Different ways to find the area of a Triangle

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.

1. Area of a triangle using base and height

2. Area of a right-angled triangle

3. Area of a triangle using Heron’s formula

4. Area of a triangle when all sides (angle) are equal

5. Area of a triangle using Trigonometric formula

6. Area of a triangle (if height is not given)

7. Area of an enclosed triangle

Now we will discuss these methods one by one.

1. 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.

2. 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

hypotenuse2 = base2 + height2

In short;

H2=B2 + P2 

Where; H is “hypotenuse”
B is “base” 

P represents “perpendicular”

3. 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.

4. 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.

1. Trigonometric formula: 

2. 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

2. 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

6. 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]

7. 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:

1. 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

2. 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

3. 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.