Area and Perimeter of Triangles: Perimeter is the total length of the three sides of any triangle. The area of a triangle is the region or surface bounded by the shape of a Triangle. We find the Perimeter when putting up Christmas lights around the house or fencing the backyard garden. Similarly, we find the area of the room floor to find the size of the carpet to be bought.

In this article, you will learn what the perimeter is and how to calculate the perimeter of various types of triangles when all side lengths are known. Furthermore, the solved cases will assist you in gaining other perspectives on the subject. Let us have a look at the article to understand the concept in a better way.

What is Perimeter?

Definition: The word perimeter is extracted from the Greek word ‘peri’ meaning around, and ‘metron’ means to measure. Perimeter is the sum of the length of the boundary of any shape.

Perimeter of a Triangle

A triangle is a polygon having three sides and three corners. The perimeter of a triangle can be obtained by simply adding the length of all three sides. The perimeter of any polygon is the sum of the lengths of the sides.
The formula for the perimeter of a closed shape figure is usually equal to the length of the outer line of the figure.

If a triangle has \(3\) sides \(a, b\) and \(c\), then the Perimeter of Triangle \((P) = (a + b + c)\,\rm{units}\).

Perimeter of Triangle Formulas

The formulas for perimeter of different types of triangles are explained below:

Perimeter of an Equilateral Triangle

Perimeter of an Equilateral triangle is given by \(= a + a + a = 3a\,\rm{units}\)

Perimeter of an Isosceles Triangle

If \(a\) and \(a\) are the two equal sides and \(b\) is the base of an Isosceles triangle, then its perimeter is given by \(= a + a + b = (2a + b)\,\rm{units}\)

Perimeter of Scalene Triangle

Perimeter of a scalene triangle is given by \(= (a + b + c)\,\rm{units}\)

What is Area?

The area is the region or surface enclosed by the shape of an object. The space covered by the figure or any geometric shapes is called the area of that shape. The area of the shapes depends upon their dimensions and properties. The areas are different for different shapes. The area of the circle is different from the area of the square.

Units of Area: In CGS system the unit of Area is \(\rm{cm}^2\) and in SI system, the unit of Area is \(\rm{m}^2\).

Area of Triangle

The area of a triangle can be defined as the total space or region occupied by the three sides of any triangle. To find the area of a triangle, we should know the base \((b)\) and height \((h)\) of it. This is applicable for Equilateral Triangle, Isosceles Triangle, and Scalene Triangle. The base and height are perpendicular to each other in a triangle.

The area is measured in square units \((\rm{cm}^2, \rm{m}^2)\).

Area of Triangle Formulas

Area of Triangle formula is given by \( = \frac{1}{2} \times {\text{base}} \times {\text{height}}\)
\(A = \frac{1}{2} \times b \times h\) square units

Proof of the Area of Triangle Formula

The Area of a rectangle is given by \(A = \rm{length} × \rm{breadth} = l × b\).

The diagonals of a rectangle divide its Area into two equal halves. So, here, the diagonal \(AC\) has divided the rectangle into two equal halves \(ABC\) and \(ADC\). In triangle \(ABC\), the length \(BC\) is now considered as the base, and the breadth \(AB\) is now considered the height of the Triangle. So, the area of the triangle \(ABC = \frac{1}{2} \times {\text{area}}\, {\text{of}}\, {\text{the}}\, {\text{rectangle}}\)
\(ABCD = \frac{1}{2} \times BC \times AB = {\text{Area}}\,{\text{of}}\,{\text{the}}\,{\text{Triangle}}\)
\(ABC = \frac{1}{2} \times BC \times AB\)
\(A = \frac{1}{2} \times {\text{base}} \times {\text{height}}\)
\(A = \frac{1}{2} \times b \times h\)
Hence, it is proved.

Area of an Equilateral Triangle

One should know the length of its side to find the Area of an Equilateral Triangle.

Area of an Equilateral Triangle is given by,
\(A = \frac{{\sqrt 3 }}{4} \times {({\text{side}})^2}\,{\text{square}}\,{\text{units}}\)

Area of Scalene Triangle (Heron’s Formula)

The Area of a Triangle with three sides of different measures can be calculated using the Heron’s formula.
Heron’s formula includes two steps,
1. Calculate “\(s\)” (half of the Perimeter of the Triangle):
\(s = \frac{ {a + b + c}}{2}\)
2. Then calculate the Area:
\(A = \sqrt {s\left({s – a} \right)\left({s – b} \right)\left({s – c} \right)} \)

Area of an Equilateral Triangle: A Special Case of Heron’s Formula

So, Heron’s formula or Hero’s formula can be used to derive a special formula applicable to calculate the Area of an Equilateral Triangle only.

In an equilateral triangle, the lengths of all the sides are the same. So, \(a = b = c\).
So, \(s = \frac{ {a + b + c}}{2} = \frac{ {a + a + a}}{2} = \frac{ {3a}}{2}\)
So, the area \( = \sqrt {s(s – a)(s – b)(s – c)} = \sqrt {\frac{{3a}}{2} \times \left( {\frac{{3a}}{2} – a} \right) \times \left( {\frac{{3a}}{2} – a} \right) \times \left( {\frac{{3a}}{2} – a} \right)} \)
\( = \sqrt {\frac{{3a}}{2} \times \frac{a}{2} \times \frac{a}{2} \times \frac{a}{2}} = \sqrt {\frac{{3{a^4}}}{{16}}} = \frac{{\sqrt 3 }}{4}{a^2}\), where the length of the side of the triangle is \(a\).
Hence, the Area of an Equilateral triangle is \( = \frac{{\sqrt 3 }}{4} \times {a^2} = \frac{{\sqrt 3 }}{4} \times {({\text{side}})^2}\)

Area of an Isosceles Triangle

Let the two same sides of an isosceles triangle \(ABC\) be given by \(AB\) and \(AC\). So, \(AB = AC = a\), and the length of the base be \(BC = b\).
Draw \(AD ⊥ BC\). So, \(D\) bisects \(AB\).
Hence, \(BD = \frac{b}{2}\)
Applying Pythagoras theorem on \(\Delta ABD\), it can be written that \(A{D^2} + B{D^2} = A{B^2}\)
\( \Rightarrow A{D^2} + {\left({\frac{b}{2}} \right)^2} = {(a)^2}\)
\( \Rightarrow A {D^2} = {a^2} – \frac{ { {b^2}}}{4}\)
\( \Rightarrow AD = \sqrt {{a^2} – \frac{{{b^2}}}{4}} \)
Hence, the Area of the Isosceles Triangle \( = \frac{1}{2} \times {\text{base}} \times {\text{height}} = \frac{1}{2} \times BC \times AD\)
\( = \frac{1}{2} \times b \times \sqrt { {a^2} – \frac{ { {b^2}}}{4}} \)

Formulas for the Area of a Triangle Involving Trigonometry

In \(\Delta ABC\), \(a, b\) and \(c\) are the sides of the Triangle. (Conventionally, \(a\) is considered as the side opposite to \(\angle A\). Similarly, \(b\) and \(c\) are considered as the sides opposite to the angles \(\angle B\) and \(\angle C\), respectively.
In this case, the formula for the Area of Triangle is given by,
\( {\text{Area}} = \frac{1}{2}ab\,\sin \,\sin C\)
\( {\text{Area}} = \frac{1}{2}bc\,\sin \,\sin A\)
\( {\text{Area}} = \frac{1}{2}ca\,\sin \,\sin B\)
Hence, the area \( = \frac{1}{2} \times {\text{product of any two sides}} \times {\text{side of the angle including those two sides}}{\text{.}}\)

Area of a Triangle with Coordinates of Vertices

The area of a triangle can be obtained if the coordinates of the three vertices of a triangle on a Cartesian plane are given.

In the triangle, \(ABC\), shown above, \(A(x_1, y_1 )\), \(B(x_2, y_2 )\) and \(C(x_3, y_3 )\) are the coordinates of the vertices.
Use the following formula,
\({\text{Area}} = \frac{1}{2}\left[ {{x_1}\left( {{y_2} – {y_3}} \right) + {x_2}\left( {{y_3} – {y_1}} \right) + {x_3}\left( {{y_1} – {y_2}} \right)} \right]\)

Solved Examples – Area and Perimeter of Triangles

Q.1. Find the Perimeter of the below Triangle.

Ans: The three sides of the Triangle are \(5\,\rm{cm}\), \(5\,\rm{cm}\) and \(4\,\rm{cm}\).
We know that Perimeter \(=\) the sum of all sides.
Now, the Perimeter of the triangle \(= 5\,\rm{cm} + 5\,\rm{cm} + 4\,\rm{cm} = 14\,\rm{cm}\)
Hence, the Perimeter of the given Triangle \(ABC\) is \(14\,\rm{cm}\).

Q.2. Find the Area of a right-angled triangle whose lengths of the sides other than the hypotenuse are \(12\,\rm{cm}\) and \(5\,\rm{cm}\)
Ans: If the lengths of the sides other than the hypotenuse are \(5\,\rm{cm}\) and \(12\,\rm{cm}\), then one of the lengths must be the height, and the other length will be the height.
So, the Area of the Triangle \( = \frac{1}{2} \times {\text{base}} \times {\text{height}} = \frac{1}{2} \times 12 \times 5 = 30\,{\text{c}}{{\text{m}}^2}.\)

Q.3. Find the Area of a triangle whose lengths of the sides are \(5\,\rm{cm}\), \(6\,\rm{cm}\) and \(7\,\rm{cm}\)
Ans: The lengths of the three sides of a triangle are given. So, we can use Heron’s formula to calculate the Area of the Triangle.

Here, \(a = 5\,\rm{cm}\), \(b = 6\,\rm{cm}\) and \(c = 7\,\rm{cm}\)
According to this formula, the Area of Triangle is given by,
\( {\text{area}} = \sqrt {s(s – a)(s – b)(s – c)} \)
where, a,b and c are the lengths of sides of the Triangle and s is the semi-perimeter of the Triangle, given by \(s = \frac{{a + b + c}}{2}.\)
So, \(s = \frac{{a + b + c}}{2} = \frac{{5 + 6 + 7}}{2} = \frac{{18}}{2} = 9\,{\text{cm}}\)
Hence, the Area of the given Triangle \( = \sqrt {s(s – a)(s – b)(s – c)} = \sqrt {9(9 – 5)(9 – 6)(9 – 7)} \)
\( = \sqrt {9 \times 4 \times 3 \times 2} = \sqrt {216} = 6\sqrt 6 \,{\text{c}}{{\text{m}}^2}\)

Q.4. Find the Area of an Equilateral Triangle of side \(4\,\rm{cm}\).
Ans: Here, \(a = 4\,\rm{cm}\)

Hence, the required Area of an Equilateral Triangle with side \(4\,{\text{cm}} = \frac{{\sqrt 3 }}{4} \times {a^2} = \frac{{\sqrt 3 }}{4} \times {(4)^2} = \frac{{\sqrt 3 }}{4} \times 16 = 4\sqrt 3 \,{\text{c}}{{\text{m}}^2}.\)

Q.5. The equal sides of an Isosceles Triangle are \(5\,\rm{cm}\) each and the base is \(2\,\rm{cm}\). Find the Area of the Triangle.
Ans: Here, \(a = 5\,\rm{cm}\) \(b = 2\,\rm{cm}\).

Hence, the Area of the Isosceles Triangle \( = \frac{1}{2} \times b \times \sqrt {{a^2} – \frac{{{b^2}}}{4}} = \frac{1}{2} \times 2 \times \sqrt {{5^2} – \frac{{{2^2}}}{4}} \)
\( = 1 \times \sqrt {25 – \frac{4}{4}} = \sqrt {25 – 1} = \sqrt {24} = \sqrt {2 \times 2 \times 2 \times 3} = 2\sqrt 6 \,{\text{c}}{{\text{m}}^2}\)

Summary

This article helps to learn about how to calculate the perimeter and area of different kinds of triangles depending on what information is available on the triangle. Knowing this one can calculate the perimeter and area of any triangular shape land or any other triangular shaped item.

It also helps to calculate the perimeter and area of any regular or irregular polygonal shaped land or any other item by dividing the polygon into triangles by diagonals and then obtaining the area of each triangle and adding them.

Frequently Asked Questions (FAQ) – Area and Perimeter of Triangles

Q.1. What is the formula of the perimeter of the triangle?
Ans: A triangle is a two-dimensional closed figure having three sides and three corners. The perimeter of a triangle is calculated by simply adding the length of all three sides.
The perimeter of a triangle \(=\) Sum of the lengths of the three sides

Q.2. What is the area and perimeter of an equilateral triangle?
Ans: Area of an equilateral triangle is given by,
\(A = \frac{{\sqrt 3 }}{4} \times {({\text{side}})^2}\,{\text{square}}\,{\text{units}}\)
And, perimeter of an equilateral triangle is given by,
\(P = a + a + a = 3a\, {\text{units}}\)

Q.3. What is the area of a triangle?
Ans: The area of a triangle is defined as the total space or region occupied by the three sides of any triangle.
The general formula to find the area of a triangle is,
Area of triangle formula is given by \(A = \left({\frac{1}{2} \times {\text{base}} \times {\text{height}}} \right){\text{square}}\,{\text{units}}\)

Q.4. What is the perimeter of \(\Delta ABC\)?
Ans: Perimeter of the triangle \(\Delta ABC\) is the sum of the length of all three sides.
\(P = (AB + BC + CA)\, {\text{units}}\)

Q.5. How do you calculate the perimeter?
Ans: The perimeter of any figure can be calculated by simply adding the length of all the sides of a given figure.