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How to Find the Area of a Triangle — Formula & Video

Written by

Malcolm McKinsey

January 11, 2023

Fact-checked by

Paul Mazzola

Area and base

First, let's cover a few definitions that will help us on our quest to finding the area of a triangle.

The area of a triangle is the interior space enclosed by its three straight sides. Though always triangle-shaped, it is measured in square units.

The base of a triangle is the side used to create the altitude, or height.

Draw an oblique triangle $\triangle RCK$ with ∠R=31°, ∠C=47°, ∠K=102°. Orient the triangle so that side RC which is 8 cm, is vertical and on left side. The other sides, side RK must be 5.98 cm and side KC must be 4.21 cm.

Normally, geometry textbook writers give you a triangle with a base drawn horizontally for you, but that is deceptive! No deceptions here -- any side can be a base!

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Height or altitude

Add a perpendicular from side RC over to $\angle K$ . That perpendicular is the altitude, or height, of the triangle from that base. We know that side RC is 8 cm long, and we can calculate the height to be about 3.08 cm. Label the point where the altitude intersects with RC as Point O. This triangle ROCKs!

Notice that our altitude is perpendicular to side RC, even when side RC is vertical, as in our drawing. The base does not have to be drawn horizontally for you.

For our same triangle, we could pick a different side to be the base. Let's use side RK, 5.98 cm, and construct an altitude to $\angle C$ . That altitude is 4.12 cm.

Try the last side and construct an altitude from side KC, 4.21 cm long, and have it meet $\angle R$ . That altitude is 5.85 cm in height.

Oblique triangle

You worked hard to recall equilateral, isosceles and scalene triangles. They are named by the lengths of their sides. Then someone told you about right, acute and obtuse triangles. They are named by their interior angles. A right triangle, for example, has one right angle in it.

Now we have to let you in on a secret about acute and obtuse triangles.

They are both types of oblique triangles, or triangles with no right angles. Acute triangles have three acute interior angles (each is less than 90°). An equilateral triangle is an example of an acute triangle. Obtuse triangles have one angle greater than 90° and two acute angles.

Can you see why you can never have a triangle with two interior angles greater than 90°? Two of the three sides would diverge, or move away from each other, instead of converging.

Now we're ready to unlock our formula!

Area of a triangle formula

A handy formula gives you the area in square units of any triangle.

Area=\frac{1}{2}(base\times height)

We already have $\triangle RCK$ ready to use, so let's try the formula on it:

Area=\frac{1}{2}bh

Area=\frac{1}{2}(8\times 3.08)

Area=\frac{1}{2}(24.64)

Area=12.3207c{m}^{2}

But we also found the altitude using the other two sides as bases. So let's see if the formula works on all three sides:

Area=\frac{1}{2}bh

Area=\frac{1}{2}(5.98\times 4.12)

Area=\frac{1}{2}(24.6376)

Area=12.3188c{m}^{2}

That is within 0.0019 of our first calculation!

Area=\frac{1}{2}bh

Area=\frac{1}{2}(4.21\times 5.85)

Area=\frac{1}{2}(24.6285)

Area=12.3142c{m}^{2}

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That is within 0.0065 of the first! These are insignificant simple rounding errors, with more exact measurements than you could achieve with a ruler.

Lesson summary

Now that you have worked your way through this lesson, you now know how to define a right triangle (having one right angle), an oblique triangle (having no right angles), the base of a triangle, and the altitude or height of a triangle.

You also should be able to recall, identify and apply the formula for finding the area of triangles: $Area=\frac{1}{2}(base\times height)$ .