An earlier section,
Area Formulas: Triangle,
Parallelogram, Trapezoid
,
showed how to find the area of a
triangle with a base/height pair.
The key results are summarized here for convenience.
With trigonometry now in hand, another useful formula for the area of a triangle is given.
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![]() Drop a perpendicular to the opposite side ... |
![]() ... or, to an extension of the opposite side. |
A derivation of the triangle area formula ($\,bh/2\,$) was given in Area Formulas: Triangle, Parallelogram, Trapezoid for the situation where the altitude ‘hits’ the base (left-most picture above). The formula also works when the altitude doesn't ‘hit’ the base, but instead hits an extension of the base, as shown at right. Here's how:
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![]() area of blue triangle is $\,\frac 12 bh\,$ |
Note that there are infinitely many triangles with a given base/height pair.
For example, all the triangles below have the same base/height pair.
Same base/height pair—same area!
A base/height pair uniquely determines the area of a triangle, but
not the shape of the triangle.
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All these triangles have the same base/height pair, hence the same area. |
By the
SAS (side-angle-side) triangle congruence theorem,
two sides and an included angle uniquely defines a triangle. Let $\,a\,$ and $\,b\,$ denote two sides of a triangle, with included angle $\,\theta\,.$ The sketches at right and below show the two situations that can occur when finding the altitude that has corresponding base $\,b\,.$ In both cases, the altitude length is found using the yellow triangle (which is a right triangle with hypotenuse $\,a\,$). In both cases, the altitude length is $\,h = a\sin\theta\,.$ Thus: $$ \begin{align} &\cssId{s45}{\text{area of triangle with sides $a$ and $b$ and included angle $\theta$}}\cr\cr &\qquad \cssId{s46}{= \frac 12(\text{base})(\text{height})}\cr\cr &\qquad \cssId{s47}{= \frac 12 b(a\sin\theta)}\cr\cr &\qquad \cssId{s48}{= \frac 12 ab\sin\theta} \end{align} $$ In words, the area is: times the sine of the included angle |
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On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. |
PROBLEM TYPES:
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