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Triangle: Any two-dimensional figure that has three sides is called a triangle. The three sides of a triangle are length, breadth, and height, denoted by symbols l, b, and h, respectively. The intersecting points where the sides of a triangle meet are called the vertices of the triangle. The angles produced at the vertices are called the angles of the triangle.
Parts of a Triangle:
There are three sides, three vertices, and three angles in a triangle.
In the triangle given above:
The three angles are ∠1, ∠2, and ∠3.
The three sides are side 1, side 2, and side 3.
The three vertices are Vertex 1 (P), Vertex 1 (Q), and Vertex 1 (R).
Essential Conditions of a triangle:
Condition on the sides
The sum of the lengths of any two sides of a triangle must be greater than or equal to the length of the third side.
Conditions on the angles
(a) Each of the angles is positive.
(b) The sum of three angles is always 180°.
The area of a triangle: It is the amount of space inside the triangle. The units are represented as square units. These units can be:
- Square millimetres (mm2)
- Square inches(in2)
- Square kilometres (km2)
Square yards
Different ways to calculate the area of a triangle:
There are seven different ways to find the area of a triangle that can be classified based on its sides and angles.
Area of a triangle using base and height
Area of a right-angled triangle
Area of a triangle using Heron’s formula
Area of a triangle when all sides (angle) are equal
Area of a triangle using Trigonometric formula
Area of a triangle (if height is not given)
Area of an enclosed triangle
Now we will discuss these methods one by one.
Area of a triangle using base and height: The general formula to find the area of a triangle using base and height is given by:
Area A=12(b h) Square units
Where;
A = area, b is base, and “h” represents the height of the triangle.
The triangles might be different but this formula can be applied to all the triangles.
Where a = length of the sides.
Area of a right-angled triangle: A triangle that has at least one angle equal to 900 is called a right-angled triangle. To find the area of a right-angled triangle, we use the Pythagoras theorem, which defines the relationship between the three sides of a right triangle
In short;
H2=B2 + P2
Where; H is “hypotenuse”
B is “base”
P represents “perpendicular”
The area of a triangle using Heron’s formula: It can be calculated if three side lengths are given.
Area= √s(s–a)(s–b)(s–c)
S=a+b+c / 2
Where;
S represents semi-perimeter
And a, b, c are the three sides of a triangle.
The area of a triangle: The area of a triangle if all three sides (angles) are equal is given by:
Area=a234
Where ‘a’ is the side of the triangle.
Trigonometric formula:
Using two sides and the angle between them: This formula is used to find the area of a triangle when the length of two sides and the angle between them is given.
Area = 1/2 b a sin()
Where; b = base of a triangle
a = length of the two equal sides
α = angle between the unequal sides
Using two angles and the length between them: This formula is also used to find the area of a triangle when two angles and the length between them is given
Area = [a2 sin(/2) sin()]
Where;
a = length of the two equal sides
α, β = angles in a triangle
Area of a triangle (if height is not known): When the height of an isosceles triangle is not known, then the following formula is used to find the height:
Height= √ (a2 – b2/4)
Where; b = base of the triangle
a = Side length of the two equal sides.
So, the area of an isosceles triangle will be
⇒Area = 1/2 [√ (a2 – b2 /4) b]
Area of an enclosed triangle: Any triangle when enclosed in a rectangle, then the area of the main triangle is given by:
Area = f*g/2
where the sides of the enclosing rectangle are f and g.
Solved examples:
Find the area of a triangle with a base of 5 cm and a height of 6 cm.
Solution: Let us calculate the area using the area of triangle formula:
Area of triangle = (1/2) b h
A = 1/2 5 6
A = 1/2 30
∴, The area of the triangle (A) = 15 cm2
Find the area of an equilateral triangle with a side of 7 cm.
Solution: We can calculate the area of an equilateral triangle using the equilateral triangle formula.
Area =34 a2 …………………………(1)
where ‘a’ is the length of one equal side.
Given : length of one side,a =7cm
Put the value of a=7cm in equation (1)
we get, Area of an equilateral triangle = (√3)/4 72
Area = 21.21 cm2
The sides of a right triangle are 4 cm and 3 cm, find its hypotenuse.
Ans. Let base, a = 4, height,b = 3cm and hypotenuse, c=?
hypotenuse2 = base2 + height2
Putting values in the formula:
42 + 32 = c2
16 + 9 = c2
25 = c2
c = 5 cm
Conclusion
In this article, we have learned about different ways to find the area of a triangle formed in mathematics and its properties. However, the triangle is a very wide topic. It governs the entire geometry. Knowing all these, the students will get a reasonably good idea about a triangle and how to use formulas to obtain the area of a triangle.
