Do you want to know how to find the height of a triangle?
That is one question that many students find difficult. There are three ways in which you can do this, and they all work for different types of triangles.
So without further delay let’s jump into the tutorial.
What is the Formula for Height?
The height of a triangle can be found using the Pythagorean theorem or by using trigonometric ratios. In this blog post, we will show you how to find the height of a triangle using three different methods.
How to Find the Height of a Triangle?
Method 1: The Pythagorean Theorem
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse equals the sum of squares of lengths.
Using this theorem, we can find height when the given a base and one side. To do so you must solve for x in terms to y using:
height = √((base² + height²)/(x))
For example, if we know the base is 12 and a side of 11, then
height = √((12² + 11²)/(x))
= √(144+121)
= 13.75 Method #: Use Trigonometric Ratios to Find Height Given Base and Side-Angle Let X be defined as the length of the side opposite angle A and Y be defined as the length of the hypotenuse.
Then, using trigonometric ratios:
sinA = opp/hypotenuse
cosA = adj/hypotenuse
tanA=opp/adj So, in order to find height we can use:
height = √((Y²-X)²+(Y)(X))
For example, if we know the side of a triangle is 11 and angle A measures 38 degrees then:
sinA = opp/hypotenuse
=11/hypotenuse
= 0.685714 (to two decimal places)
cosA = adj/hypotenuse
= 11/13
= 0.84 (to two decimal places)
tanA=opp/adj
=11/x
height = √((Y²-X)²+(Y)(X))
So, the height of the triangle is 13.14 (to two decimal places).
Method 2: Use Law of Sines to Find Height Given Base and Angle
The law of sines states that in any triangle, the ratio of the length of a side to the sine of its angle is constant.
This means that we can use this equation to solve for height:
height = (opposite side)/(sin of angle opposite)
For example, if we know the base is 12 and angle A measures 38 degrees then:
height = (opposite side)/(sin of angle opposite)
= (12)/(.685714)
= 17.46 (to two decimal places)
Method 3: Finding an Equilateral Triangle’s Height
An equilateral triangle is a type of triangle where all three sides have the same length.
Using this fact, we can find height by taking one side and dividing it by two:
height = (side)/(√((s-a)²+(s-b)²+ (s-c)))
For example, if we know that side = 11 cm then height would equal (11cm)/(√((s-a)²+(s-b)²+ (s-c)))
height= 11/(√((11-(x))²+(y)²+(-z))))
This is because, in an equilateral triangle, the height is equal to the side divided by two.
If you don’t know any of the lengths of the sides, you can still find the height using Pythagoras’ theorem.
The equation for this is: a²+b²=c²
So, there’s still a way to find the height! All you need is the angles of the triangle.
The Law of Sines states that: sin(A)/a = sin(B)/b = sin(C)/c
We can solve for a, b, and c. Then we know the height is: sin(A)cos(B). So if you have two angles of a triangle, you can find its height!
Conclusion
We hope you now understand how to find the height of a triangle by using 3 different ways. If you have any questions, please let us know in the comments below!
Frequently Ask Questions
Q: How do you find the height of an oblique triangle?
A: Unfortunately, there is no one definitive answer to this question. Some methods include using trigonometry or the law of sines, or you could use the Pythagorean theorem.
Q: What if you don’t know the angles of a triangle but do know some sides?
A: You can still find the height using Pythagoras’ theorem. The equation for this is: a²+b²=c²
Q: What is the shortest side of a 30 60 90 triangle?
A: The shortest side is the side opposite of the 60° angle. This would be the side with length x.
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