Heron’s Formula ( Part – 5 )

1. An umbrella is made by stitching 10 triangular pieces of cloth of two different colours each piece measuring 10 cm , 25 cm and 25 cm . How much cloth of each colour is required for the umbrella ?

Sol. The sides of triangle are

a = 10 cm b = 25 cm c = 25 cm

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

S = ( 10 + 25 + 25 ) / 2

S = 60 / 2

S = 30 cm

Using Heron’s Formula , Area of triangle = √ s ( s – a ) ( s – b ) ( s – c )

Area of triangle = √ 30 ( 30 – 10 ) ( 30 – 25 ) ( 30 – 25 )

Area of triangle = √ 30 * 20 * 5 * 5

Area of triangle = √ 2 * 3 * 5 * 2 * 2 * 5 * 5 * 5

Area of triangle = 2 * 5 * 5 √ 2 * 3

Area of triangle = 50 √ 6

Area of triangle = 50 * 2.45

Area of triangle = 122.47 cm^2

Area of 10 triangular pieces of cloths = 10 * 122.47 = 1224.70 cm^2

The area of cloths of each other colour = 1224.70 / 2 = 612.37 cm^2

2. A kite in the shape of square with a diagonal 16 cm and isosceles triangle of base 4 cm and side 3 cm each is to be made of three different shades . How much paper of each shade has been used in it ?

Sol. In the triangle ,

Base = 16 cm height = 8 cm

Area of triangle = ½ * base * height

Area of triangle = ½ * 16 * 8

Area of triangle = 8 * 8

Area of triangle = 64 cm^2

Kite is in shape of square , so area in both sides is same

Area of kite of square shape = 2 * 64 = 128 cm^2

And area of isosceles triangle a = 3 cm b = 3 cm c = 4 cm

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

S = ( 3 + 3 + 4 ) / 2

S = 10 / 2

S = 5 cm

Using Heron’s Formula , Area of triangle = √ s ( s – a ) ( s – b ) ( s – c )

Area of triangle = √ 5 ( 5 – 3 ) ( 5 – 3 ) ( 5 – 4 )

Area of triangle = √ 5 * 2 * 2 * 1

Area of triangle = 2 √ 5

Area of triangle = 2 * 2.24

Area of triangle = 4.48 cm^2

Total area of the kite of shape square with isosceles triangle = 128 + 4.48

Total area of the kite of shape square with isosceles triangle = 132.48 cm^2

maths ncert solutions

3. An umbrella is made by stitching 12 triangular pieces of cloth of two different colours each piece measuring 10 cm , 25 cm and 25 cm . How much cloth of each colour is required for the umbrella ?

Sol. The sides of triangle are

a = 10 cm b = 25 cm c = 25 cm

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

S = ( 10 + 25 + 25 ) / 2

S = 60 / 2

S = 30 cm

Using Heron’s Formula , Area of triangle = √ s ( s – a ) ( s – b ) ( s – c )

Area of triangle = √ 30 ( 30 – 10 ) ( 30 – 25 ) ( 30 – 25 )

Area of triangle = √ 30 * 20 * 5 * 5

Area of triangle = √ 2 * 3 * 5 * 2 * 2 * 5 * 5 * 5

Area of triangle = 2 * 5 * 5 √ 2 * 3

Area of triangle = 50 √ 6

Area of triangle = 50 * 2.45

Area of triangle = 122.47 cm^2

Area of 10 triangular pieces of cloths = 12 * 122.47 = 1469.64 cm^2

The area of cloths of each other colour = 1469.64 / 2 = 734.82 cm^2

4. A kite in the shape of square with a diagonal 48 cm and isosceles triangle of base 12 cm and side 9 cm each is to be made of three different shades . How much paper of each shade has been used in it ?

Sol. In the triangle ,

Base = 48 cm height = 24 cm

Area of triangle = ½ * base * height

Area of triangle = ½ * 48 * 24

Area of triangle = 24 * 24

Area of triangle = 576 cm^2

Kite is in shape of square , so area in both sides is same

Area of kite of square shape = 2 * 576 = 1152 cm^2

And area of isosceles triangle a = 9 cm b = 9 cm c = 12 cm

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

S = ( 9 + 9 + 12 ) / 2

S = 30 / 2

S = 15 cm

Using Heron’s Formula , Area of triangle = √ s ( s – a ) ( s – b ) ( s – c )

Area of triangle = √ 15 ( 15 – 9 ) ( 15 – 9 ) ( 15 – 12 )

Area of triangle = √ 15 * 6 * 6 * 3

Area of triangle = √ 3 * 5 * 6 * 6 * 3

Area of triangle = 3 * 6 √ 5

Area of triangle = 18 * 2.24

Area of triangle = 40.32 cm^2

Total area of the kite of shape square with isosceles triangle = 1152 + 40.32

Total area of the kite of shape square with isosceles triangle = 1192.32 cm^2

Milan Tomic

Hi. I’m Designer of Blog Magic. I’m CEO/Founder of ThemeXpose. I’m Creative Art Director, Web Designer, UI/UX Designer, Interaction Designer, Industrial Designer, Web Developer, Business Enthusiast, StartUp Enthusiast, Speaker, Writer and Photographer. Inspired to make things looks better.