Aim: How can we find the area of a Triangle using Heron’s Formula Do Now: A Find the area of triangle ADC. Round to the nearest tenth. 12 X Answer: Area=35.8 C 8 B

Facts About Heron’s Formula The formula is credited to Heron of Alexandria, who was an ancient Greek mathematician and engineer who was active in his native city of Alexandria, Roman Egypt. Hero also described a method of iteratively computing the root. Today however, his name is most closely associated with Heron’s Formula for finding the area of a triangle from its side lengths and a proof can be found in his book, Metrica, which was written c. A.D. 60. It has been suggested that Archimedes knew the formula, and since Metrica is a collection of the mathematical knowledge available in the ancient world, it is possible that it predates the reference given in the work. A formula equivalent to Heron's namely: , where  was discovered by the Chinese independently of the Greeks.

HERON’S FORMULA: Heron’s Formula is used to get the area of a triangle when you know the sides of the triangle, but you do not know the height. Step 1: Find the semi-perimeter – half the perimeter of the triangle. Step 2: Plug “s” into the formula and solve. Heron’s Formula is also used as an equivalent to the Pythagorean theorem. A C B

Example 1: What is the area of a triangle where every side is 5 long? Answer- Step 1: S = (5+5+5)/2 = 7.5 Step 2: A = √(7.5 × 2.5 × 2.5 × 2.5) = √(117.1875) = 10.825...

Example 2: Use Heron’s Formula to find the area. Round to the nearest tenth. 3 7 6 Answer- Step 1: 7+3+6/2=8 Step 2:√8(8-7)(8-6)(8-3)=√(80)=8.9

Example 3: Find the Area using Heron’s Formula. 30 Answers 18 Step 1:27.5 Step 2: 141.989 18 24

Proof With Heron’s Formula: Area Law of Cosines

Proof With Heron’s Formula: Pythagorean Theorem