How to find the perimeter of a 30-60-90 Triangle?

Geometry is a branch of mathematics that studies the sizes, shapes, positions angles and dimensions of things. Flat shapes like squares, circles, and triangles are a part of flat geometry and are called 2D shapes. These shapes have only 2 dimensions, the length and the width.

Geometry is one of the oldest branches of mathematics. It is concerned with properties of space that are related to distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a geometer.

What is a Triangle?

A triangle is a simple closed polygon with three edges and three vertices. It is one of the basic shapes in geometry.

The basic shape of a triangle

Properties:

1. It has three sides.
2. It has three angles.
3. It has three vertices.

Formulae of Triangles

Area of Triangle = 1/2 * base * height

Perimeter of Triangle = Sum of all three sides

Types of Triangles

Based on sides

1. Equilateral Triangle: When all three sides of the triangle are equal then the triangle is known as an equilateral triangle. Each of the angles of an equilateral triangle measures 60°.
2. Isosceles Triangle: When two of the sides of the triangle are equal then the triangle is known as the Isosceles Triangle. Angles opposite to equal sides are equal in the isosceles triangle.
3. Scalene Triangle: In this type of triangle no two sides are equal to each other and no two angles are equal to each other.

Based on the angle

Acute angled triangle: When all the angles of a triangle are acute, that is, they measure less than 90°, it is called an acute-angled triangle or acute triangle.

acute-angled triangle

Obtuse angled triangle: When one of the angles of a triangle is 90°, it is called a right-angled triangle or right triangle.

obtuse-angled triangle

Right-angled triangle: When one of the angles of a triangle is an obtuse angle, that is, it measures greater than 90°, it is called an obtuse-angled triangle or obtuse triangle.

right-angled triangle

30-60-90 Triangle

It is a special type of right-angled triangle in which other than 90° angle one angle is 30° and the other angle is 60°.

Or in other words, we can say that angles are in ratio 30:60:90

                                                                                   = 1 : 2 : 3

As we know this triangle is special so we can easily calculate its other sides through angles using basic trigonometry and Pythagoras Theorem. 

We know that,

Perimeter of triangle = sum of all three sides

So first of all we will calculate its sides.

let the base be x.

tanθ = P/B

tan 30° = P/x

P = x * tan 30°

P = x * 1/√3

P = x / √3

Now,

secθ = H/B

sec 30° = H/x

H = x * sec 30°

H = x * 2/√3

H = 2x / √3

So now we have calculated the sides 

first side=x

second side=2x/√3

third side = x / √3

Now we can generalize a formulae for calculating the perimeter,

perimeter of triangle = sum of three sides

                                 = first side + second side + third side

                                 = x + 2x/√3 + x/√3

                                 = (√3x + 2x + x) / √3

                                 = (3x + √3x) / √3

                                 = √3x + x

                                 = (√3 + 1)x 

Sample Questions

Question 1: Find the perimeter of a 30 60 90 triangle whose base is 5m.

Solution: 

Perimeter of 30-60-90 triangle = (√3+1)x

                                                 = (√3+1)*5 

                                                 = (5√3+5)m.

So, perimeter of 30-60-90 triangle with base 5 m is  (5√3+5)m.

Question 2: The perimeter of a 30 60 90 triangle is (12√3 + 12) m. Find its height.

Solution: 

Perimeter of 30-60-90 triangle = (√3+1)x

                      (12√3 + 12) m   = (√3+1)*x

                                             x = 12 (√3+1) /  (√3+1)

                                             x = 12 m

So, base is 12 m.

height of triangle is x / √3 = 12 / √3

                                          = 12√3 / 3

                                          = 4√3 m

So, height of the triangle is 4√3 m.

Question 3: Find the perimeter of the 30-60-90 triangle. Given the sum of two sides other than hypotenuse as 7 + 7 √3 cm.

Solution: 

sum of two sides other than hypotenuse = base + perpendicular 

                                               7 + 7 √3  =  x + x/√3    

                                      multiplying whole equation by √3,

                                             21 + 7 √3  = x + √3x

                                             x ( 1 + √3 ) =  7√3 ( 1 + √3 )

                                                            x  =  7√3  cm

Perimeter of triangle = x ( 1 + √3 ) 

                                  = 7√3  ( 1 + √3 ) 

                                  =  21 + 7 √3 cm

So, perimeter of triangle is (21 + 7√3 ) cm.

Question 4: Find the perimeter of triangle ABC. If AB = 4 cm , BC = 5 cm and CA = 6 cm.

Solution: 

perimeter of triangle = sum of sides 

                                 = AB + BC + CA 

                                 = 4 + 5 + 6 

                                = 15 cm

So, perimeter of triangle ABC is 15cm.

Question 5: Given perimeter of a triangle is 15 cm. If the sum of two adjacent sides is 8 cm then find the third side.

Solution: 

Perimeter of triangle = sum of sides

Let the third side be x

x + 8 = 15

x = 7 cm

So, the length of third side is 7 cm.

Question 6: If the hypotenuse of the right-isosceles triangle is 6√2 m. Find its perimeter.

Solution: 

Perimeter of triangle = sum of sides

let the lengths of isosceles triangle be l , l , √2l as they satisfy the conditions for right-isosceles triangle.

hypotenuse = √2l 

6√2  = √2l 

l = 6m

perimeter of triangle = l + l + √2l

                                 = 6 + 6 + 6√2

                                 = 12 + 6√2 m

 So, the perimeter of given right-isosceles triangle is 12 + 6√2 m