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Area of triangle: The introduction

The area of triangle in any polygon is the total amount of square units inside that polygon. By area, we mean a 2-dimensional carpet or an area rug. Likewise, a triangle as we already know is a 3-sided polygon. There are so many types of triangles in mathematics as well as in geometry. Finding the area of triangle can be quite interesting and somewhat challenging for students. This is because each of these triangles has its own properties as well as its own formula for finding the area of triangle

There are few criteria we must know or understand in order to find the area of triangle. Some of which include the following:

  • We must first know what type or kind of triangle it is, as this gives you the insight or foresight of what the area of triangle might be
  • What angle was given which determines how and what formula must be applied and used to find the area of triangle
  • The length and sides given that must be used in finding the area of triangle

 

The use of trigonometry to determine the height h of a triangle

 

The area of the triangle can be expressed as: T=12 ab sin γ 12 bc sin α = 12 ac sin β, where α represents the interior angle at A; β represents the interior angle at B, γ represents the interior angle at C and c is line AB. Furthermore, since sin α = sin (π − α) = sin (β + γ), and similarly for the remaining two angles: T=12 ab sin (α + β) = 12 bc sin (β + γ) = 12 ca sin (γ+α). Also, T=b.b(sin α )(sin⁡(α+ β))2sin β and analogously if the known side is a or c. Finally, Knowing ASA: T= a.a2(cot β+cot γ) = a.a(sin β )(sin γ )2sin⁡(β+γ) and analogously if the known side is b or c

Using Heron's formula to determine the area of triangle: The shape of any triangle can be determined by its lengths of the sides. Therefore, the area of triangle can be obtained from the lengths of the sides. By Heron's formula:

T= √s(s-a)(s-b)(s-c); where s = a+b+c2 is the semiperimeter of the triangle. Three other similar ways of writing Heron's formula are as follows:

 

  • T= √ (a2+b2+c2)-2(a4+b4+c4)
  • T=14 √2 (a2 b2+a2 c2+ b2c2)-(a4+b4+c4)
  • T=14 √ (a+b-c) (a-b+c)(-a+b+c)(a+b+c)
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