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To make a puzzle piece, a cardboard block of dimensions 9 cm × 5 cm is cut along the dotted lines as shown below. Each semicircle cut from the length of the cardboard have an area of 2π cm2, while each semicircle cut from the breadth of the cardboard have an area of π/2 cm2. If the four triangles cut from the corners are equilateral triangles, each having an area of √3/4 cm2, then find the perimeter of the puzzle piece so formed?

To make a puzzle piece, a cardboard block of dimensions 9 cm × 5 cm is cut along the dotted lines as shown below. Each semicircle cut from the length of the cardboard have an area of 2π cm2, while each semicircle cut from the breadth of the cardboard have an area of π/2 cm2. If the four triangles cut from the corners are equilateral triangles, each having an area of √3/4 cm2, then find the perimeter of the puzzle piece so formed? Correct Answer (12 + 6π) cm

Given:

Dimension of board = 9 cm × 5 cm

Area of bigger semi circle = 2π cm2

Area of smaller semi circle = π/2 cm2

Area of equilateral triangle = √3/4 cm2

Formula used:

Area of circle = π × (radius)2

Area of equilateral triangle = √3/4 (side)2

Calculation:

Given, area of bigger semi-circle cut from length of cardboard = 2π cm2

Radius of bigger circle = 2 × 2π = π × (radius)2

Radius of circle along the length = 2 cm

⇒ Diameter of bigger circle along the length = 4 cm

Area of smaller semi-circle cut from breadth of cardboard = π/2 cm2

Radius of smaller circle = 2 × π/2 = π × (radius)2

Radius of circle along the length = 1 cm

⇒ Diameter of circle along the length = 2 cm

Area of equilateral triangle cut from corners of cardboard = √3/4 cm2

∵ Area of equilateral triangle = (√3/4) × (side)2

⇒ Side of equilateral triangle = √1 = 1 cm

Now,

Perimeter of puzzle piece = Perimeter of rectangular cardboard + + (2 × circumference of bigger semi-circle) – (2 × diameter of bigger semi-circle) + (2 × circumference of smaller semi-circle) – (2 × diameter of smaller semi-circle)

⇒ Perimeter of puzzle piece = 2(9 + 5) – 4(1) + 2(2π) – 2(4) + 2(π) – 2(2)

⇒ Perimeter of puzzle piece = 28 – 4 + 4π – 8 + 2π – 4

⇒ Perimeter of puzzle piece = (12 + 6π) cm

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