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Determining a Triangle's Area from Its Three Sides

When you know the length of a triangle’s three sides and you don’t know an altitude, you can use Hero’s formula to find the area. Check it out:


where a, b, and c are the lengths of the triangle’s sides and S is the triangle’s semiperimeter


Now use Hero’s formula to calculate the area of a triangle with sides of length 5, 6, and 7. First, you need the triangle’s perimeter (the sum of the lengths of its sides), and from that you get the semiperimeter. The perimeter is 5 + 6 + 7 = 18, so you get 9 for the semiperimeter. Now just plug 9, 5, 6, and 7 into the formula:

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