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

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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:
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, where was discovered by the Chinese independently of the Greeks.

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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.
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Step 1: Find the semi-perimeter – half the perimeter of the triangle.
A
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Step 2: Plug “s” into the formula and solve.
C
B
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Heron’s Formula is also used as an equivalent to the Pythagorean theorem.

What is the area of a triangle where every side is 5 long?
5 5
5
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...

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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

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

Law of Cosines

Proof With Heron’s Formula: Pythagorean Theorem