Understanding Areas of Shaded Regions

The video tutorial explains how to find the area of a shaded region by calculating the area of four semicircles and subtracting the area bounded by a circle and a square. It uses geometric formulas and the Pythagorean theorem to determine necessary dimensions and solve the problem step-by-step, concluding with the shaded area being 16 square units.

What is the initial strategy to find the area of the shaded region?

• Use the Pythagorean theorem to find the area directly.

• Divide the square into smaller squares and sum their areas.

• Calculate the area of four semicircles and subtract the area of a circle and square.

• Find the area of the square and multiply by four.

How is the area of the four semicircles calculated?

• By subtracting the area of a square from a circle.

• By using the formula for the area of a square.

• By finding the area of two full circles.

• By finding the area of one semicircle and multiplying by four.

What is the diameter of the semicircle if the radius is 2 units?

• 1 unit

• 4 units

• 2 units

• 8 units

Which theorem is used to find the diagonal of the square?

• The Law of Cosines

• The Area theorem

• The Law of Sines

• The Pythagorean theorem

What is the relationship between the sides of a 45-45-90 triangle?

• 2:2:√2

• 1:1:√2

• 1:2:√3

• 1:1:1

How is the radius of the circle determined from the diagonal of the square?

• By subtracting the side length from the diagonal

• By adding the diagonal length to the side of the square

• By halving the diagonal length

• By doubling the diagonal length

What is the formula used to find the area of the circle?

• πr

• πr²

• 2πr

• r²

What is the final step to find the area of the shaded region?

• Subtract the area of the bounded region from the semicircles' area.

• Add the area of the circle and the square.

• Multiply the area of the semicircles by two.

• Divide the area of the circle by the area of the square.

What is the area of the shaded region in square units?

• 64 square units

• 32 square units

• 16 square units

• 8 square units

What is the result of subtracting the area of the bounded region from the semicircles' area?

• 32

• 16

• 8π

• 0