Heron’s Formula
SSS triangle area formula
12.25 Identify the formula used to calculate the area of a triangle given only the lengths of its sides.
Heron’s formula allows you to calculate the area of a triangle given only the lengths of its sides, x, y, and z. In order to apply the formula, you must first calculate the semiperimeter, s, of the triangle using the following formula.
The perimeter of a triangle is the sum of the lengths of its sides. The SEMIperimeter is equal to half of the perimeter, just like a SEMIcircle is half of a circle.
Once you have calculated the semiperimeter, you can apply Heron’s formula to calculate the area A.
Note: Problems 12.26–12.28 refer to isosceles triangle XYZ below.
12.26 Calculate the semiperimeter of XYZ.
Triangle XYZ is isosceles because at least two of its sides have the same length (XY = XZ = 5). The semiperimeter is equal to half the sum of the lengths of the sides.
Note: Problems 12.26–12.28 refer to isosceles triangle XYZ, as illustrated in Problem 12.26.
12.27 Apply Heron’s formula to calculate the area of triangle XYZ.
Heron’s formula is the square root of the product of the semiperimeter and the differences of the semiperimeter and each of the side lengths.
Note: Problems 12.26–12.28 refer to isosceles triangle XYZ, as illustrated in Problem 12.26.
12.28 Verify your solution to Problem 12.27 by dividing triangle XYZ into two congruent right triangles and calculating their areas.
The segment connecting X and the base 
of the isosceles triangle bisects that base, dividing it in half, and is perpendicular to the base. In the diagram below, 
and WY = WZ.
Apply the Pythagorean theorem to calculate WX.
If a triangle is drawn in the coordinate plane, you may have to worry about signed length, meaning the side of a triangle can have a positive or negative value. This triangle is not drawn in the coordinate plane, so its side lengths MUST be positive. In other words, the answer WX = ±4 is wrong.
Both right triangles have a base of 3 and a height of 4. Use the standard triangle area formula to calculate their areas.
Each of the right triangles has an area of 6. Together they have a total area of 12, which verifies the solution to Problem 12.27.
Note: Problems 12.29–12.30 refer to a triangle with side lengths 7, 9, and 10.
12.29 Calculate the semiperimeter of the triangle.
Add the lengths of the sides and divide by 2.
Note: Problems 12.29–12.30 refer to a triangle with side lengths 7, 9, and 10.
12.30 Calculate the exact area of the triangle.
Apply Heron’s formula, setting x = 7, y = 9, z = 10, and s = 13 (as calculated in Problem 12.29).
Note: Problems 12.31–12.32 refer to triangle LMN, illustrated below.
12.31 Calculate the semiperimeter of the triangle.
You are given the lengths of two sides of the right triangle; apply the Pythagorean theorem to calculate the third.
Calculate the semiperimeter of the triangle.
Note: Problems 12.31–12.32 refer to triangle LMN, illustrated in Problem 12.31.
12.32 Calculate the exact area of the triangle.
Uh oh. That means no calculator and a bunch of arithmetic. Bring it on!
According to Problem 12.31, 
Apply Heron’s formula.
Group the factors to simplify the product.
Multiply the factors within each bracketed group using the FOIL method.
Note: Problems 12.33–12.34 refer to the diagram below.
12.33 Calculate the lengths of the sides of triangle QRS.
Note that QST is a right triangle, so apply the Pythagorean theorem to calculate ST (which is also equal to x).
Now that you know ST = x = 5, you can substitute x = 5 into the expressions that represent the lengths of QR and RS.
The lengths of the sides of triangle QRS are RS = 6, QS = 13, and QR = 16.
Note: Problems 12.33–12.34 refer to the diagram in Problem 12.33.
12.34 Apply Heron’s formula to calculate the area of triangle QRS. Report the answer accurate to the thousandths place.
According to Problem 12.33, the lengths of the sides of triangle QRS are RS = 6, QS = 13, and QR = 16. Calculate the semiperimeter.
Apply Heron’s formula to calculate the area of triangle QRS.
