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{{ printedBook.courseTrack.name }} {{ printedBook.name }} The area of a triangle is half the product of the lengths of any two sides and the sine of the included angle.

Based on the diagram above, the area of $△ABC$ can be found using any of the three formulas below.

$AreaAreaArea =21 absinC=21 bcsinA=21 acsinB $

To find the first formula, start by drawing the altitude from $B$ and let $h$ be its length. Since the altitude is perpendicular to the base, it generates two right triangles.

Because $△BCD$ is a right triangle, the height of the triangle can be related to the sine of $∠C$ using the sine ratio. $sinC=ah ⇒h=asinC $ Finally, substitute the expression found for $h$ into the general formula for finding the area of a triangle.

$Area=21 bh⇓Area=21 absinC $

To obtain the second formula, notice that $△ABD$ is also a right triangle and then, the sine ratio can be applied. $sinA=ch ⇒h=csinA $ Substituting this expression into the general formula for finding the area of a triangle gives the first formula.

$Area=21 bh⇓Area=21 bcsinA $

To deduce the third formula, it is needed to draw the height from $C$ $($or $A).$

In this case, the length of the base is $c$ and the height is $h.$ Since $△BDC$ is a right triangle, the sine ratio can be used. $sinB=ah ⇒h=asinB $ Lastly, substitute the expression for $h$ into the formula to find the area of $△ABC.$

$Area=21 ch⇓Area=21 acsinB $