How To Find A Triangle's Height

Triangles each take iii heights, each related to a separate base. Regardless of having up to iii different heights, one triangle will always take only ane mensurate of area. In some triangles, like right triangles, isosceles and equilateral triangles, finding the top is easy with 1 of two methods.

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

How to Notice the Height of a Triangle

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

In an equilateral triangle, like $△ S U N$ below, each elevation is the line segment that splits a side in half and is too an angle bisector of the opposite bending. That will only happen in an equilateral triangle.

$Three heights of equilateral triangle$

By definition of an equilateral triangle, y'all already know all iii sides are congruent and all 3 angles are $60 °$ . If a side is labelled, you know its length.

Our brilliant lilliputian $△ Southward U North$ has one side labelled $24 c m$ , and so all 3 sides are $24 c m$ . Each line segment showing the pinnacle from each side also divides the equilateral triangle into two right triangles.

Height of a triangle formula

Your ability to divide a triangle into correct triangles, or recognize an existing right triangle, is your fundamental to finding the mensurate of height for the original triangle. You tin can take any side of our fantabulous $△ S U North$ and run across that the line segment showing its pinnacle bisects the side, then each short leg of the newly created correct triangle is $12 c m$ . We already know the hypotenuse is $24 c g$ .

Knowing all iii angles and 2 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:

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

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

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

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

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

$b = 20.7846096908 c 1000$

$Find height of a triangle using pythagorean theorem$

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

You tin can decide for yourself how many pregnant digits your answer needs, since the decimal will go on and on. Do not forget to use linear measurements for your reply!

The Pythagorean Theorem solution works on right triangles, isosceles triangles, and equilateral triangles. It will non piece of work on scalene triangles!

Using the area formula to notice height

The formula for the expanse of a triangle is $\frac{1}{2} b a s due east \times h e i one thousand h t$ , or $\frac{1}{ii} b h$ . If you know the surface area and the length of a base, and so, yous tin calculate the height.

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

Here we have scalene $△ Z I 1000$ with a base of operations shown as $56 y a r d southward$ and an area of $987 s q u a r due east y a r d south$ , but no clues about angles and the other ii sides!:

$Using area formula to find height of a triangle$

Recalling the formula for area, where $A$ means area, $b$ is the base of operations and $h$ is the height, we call up

$A = \frac{1}{2} b h$

This tin be rearranged using algebra:

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

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

Put in our known values:

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

$h = two (17.625 y a r d s)$

$h = 35.25 y a r d due south$

Call up how nosotros said every triangle has iii heights? If nosotros accept $△ Z I G$ and rotate information technology clockwise and so side $One thousand Z$ is horizontal, and construct a height up to $∠ I$ , we can get the peak for that side, too.