Heron's Theorem

An Example of Triangulation

Step 1

Separate the quadrilateral

into two triangles

-----------------------------------

Step 2

Quadrilaterals and Triangulation

Find the lengths of the sides

of each of the

triangles

B

side a = 8.5

A

side b = 8.5

-----------------------------------

side e = 13.4

side d = 7.2

D

C

side c = 13

• All quadrilaterals can be separated into two triangles. This makes finding the area of an irregular quadrilateral fairly simple

• Steps to finding the area of a quadrilateral with triangulation and Heron's Theorem:

Step 3

• Use Heron's Theorem to find the areas of triangles ABC and CDA

B

side a = 8.5

A

Triangle ABC:

s = 1/2(8.5+8.5+13.4)

s = 1/2(30.4)

s = 15.2

----------------------------------------------------------

area = √15.2(15.2-8.5)(15.2-8.5)(15.2-13.4)

area = √(231.04-129.2)(6.7)(1.8)

area = √(101.84)(6.7)(1.8)

area = √1228.1904

area = 35.05 square units

side b = 8.5

A Quick Overview of Triangulation

-----------------------------------

side e = 13.4

side d = 7.2

Step 4

Triangle CDA:

s = 1/2(7.2+13+13.4)

s = 1/2(33.6)

s = 16.8

----------------------------------------------------

area = √16.8(16.8-7.2)(16.8-13)(16.8-13.4)

area = √(282.24-120.96)(3.8)(3.4)

area = √(161.28)(3.8)(3.4)

area = √2083.7376

area = 45.65 square units

D

C

side c = 13

Add the areas of the two triangles to find the total area of the

quadrilateral

35.05

square units

35.05+45.65=80.7

-----------------------------------

45.65

square units

The total area of the quadrilateral is 80.7 square units

• Triangulation is a tactic for finding the area of an irregular shape.
• This method is commonly used by architects an landscapers for finding the area of buildings or plots of land.
• Every shape is essentially made out of triangles, the only shape that can be irregular and still easily have it's area be found.
• Some shapes are made up of an infinite amount of triangles. These are called compact shapes. Shapes that have curved lines are compact

What if you separate the quadrilateral into different triangles?

The area remains the same the two triangles have different areas by still add up to the same area for the whole quadrilateral.

B

side a = 8.5

A

side b = 8.5

side e = 13.2

Citations

-----------------------------------------

Triangle ABD:

s = 1/2(7.2+8.5+13.2)

s = 1/2(28.9)

s = 14.45

area = √14.45(14.45-7.2)(14.45-8.5)(14.45-13.2)

area = √(208.8025-104.04)(5.95)(1.25)

area = √(104.7625)(5.95)(1.25)

area = √779.1711

area = 27.91 square units

Triangle CDB

s = 1/2(8.5+13+13.2)

s = 1/2(34.7)

s = 17.35

area = √17.35(17.35-8.5)(17.35-13)(17.35-13.2)

area = √(301.0225-147.475)(4.35)(4.15)

area = √(153.5475)(4.35)(4.15)

area = √2771.9162

area = 52.65 square units

Adding:

27.91+52.65 = 80.56 square units

The area with the other diagonal was 80.7 square units. There was a slight difference due to human error while measuring. If the measurements were exact, the areas would be exactly the same

side d = 7.2

D

C

side c = 13

Triangulation and Heron's Theorem

• Heron's Formula. (2012). . Retrieved April 19, 2014, from http://www.mathsisfun.com/geometry/herons-formula.html
• Area of a Triangle Given Three Sides- Heron's Formula. (2009). . Retrieved April 20, 2014, from http://www.mathopenref.com/heronsformula.html
• Heron's Formula l Khan Academy. (2014). . Retrieved April 17, 2014, from http://www.khanacademy.org/math/geometry/basic-geometry/heron_formula_tutorial/v/heron-s-formula

By: Lily Gido

Does Heron's Theorem

work for all Triangles?

History

Heron's Theorem

In order to find out, I tried out Heron's Theorem on three different types of triangles

Right Isosceles (45, 45, 90) Triangle

• Heron's Theorem was first proven by a mathematician, Heron of Alexandria around 200 A.D.
• It is unknown whether Heron was Greek or Egyptian but he did study in Alexandria Egypt.
• Heron wrote many pieces on geometric and mechanical logic including the Metrica, one of the greatest contributions ever to the mathematical world.

A

Heron's Theorem:

s = 1/2(3+3+4.24)

s = 1/2(10.24)

s = 5.12

area = √5.12(5.12-3)(5.12-3)(5.12-4.24)

area = √(26.2144-15.36)(2.12)(0.88)

area = √(10.8544)(2.12)(0.88)

area = √20.249

area = 4.5 square units

Base Height Formula:

area = 1/2(3*3)

area = 4.5 square units

side c = 4.24

side a = 3

Same Area

Heron's Theorem works

for this triangle

What is Heron's Theorem?

B

C

side b = 3

• Heron's Theorem is a two step formula that can be used to find the area of any triangle by just knowing the three side lengths of said triangle.
• In basic math we are taught that the formula for the area of a triangle is: A=(1/2)(base*height). The problem with this formula is that we don't always know or are able to find the height of a triangle.
• With Heron's Theorem, it is not necessary to know the height of the triangle, just it's side lengths.
• The formula for Heron's Theorem is:

Acute Equilateral

Heron's Theorem:

s = 1/2(5+5+5)

s = 1/2(15)

s = 7.5

area = √7.5(7.5-5)(7.5-5)(7.5-5)

area = √(56.25-37.5)(2.5)(2.5)

area = √(18.75)(2.5)(2.5)

area = √117.1875

area = 10.83 square units

Base Height Formula:

area = 1/2(5*4.33)

area = 10.83 square units

side c = 5

side a = 5

------------------------------

height = 4.33

Step 1:

Calculate half of the triangle's perimeter. This can be symbolized by "s."

The Equation is: s = 1/2(side a+side b+side c)

An Example of Heron's Formula

VS

The traditional area of a triangle formula

side b = 5

Same area

Heron's Theorem Works for This Triangle

Step 2:

You can calculate the area of the triangle using the value of "s" that was found in step 1.

The Equation is: Area = √( s(s-a)(s-b)(s-c) )

The Problem

A

Obtuse Scalene

What is the area of triangle ABC?

*Not drawn

to scale

Base Height Formula:

area = 1/2(6*1)

area = 1/2(6)

area = 3 square units

side b = 5

side a = 5

l-----------------------------------------l

Heron's Theorem:

s = 1/2(5.1+1.41+6)

s = 1/2(12.51)

s = 6.255

area = √6.255(6.255-5.1)(6.255-1.41)(6.255-6)

area = √(39.13-31.9)(4.85)(0.255)

area = √(7.23)(4.85)(0.255)

area = √8.9417

area = 3 square units

l-----------------------------------------l

l

l

l

l

l

l

l

l

l

l

l

l

l

l

l

l

Height = 4

Given:

• Side a = 5
• Side b = 5
• Side c = 6
• The height of triangle ABC is 4

Same Area

l-------------------------------------------l

B

C

side c = 6

Heron's Theorem Works for this

Triangle

side b = 5.1

side c = 1.41

-----------

height = 1

side c = 6

Solution with Heron's Theorem

Solution with Base Height Formula

Step 1:

Plug triangle ABC's values into the first equation

s = 1/2 (5+5+6)

s = 1/2 (16)

s = 8

Step 1 (Only step): Plug triangle ABC's values into the base height equation

area = (1/2)(6*4)

area = 1/2(24)

area = 12 square units

Step 2:

Plug triangle ABC's values and the value of "s" into the second equation

area = √8(8-5)(8-5)(8-6)

area = √(64-40)(3)(2)

area = √(24)(3)(2)

area = √144

area = 12 square units

Final answer:

12 square units

Final Answer:

12 square units

The answer is the same with both methods