AIM: WHAT IS HERON’S FORMULA AND HOW DO WE
USE IT?
Do Now: Find the area.
1)
1)
2)
Area of triangle = b h
2
A =86
2
6 in
A = 48
2
8 in
A = 24 in2
3 in
6
in
8 in
THE HERON’S FORMULA
IS USED TO FIND THE
AREA OF A TRIANGLE
USING ITS SIDES.
The formula is credited to Heron, who was the “Hero of
Alexandria”; a proof can be found in his book, Metrica written
in 60 A.D. It was discovered by the Chinese published in Shushu
Jiuzhang.
A, B, and C are the
sides of the
triangle.
“S” is half the
triangle’s
perimeter
THE HERON’S FORMULA:
After using the formula to find “s,” you plug
it into the Heron’s formula and again a, b,
and c refer to the sides of the triangle.
EXAMPLES:
1) What is the area of the triangle with sides of
length 10 feet, 15 feet, and 17 feet?
𝟏
𝟐
S = (10 + 15 + 17) = 21
area = 𝒔(𝒔 − 𝒂)(𝒔 − 𝒃)(𝒔 − 𝒄)
area = 𝟐𝟏(𝟐𝟏 − 𝟏𝟎)(𝟐𝟏 − 𝟏𝟓)(𝟐𝟏 − 𝟕)
area = 𝟐𝟏 𝟏𝟏 𝟔 𝟒 = 𝟓, 𝟓𝟒𝟒 ≈ 𝟕𝟒. 𝟒𝟓𝟖 square ft
2) What is the area of an equilateral triangle with
all sides 6 inches in length?
s=
𝟏
(6
𝟐
area =
area =
area =
+ 6 + 6) = 9
𝒔(𝒔 − 𝒂)(𝒔 − 𝒃)(𝒔 − 𝒄)
𝟗(𝟗 − 𝟔)(𝟗 − 𝟔)(𝟗 − 𝟔)
𝟗(𝟑)(𝟑)(𝟑)= 𝟗 𝟏𝟑 ≈15.588 square in.
NOW TRY THE DO NOW QUESTION
2)
3 in
6
in
8 in
2) semiperimeter = a + b + c
2
s = 3+6+8
2
s = 17
2
s = 8.5
A = √s(s – a)(s – b)(s – c)
A = √8.5(8.5 – 3)(8.5 – 6)(8.5 – 8)
A = √8.5(5.5)(2.5)(0.5)
A ≈ 7.64 in2
EXAMPLE :
Jack has a garden, which is triangular in shape. The
sides of the garden are 13 m, 14 m, and 15 m
respectively. He wants to spread fertilizer in the garden
and the total cost required for doing it is 10 Dirhams per
m2. He is wondering how much money will be required to
spread the fertilizer in the garden.
SOLUTION
• Given a = 13 m , b = 14 m and c = 15 m
So , we will find
the area of the
triangle by
using Heron’s
formula.
CONTINUE..
21(21  13)(21  14)(21  15)
=
21*8* 7 * 6
CONTINUE …
Given the rate = 10 Dirhams per square meter
Now :
Total cost = 10 * 84 = 840 Dirhams