Area and Perimeter of the Triangle

Here we will discuss about the area and perimeter of the triangle.

● If a, b, c are the sides of the triangle, then the perimeter of triangle = (a + b + c) units.

● Area of the triangle = √(s(s - a) (s - h) (s - c))

The semi-perimeter of the triangle, s = (a + b + c)/2

● In a triangle if 'b' is the base and h is the height of the triangle then

Area of triangle = 1/2 × base × height

Similarly,

$area and perimeter of the triangle$

1/2 × AC × BD 1/2 × BC × AD

● Base of the triangle = (2 Area)/height

● Height of the triangle = (2 Area)/base

Area of right angled triangle

● If a represents the side of an equilateral triangle, then its area = (a²√3)/4

$perimeter of an equilateral triangle$

● Area of right angled triangle

A = 1/2 × BC × AB

= 1/2 × b × h

$area of right angled triangle$

Worked-out examples on area and perimeter of the triangle:

1. Find the area and height of an equilateral triangle of side 12 cm. (√3 = 1.73).

Solution:

Area of the triangle = $\frac{√3}{4}$ a² square units

= $\frac{√3}{4}$ × 12 × 12

= 36√3 cm²

= 36 × 1.732 cm²

= 62.28 cm²

Height of the triangle = $\frac{√3}{2}$ a units

= $\frac{√3}{2}$ × 12 cm

= 1.73 × 6 cm

= 10.38 cm

2. Find the area of right angled triangle whose hypotenuse is 15 cm and one of the sides is 12 cm.

Solution:

AB² = AC² - BC²

= 15² - 12²

= 225 - 144

= 81

Therefore, AB = 9

Therefore, area of the triangle = ¹/₂ × base × height

= ¹/₂ × 12 × 9

= 54 cm²

3. The base and height of the triangle are in the ratio 3 : 2. If the area of the triangle is 243 cm² find the base and height of the triangle.

Solution:

Let the common ratio be x

Then height of triangle = 2x

And the base of triangle = 3x

Area of triangle = 243 cm²

Area of triangle = 1/2 × b × h 243 = 1/2 × 3x × 2x

⇒ 3x² = 243

⇒ x² = 243/3

⇒ x = √81

⇒ x = √(9 × 9)

⇒ x = √9

Therefore, height of triangle = 2 × 9

= 18 cm

Base of triangle = 3x

= 3 × 9

= 27 cm

4. Find the area of a triangle whose sides are 41 cm, 28 cm, 15 cm. Also, find the length of the altitude corresponding to the largest side of the triangle.

Solution:

Semi-perimeter of the triangle = (a + b + c)/2

= (41 + 28 + 15)/2

= 84/2

= 42 cm

Therefore, area of the triangle = √(s(s - a) (s - b) (s - c))

= √(42 (42 - 41) (42 - 28) (42 - 15)) cm²

= √(42 × 1 × 27 × 14) cm²

= √(3 × 3 × 3 × 3 × 2 × 2 × 7 × 7) cm²

= 3 × 3 × 2 × 7 cm²

= 126 cm²

Now, area of triangle = 1/2 × b × h

Therefore, h = 2A/b

= (2 × 126)/41