Question

Please help. Find the area and the perimeter of the shaded regions below. Give your answer as a completely simplified exact value in terms of π (no approximations). The figures below are based on semicircles or quarter circles and problems b), c), and d) are involving portions of a square. I will give Brainiest! Thank you! <3

Answer 1

Answer:

Area = (18 + 4.5π) cm²

Perimeter = (6√2 + 12 + 3π) cm

Step-by-step explanation:

The shaded region given is made up of triangle ABC and a semicircle

AB = BC = 6 cm (note that the triangle is a portion of a square)

Diameter of semi-circle (d) = BC = 6cm

Radius (r) = ½*6 = 3 cm

==>Area of the shaded region in terms of π

Area of shaded region = area of triangle + area of semicircle

Area = ½*a*b + ½*πr²

Area = ½*6*6 + ½*π3²

Area = 18 + ½*π9

Area = 18 + 4.5π

Area of the shaded portion = (18 + 4.5π) cm²

==>Perimeter of shaded region in terms of π

Perimeter of shaded region = perimeter of triangle + perimeter of semicircle

= Sum of all sides of the triangle + ½πd

Sides of triangles are AB = 6 cm, BC = 6 cm

Use Pythagorean theorem to find side AC:

AC² = AB² + BC²

AC² = 6² + 6² = 36 + 36 = 72

AC = √72 cm

Perimeter of shaded triangle = √72 + 6 + 6 = √72 + 12 = (6√2 + 12) cm

Perimeter of semicircle = ½*πd = ½π6

= 3π

Perimeter of the whole shaded region in terms of π = (6√2 + 12 + 3π) cm