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FM Herons Formula

1.6 Solving Triangles in 2 and 3 dimensions

Note: if you cannot remember the trigonometric identities, revise notes for 1.5 Methods for Solving Triangles.

Solving for Area without Height

  • In some cases, triangles will be given with the edge lengths rather than the height. In this case our regular formula for the area of a triangle (A=\frac{1}{2} b h) needs some reworking. If we draw a line down the middle of a triangle, we can effectively split it into two right-angled triangles from which we can use our trigonometric identities to solve for the height. Using our formula for sine, we find:

\sin (A) =\frac{h}{c}

h =c \sin (\theta)

Consequently:

A=\frac{1}{2} b c \sin (\theta)

Picture 8
  • This formula works for non-right angled triangles as well.

Note: trigonometry is explored in more detail in notes 2.3 Solving Triangles using Trigonometry and 2.4 Applications of Trigonometry and Pythagoras Theorem.

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1.5 Methods for Solving Triangles

The 180o Rule

  • For any given triangle, the sum of the inside angles is equal to 180o:

A+B+C=180^{\circ}

Where A, B and C are the inside angles of a triangle.

Example

Picture 1

We wish to find the unknown angle, X, in the above triangle. As the other angles are known, we can use the 180o rule. First, we substitute in the known values:

X+35+45=180

Now we solve for the unknown angle:

X+80 &=180

X &=100^{\circ}

The Sine Rule

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