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

${\sf{=}}{\sf{\dfrac{1}{2}}}{\sf{ {\times} base {\times} height }}$

Subsitute the required values,

${\sf{=}}{\sf{\dfrac{1}{2}}}{\sf{ {\times} 3 {\times} 6}}$

$\sf{ = 3 {\times} 3 }$

$\bold{ = 9 }$

Therefore,

Area covered by 4 right angled triangles

$\sf{ = 4 {\times} 9 }$

$\bold{ = 36}$

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

$\sf{ = Side {\times} Side }$

Subsitute the required values,

$\sf{ = 9 {\times} 9 }$

$\bold{ = 81 }$

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

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

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

$\sf{ x = \sqrt{45}}$

$\bold{ x = 6.7 }$

That means,

• The length of the small square = 6.7

We know that ,

Area of square

$\sf{ = Side {\times} Side }$

Subsitute the required values,

$\sf{ = 6.7 {\times} 6.7 }$

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

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 .