Video Transcript
Find the area of an equilateral triangle with a side length of three centimeters, giving the answer in an exact form.
Let’s begin by sketching the equilateral triangle in this question. As the triangle is equilateral, all three of its sides are the same length. And we’re told in the question that this length is three centimeters. We’re asked to find the area of this triangle, which usually we may do using the formula area equals base multiplied by height over two. We aren’t given the perpendicular height of this triangle though. And although it would be possible to work it out, let’s consider an alternative method for finding the area of any triangle.
We recall that the trigonometric formula for the area of a triangle 𝐴𝐵𝐶, in which the uppercase letters 𝐴, 𝐵, and 𝐶 represent the vertices of the triangle and the lowercase letters 𝑎, 𝑏, and 𝑐 represent the three opposite side lengths, is area is equal to one-half 𝑎𝑏 sin 𝐶. Here, 𝑎 and 𝑏 represent the lengths of any two sides in the triangle and capital 𝐶 represents the measure of their included angle. That’s the angle between the two sides whose lengths we know. We don’t need to be overly concerned about the letters 𝑎, 𝑏, and 𝑐. We can use any two side lengths in the triangle and the measure of their included angle.
Returning to this problem, in which we have an equilateral triangle, we know the lengths of all three sides. They’re each three centimeters. We haven’t been given the measures of the angles. But in fact because this triangle is equilateral, the measures of all three angles are the same. They’re each one-third of 180 degrees, which is 60 degrees.
It doesn’t matter at all which two sides we use. But for the sake of argument, let’s suppose we’re using the two sides marked in orange and their included angle of 60 degrees. Substituting three for the values of both 𝑎 and 𝑏 and 60 degrees for the value of the angle 𝐶, we have that the area of this triangle is one-half multiplied by three multiplied by three multiplied by sin of 60 degrees. That’s nine over two multiplied by sin of 60 degrees.
Now, the question asks us to give our answer in an exact form. So at this point, we should recall that 60 degrees is one of the special angles for which the sine, cosine, and tangent ratios can be expressed exactly in terms of fractions and surds. sin of 60 degrees is exactly equal to the square root of three over two. And so we can substitute this value for sin of 60 degrees to give an exact answer. We have nine over two multiplied by root three over two, which simplifies to nine root three over four. Equivalently, we can write this as nine over four multiplied by root three, and the units for this area will be square centimeters.
So, by recalling the trigonometric formula for the area of a triangle, we found that the area of an equilateral triangle with a side length of three centimeters in exact form is nine over four root three square centimeters.
