Expert: Josh - 1/29/2009

Question
QUESTION: "Hi, is there a formula to work out the hypotenuse length in a right angle triangle when you know the length of the other two sides. thanks"



ANSWER: Hi Mark,

If the length of the two shorter sides (adjacent to the right angle) are X and Y, respectively, then the length of the hypotenuse (d) is given by the square root of (X^2 plus Y^2).

Here, the superscript notation X^2 means X times X. Similarly, Y^2 is the same as Y times Y.

To put this simply, the length of the hypotenuse d=sqrt(X^2+Y^2).

e.g., If the two shorter sides are 3 and 4 units respectively, then the hypotenuse has length d = sqrt(3*3+4*4) = sqrt(9+16) = sqrt(25) = 5.

---------- FOLLOW-UP ----------

QUESTION: This is Pythagoras Theory. what if you only know the length of one side not the hypotenuse of a 30,60,90 degree triangle. how would you find the length of the other side closest to the hypotenuse? is their a theroy? thanks

Answer
Let me draw a diagram so that we are on the same page.

=== CASE 1 ===

First, let us consider a right angle triangle ABC where the angle subtended by BCA is x. Since the sum of angles within a triangle (^ABC+^BCA+^CAB) is always 180 degrees, knowing that ^ABC=90, the angle ^CAB=90-x.

A
|
|
|
|
B--------------C

So long as we know the value of "x" for angle ^BCA (or "90-x" for ^CAB) and the length of any side in a right angle triangle, we can always use the trigonometric identity cos(x)=sin(x+90) to find out the length of the two remaining sides.

Example 1: Referring to the diagram, let |AB| denotes the length of side AB. Suppose |AB|=5 and angle ^BCA=x=30 degrees. From tan(^BCA)=|AB|/|BC|, we obtain |BC|=|AB|/tan(^BCA)=5/tan(30)=5/(1/sqrt(3))=5*sqrt(3).

Example 2: Next, suppose the hypotenuse measures 10 units and ^CAB=60. Since sin(^CAB)=|BC|/|AC| and cos(^CAB)=|AB|/|AC|, knowing |AC|, we conclude that |BC|=|AC|*sin(^CAB) and |AB|=|AC|*cos(^CAB).

=== CASE 2 ===
Let us draw a triangle by joining vertices A, B and C.
A
.
.
.
......B.............C

If ^ABC is not equal to 90 deg, then, we must either have
a) 2 sides plus an angle OR
b) 2 angles plus a side
to fully determine the unknowns in a triangle.

For a) Suppose that we know the following:
|AB|=c, |BC|=a and angle ^ABC=x. The unknown is |AC|=b.
The law of cosine states that b^2 = a^2 + c^2 - 2*a*c*cos(x).
The key to the formula is that the two known sides are adjacent to a known angle x, while the unknown side is opposite to angle x.

For b) Suppose that two angles and one side are known. In particular, ^CAB=x, ^BCA=y and |BC|=a. Let |AC|=b be the unknown. The law of sine states that a/sin(^CAB) = b/sin(^ABC). The key point is that we always consider two sides which are directly opposite to their respective angle.

Summary
=======
Knowing 2 sides + 1 angle:
Use b^2 = a^2 + c^2 - 2*a*c*cos(x)

Knowing 2 angles + 1 side:
Use a/sin(A)=b/sin(B)=c/sin(C),
where side "a" is opposite to angle A, side "b" is opposite to angle B and so forth.

Don't forget the angle sum of a triangle is 180. Knowing any two angles, you know all of them.