triangle (not drawn to scale):

body_triangle_area_sample_problem_1

Here, we're told that the base of our triangle is 5 and the height is 6. To find the area, then, all we need to do is plug these numbers into the area formula as so:

$A=1/2bh$
$A=1/2(5)(6)$
$A=15$

This gives us an area of 15 square units for the triangle. This is a fairly straightforward example of how the area of triangle formula works. Keep reading for more sample problems!

 

body_kids_math_blackboardTime for sample problems—that aren't nearly as easy as these.

 

Area of a Triangle: Sample Problems

Try your hand at finding the area of a triangle with these three sample problems. We’ll then go over the answers for each problem. Note that none of these triangles are drawn to scale.

#1

body_scalene_triangle_area_sample

 

#2

body_right_triangle_area_sample

 

#3

body_equilateral_triangle_area_sample

 

Answers

  1. 18
  2. 12.5
  3. 16√3

 

Answer Explanations

#1

body_scalene_triangle_area_sample

You should be able to tell right away that this is a scalene triangle, meaning that all the sides are different lengths and that there is no right angle (unlike the triangle in #2 below).

Fortunately, you’re given all the information you need to find the area of the triangle. We are told that the base is 9 and the height is 4 (remember that the height is the line that's perpendicular to the apex, or highest point, of the triangle and the base).

Now, just plug these numbers into the area of triangle formula:

$A=1/2bh$
$A=1/2(9)(4)$
$A=18$

The area of the triangle here is 18 square units.

 

#2

body_right_triangle_area_sample

You should notice two things before you even attempt to solve for the area:

  • It’s a right triangle, as noted by the small square in the lower-left corner
  • It’s an isosceles triangle since it has two sides of equal lengths (5 and 5)

Remember that with right triangles, the base and the height are always the two sides that are not the hypotenuse. In other words, they're the two sides that connect to form a right angle. This means that, for this problem, our base is 5 and our height is also 5.

Don’t get thrown off by the square root here—that’s just extra information you don’t even need to find the area of a triangle.

Now that we know our base and height, we can simply plug these numbers into the area of a right triangle formula:

$A=1/2bh$
$A=1/2(5)(5)$
$A=12.5$

The area of this triangle is 12.5 square units.

 

body_tricky_rubiks_cubes_puzzlesThis next sample problem is a bit trickier than the other two.

 

#3

body_equilateral_triangle_area_sample

With this problem, you likely noticed immediately that you’re not told the height of the triangle—only the lengths of the three sides; you should also have recognized this particular triangle as an equilateral triangle, since all three sides are the same length (8 units).

So how do you approach this problem to find the area of the triangle?

In order to find the area, we need to know the base and height. Since we have the base already (8), we'll have to find the height.

Start by drawing a vertical line that dissects the triangle from its apex to the base:

body_equilateral_triangle_area_height

As this is an equilateral triangle, the line we drew for the height will dissect the triangle in half, thereby cutting the base in half as well. This means we'll have 4 on one side of the height line and 4 on the other:

body_equilateral_triangle_area_height_base

Now, look closely at this equilateral triangle. It kind of looks like two right triangles pieced together, right? That’s because it is! Thus, we can use one of these right triangles to find the length of the height line.

Let’s use the right side (it doesn’t matter which one you choose):

body_equilateral_triangle_area_right_triangle

We know the length of the base (what we’ll call a) of this right triangle, as well as the length of the hypotenuse, or c. The height, or b, is unknown, however.

So how can we use the information we have to calculate it?

This is where another formula comes into play: the Pythagorean theorem. Per the Pythagorean theorem, by squaring sides a and b (i.e., the two shorter sides) of a right triangle and then adding them together, you'll get a sum equal to the square of the hypotenuse (side c). 

In other words:

$a^2+b^2=c^2$

As mentioned, we have a and c, so we just need to use this theorem to find b (or h in our original equilateral triangle). Here's how to do that:

$a^2+b^2=c^2$
$4^2+b^2=8^2$
$16+b^2=64$
$b^2=48$
$b=√{48}$

You can further simplify the square root by factoring out 48 like this:

$b=√{48}$
$b=√{(3*4*4)}$
$b=√{(3*4^2)}$
$b=4√3$

Now, we finally have the height of our equilateral triangle (4√3):

body_equilateral_triangle_area_final

All that’s left to do is plug the base (8) and height (4√3) into our area of triangle formula:

$A=1/2bh$
$A=1/2(8)(4√3)$
$A=4(4√3)$
$A=16√3$

The area of this equilateral triangle is 16√3 square units.

 

body_pen_paper_ruler_calculator_math

 

How to Find the Area of a Triangle: 3 Tips

Finding the area of a triangle can be tricky, even if you know the formula. Sometimes it can be hard to understand the logic behind it or to figure out what information you need.

In this section, we give you three tips to help you find the area of a triangle with ease.

 

#1: Visualize a Rectangle (or Parallelogram)

It can often help to visualize the rectangle or parallelogram whose area is equal to double that of the triangle you’re analyzing.

The following diagram captures this idea nicely, indicating how if you copy the triangle and flip it, you'll get a parallelogram that equals double the area of the original triangle (since $A=bh$ for both parallelograms and rectangles):

 

body_triangle_geometry_areaHerbee and Limaner/Wikimedia Commons

 

Furthermore, if you rearrange the parts of the second triangle, you can form a rectangle (instead of a parallelogram) using both the original triangle and the new one.

Once more, what this proves is that the area of a triangle will always equal half the area of the rectangle in which it is inscribed.

 

#2: Know How (and When) to Use the Pythagorean Theorem

Many math problems dealing with the area of a triangle aren’t just about finding the area itself but rather finding a missing side or length (typically the height), too. In these cases you'll need to know more than just how to find the area of a triangle; you'll have to know the Pythagorean theorem.

As mentioned above, the Pythagorean theorem is a formula used to find the lengths of the sides of a right triangle.

Here is the theorem once again:

$a^2+b^2=c^2$

In this, a is the length of one short side, b is the length of the other short side, and c is the length of the hypotenuse (that is, the longest side of a triangle). You’ll need to use the Pythagorean theorem if you’re looking for both the height and area of an isosceles or equilateral triangle.

 

#3: When in Doubt, Review What You’ve Learned

There’s no shame in refreshing your memory or getting help! If you’re struggling to understand how to find the area of a triangle or when and how to use the Pythagorean theorem, I strongly recommend watching some math tutorial videos.

Here’s an example of a video I found particularly helpful while writing this article:

 

You can also check out Khan Academy, a free website that offers a bunch of videos, lessons, and practice problems for an array of math subjects, including triangles and area.

Finally, read our SAT Math/ACT Math triangle guides to learn more about triangles in general.

 

What’s Next?

Got other questions about math? Our expert guides on the distributive property, PEMDAS, and SOHCAHTOA can lend you the hand you need!

Getting ready to take the SAT/ACT? Learn what's tested on SAT Math/ACT Math. You should also start memorizing the most important SAT/ACT math formulas so you can ace the Math section.

 

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Hannah Muniz
About the Author

Hannah received her MA in Japanese Studies from the University of Michigan and holds a bachelor's degree from the University of Southern California. From 2013 to 2015, she taught English in Japan via the JET Program. She is passionate about education, writing, and travel.



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