Document Type : Original Article
Author
Cairo University
Abstract
Keywords
Article Title [العربیة]
Author [العربیة]
الملخص عربى
هذه الصیغ تطویر لنظریات قدیمة لتسهیل الحل استخدام المساحات الهندسیة هام جدا فى حیاتنا قدیما کان یستخدم خطوات کثیرة لحل المساحة للمثلث ولکن هذا البحث یسهل حل المثلث ومثال ذلک مساحة المثلث بضلع واحد فقط وزاویتین وذلک یوفر الوقت والجهد والتفکیر قدیما یتم حل مساحة المعین بطریق صعبة ولکن فى هذا البحث یتم الحل بسهولة مساحة المعین نصف حاصل ضرب القطرین ولکن فى القانون الجدید المساحة بدلالة قطر وزاویة فقط مساحة متوازى الاضلاع تساوى القاعدة فى الارتفاع ولکن القانون الجدید بدلالة قطر وزاویتى القطر فقط مساحة المستطیل تساوى الطول فى العرض ولکن القانون الجدید بدلالة قطر وزاویتى القطر فقط وبذلک یسهل الاجابة فى ایجاد مساحة المثلث والمعین والمستطیل والمربع بطرق جدیدة وهذا یفید الطلاب فى هندسة مدنى کلیة الهندسة
New formulas in area
Ehab esmail ahmed
Bank el taslef Street, Tahta, Sohag, egypt
mathehab75@yahoo.com E-mail
Abstract
This formula develops old theories to facilitate the solution The use of geometric spaces is very important in our live. In ancient times it took many steps to resolve the area of the triangle, but in this research, it will be easy to solve the area of triangle with one side and two angles only and this saves time , effort and thinking.
In Old times, solving rhombus area took a number of difficult steps, but in this research it will be easily solved..
Generally, rhombus area is calculated by half multiplied the lengths of the two diagonals, but in but in this research area is calculated in terms of one diagonal length and an angle only
Parallelogram area is equal to multiplication of the base length by the perpendicular , but the new method uses diagonal length and angles of the diagonal only
Area of the rectangle equal multiplication of length and width, Using the new method diagonal length and angles of the diagonal are used only
Uses
1- triangle area in a new method
2- parallelogram area a new method
3 – rectangle area in a new method
4 - rhombus area in a new method
5 - square area in a new method
Keyword
The area; Triangle; rhombus (diamond); Parallelogram; rectang
الملخص عربى
هذه الصیغ تطویر لنظریات قدیمة لتسهیل الحل
استخدام المساحات الهندسیة هام جدا فى حیاتنا قدیما کان یستخدم خطوات کثیرة لحل المساحة للمثلث ولکن هذا البحث یسهل حل المثلث ومثال ذلک مساحة المثلث بضلع واحد فقط وزاویتین وذلک یوفر الوقت والجهد والتفکیر قدیما یتم حل مساحة المعین بطریق صعبة ولکن فى هذا البحث یتم الحل بسهولة مساحة المعین نصف حاصل ضرب القطرین ولکن فى القانون الجدید المساحة بدلالة قطر وزاویة فقط مساحة متوازى الاضلاع تساوى القاعدة فى الارتفاع ولکن القانون الجدید بدلالة قطر وزاویتى القطر فقط مساحة المستطیل تساوى الطول فى العرض ولکن القانون الجدید بدلالة قطر وزاویتى القطر فقط وبذلک یسهل الاجابة فى ایجاد مساحة المثلث والمعین والمستطیل والمربع بطرق جدیدة وهذا یفید الطلاب فى هندسة مدنى کلیة الهندسة
Introduction
Calculating the area of a triangle using the length of one side and two angles is better than using the old solution solving rhombus area took a number of difficult steps, but in this research it will be easily solved.
Generally, rhombus area is calculated by half multiplied the lengths of the two diagonals, but in but in this research area is calculated in terms of one diagonal length and an angle only
Parallelogram area is equal to multiplication of the base length by the perpendicular , but the new method uses diagonal length and angles of the diagonal only
Area of the rectangle equal multiplication of length and width, Using the new method diagonal length and angles of the diagonal are used only
Idea of the research
"The new methods"
The area of :
(1) Triangle (2) rhombus (diamond)
(3) Parallelogram (4) rectangle
The area of triangle
A
b c Figure(6.1)
proof
the area of the triangle
= × (a b)× (a c)× sin ȃ
000 =
000 ac = (ab) × sin (b) / sin (c)
000 sin c = sin(180- ( a + b ))
= sin ( a + b )
from (1), (2), (3)
000 The area at triangle =
Example (1)
c
2 cm 45
1cm
45
a 1 cm b
Solution (1)
The Area of triangle abc = a b × b c × sin b
= × 1 × 1 × 1 = cm2
Solution (2) by new method
The Area of triangle abc =
=
= = cm2
Example (1)
a
30 5 cm
70
b c
The Area oF abc =
= = 5.96 cm2
The area of parallelogram
d c
3
2
a b
Figure (6.2)
The proof
The area of parallelogram a b c d
= 2 × area abd
= 2 ×
000 ab // bc
000 m ( 3 ) = m ( 1 )
from (1) , (2)
000 area a b c d =
\
example (1)
a b
5 cm
40 30
d c
Area of a b c d =
= 8.55 cm2
The area of rhombus (diamond)
b
a c
2 1
d Figure (6.3)
Proof
000 Area of a b c d =
000 m(1) = m(2) = m(
from (1), (2)
000 Area of a b c d =
new method
example
b
a
4 cm c
30 30
d
Area of a b c d =
= = = cm2
(2.4)The area of rectangle
a b
1 2
d c Figure (6.1)
Proof
= = (bd)2 × sin(2) cos(2)
Because
sin(90) = 1
Sin (1) = cos(2) because 1+2 = 90
References
Book calculate areas and volumes of geometric shapes
Carson, P. B. (2012). Effects of levels of formal educational training in mathematics on teacher self-efficacy (Doctoral dissertation, Piedmont College).
Gardner, R. J. (1995). Geometric tomography (Vol. 1). Cambridge: Cambridge University Press
http://www.mathcentre.ac.uk/resources/uploaded/mc-ty-triangleformulae-2009-1.pdf
httpfiles.books.elebda3.netdownload-pdf-ebooks.org-ku-9199.pdfhttp://www.mathcentre.ac.uk/resources/uploaded/mc-ty-triangleformulae-2009-1.pdf
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