Question

Why is the Area for my shape not correct and can you explain why?

Answer 1

`\bold{\huge{\underline{ Solution }}}`

Here, We have given

• 2 squares , In which 1 square is enclosed within the another square and it arranged in a form that it forms 4 right angled triangle
• The height and base of the given right angled triangles are 6 and 3 each.

We know that,

Area of triangle

`{\sf{=}}{\sf{(1)/(2)}}{\sf{ {*} base {*} height }}`

Subsitute the required values,

`{\sf{=}}{\sf{(1)/(2)}}{\sf{ {*} 3 {*} 6}}`

`\sf{ = 3 {*} 3 }`

`\bold{ = 9 }`

Therefore,

Area covered by 4 right angled triangles

`\sf{ = 4 {*} 9 }`

`\bold{ = 36}`

Now,

We have to find the area of the big square

• The length of the side of the big square
• 
  `\sf{ = 6 + 3 = 9 }`

We know that,

Area of square

`\sf{ = Side {*} Side }`

Subsitute the required values,

`\sf{ = 9 {*} 9 }`

`\bold{ = 81 }`

Therefore,

The total area of shaded region

= Area of big square - Area covered by 4 right angled triangle

`\sf{ = 81 - 36 }`

`\bold{ = 45 }`

Hence, The total area of shaded region is 45 .

Part 2 :-

Here,

We have to find the area of non shaded region

According to the question

• Hypotenuse = The length of square

Let the hypotenuse of the given right angled triangle be x

Therefore,

By using Pythagoras theorem,

• This theorem states that the sum of the squares of the base and perpendicular height is equal to the square of hypotenuse.

That is,

`\sf{ (Perpendicular)^(2) + (Base)^(2) = (Hypotenuse)^(2) }`

Subsitute the required values

`\sf{ (6)^(2) + (3)^(2) = (x)^(2) }`

`\sf{ 36 + 9 = (x)^(2) }`

`\sf{ x = √(45)}`

`\bold{ x = 6.7 }`

That means,

• The length of the small square = 6.7

We know that ,

Area of square

`\sf{ = Side {*} Side }`

Subsitute the required values,

`\sf{ = 6.7 {*} 6.7 }`

`\bold{ = 44.89 \:\: or \:\: 44.9 }`

Therefore ,

Area of non shaded region

= Area of big square - Area of small square

`\sf{ = 81 - 44.9 }`

`\bold{ = 36.1 }`

Hence, The total area of non shaded region is 36.1 or 36 (approx) .

Part 3 :-

Here, we have to

• find the total area of the figure

Therefore,

The total area of the figure

= Non shaded region + Shaded region

`\sf{= 36 + 45 }`

`\bold{= 81}`

Hence, The total area of the given figure is 81 .