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### Course: Geometry (all content)>Unit 7

Lesson 9: Advanced area with triangles

# Area of equilateral triangle

Finding the formula for the area of an equilateral triangle with side s. Created by Sal Khan.

## Want to join the conversation?

• Couldn't we use the trig functions to find this too? We have all the angles, so we could have found the height using one of the sides of the Triangle, then plugged that into the area formula.
• what if the equalateral triangles has unequal sides?
• Well, then, it won't be equilateral! An equilateral triangle is, by definition, a triangle with all equal sides. It would be a scalene or isosceles instead.
• How do you figure out the height of an equilateral triangle?
• First draw the height - due to the symmetry of equilateral triangles, it will start at the midpoint of whichever side you choose to start from, and end at the opposite vertex (point/corner). In effect you will split the equilateral triangle into two congruent right triangles.

Now consider one of these two right triangles by itself. If the equilateral triangle has sides of length x, then the hypotenuse of our right triangle will also be x. We also know that the side opposite the 30 degree angle has length x/2, since we split that side of the triangle in half to construct this right triangle. Since you know two of the sides of a right triangle, you can use the Pythagorean theorem to find the length of the 3rd.

(x/2)^2 + m^2 = x^2
x^2/4 + m^2 = x^2
m^2 = (3*x^2)/4
m = (x*sqrt(3))/2

Where m is the height of the right triangle, which is equal to the height of the equilateral triangle. Incidentally, this derivation also proves the shortcut for the ratio of the sides in a 30-60-90 triangle, since the effect of cutting an equilateral triangle in half is to create 2 30-60-90 triangles.
• Isn't it easier to just multiply the 1/2 with the base and height rather than to just go through all of those square roots because he lost me there
(1 vote)
• We don't know what the height is without working it out. Pythagoras' theorem says that in a right triangle, the area of a square drawn on the longest side (the hypotenuse) is equal in area to the areas of the two squares drawn on the other two sides (the legs), added together. The square on the side with length s/2 is 1/4 of the area of the square on the side with length s, so a square on the height must be 3/4 of it. This tells us that the height is s times the square root of 3/4, which is equal to s times a half of the square root of three.
• what does he mean by 30,60,90 video
• Why is the square root of 3 used? I Don't see how it came up. I don't understand that bit. Help me please.
(1 vote)
• The √3 comes from a special property that applies only to 30°-60°-90° right triangles. For just that type of triangle, if the hypotenuse is h, then the shortest side (the side opposite the 30°) will have a length of ½ h. The second longest leg, the one opposite the 60° will have a length of ½ h√3 .

Another way of saying the same thing is that the ratio of the sides of a 30°-60°-90° are 1L : L√3 : 2L where L is the length of the shortest side, the side opposite 30°. L√3 is the length of the second shortest side and 2L is the length of the hypotenuse.

If you cannot remember this, don't worry, you can always do solve it with the usual sin, cos, tan functions.
• One of the sides of an equilateral triangle is 22cm and the height is 16 cm what is the area
(1 vote)
• This is not a possible situation. It follows from the 30-60-90 right triangle side length ratio that the ratio of the height to the side length of an equilateral triangle is sqrt(3) / 2, or approximately 0.866. The ratio 16/22 is approximately only 0.727.

Have a blessed, wonderful day!
• Can you use the herons formula for equilateral triangles?
(1 vote)
• Yes, but I would suggest using the formula Sal gave:

3s² / 4

where 's' is any one side of an equilateral triangle.

The reason why is because it saves a lot of time and effort. In fact, I have never heard of Heron's formula until now, but the formula I provided above is one of the easiest formulas to memorize AND use in my opinion. However, you can always use the Heron's formula if that is easier for you.