How To Find The Area Of An Equilateral Triangle

Triangles each have 3 heights, each related to a separate base. Regardless of having upward to iii dissimilar heights, 1 triangle will always have only one measure out of area. In some triangles, like correct triangles, isosceles and equilateral triangles, finding the height is like shooting fish in a barrel with one of ii methods.

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How to find
Formula
- Using Pythagorean Theorem
- Using area formula

How to Find the Superlative of a Triangle

Every triangle has 3 heights, or altitudes, because every triangle has three sides. A triangle'southward height is the length of a perpendicular line segment originating on a side and intersecting the opposite bending.

In an equilateral triangle, similar $△ S U N$ below, each height is the line segment that splits a side in half and is as well an bending bisector of the opposite angle. That volition just happen in an equilateral triangle.

$Three heights of equilateral triangle$

By definition of an equilateral triangle, yous already know all iii sides are congruent and all three angles are $60 °$ . If a side is labelled, you know its length.

Our bright little $△ S U North$ has one side labelled $24 c yard$ , and so all 3 sides are $24 c m$ . Each line segment showing the peak from each side likewise divides the equilateral triangle into two correct triangles.

Tiptop of a triangle formula

Your ability to divide a triangle into right triangles, or recognize an existing right triangle, is your key to finding the measure out of meridian for the original triangle. You can accept any side of our splendid $△ S U N$ and run into that the line segment showing its height bisects the side, so each short leg of the newly created correct triangle is $12 c m$ . Nosotros already know the hypotenuse is $24 c one thousand$ .

Knowing all three angles and ii sides of a right triangle, what is the length of the third side? This is a job for the Pythagorean Theorem:

Using Pythagorean Theorem

Focus on the lengths; angles are unimportant in the Pythagorean Theorem. Plug in what you lot know:

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

$12^{2} + b^{2} = 24^{2}$

$144 + b^{2} = 576 c m^{2}$

$b^{2} = 432 c m^{2}$

$\sqrt{b^{2}} = \sqrt{432 c m^{2}}$

$b = twenty.7846096908 c m$

$Find height of a triangle using pythagorean theorem$

Most people would be happy to say the height (side $b$ ) is approximately $20.78$ , or $b \approx 20.78$ .

Y'all tin decide for yourself how many meaning digits your answer needs, since the decimal will go on and on. Do non forget to use linear measurements for your answer!

The Pythagorean Theorem solution works on correct triangles, isosceles triangles, and equilateral triangles. It will not work on scalene triangles!

Using the area formula to find height

The formula for the area of a triangle is $\frac{1}{two} b a s east \times h e i g h t$ , or $\frac{1}{2} b h$ . If yous know the surface area and the length of a base, and so, you tin can summate the tiptop.

In contrast to the Pythagorean Theorem method, if you have two of the three parts, you can find the height for any triangle!

Hither we have scalene $△ Z I M$ with a base shown as $56 y a r d southward$ and an area of $987 due south q u a r east y a r d due south$ , merely no clues about angles and the other two sides!:

$Using area formula to find height of a triangle$

Recalling the formula for expanse, where $A$ means expanse, $b$ is the base and $h$ is the height, we remember

$A = \frac{ane}{2} b h$

This tin be rearranged using algebra:

$A = \frac{b h}{2}$

$h = ii (\frac{A}{b})$

Put in our known values:

$h = 2 (\frac{987 due south q u a r e y a r d s}{56 y a r d s})$

$h = ii (17.625 y a r d southward)$

$h = 35.25 y a r d s$

Retrieve how we said every triangle has three heights? If nosotros take $△ Z I Yard$ and rotate information technology clockwise so side $1000 Z$ is horizontal, and construct a summit upwardly to $∠ I$ , nosotros can get the peak for that side, too.