Two squares are enclosed in a large square as shown. The area of the smallest square is 5cm^2, and the area of the middle square is 20cm^. Determine the area and perimeter of the shaded region.
Correct
To find the area and perimeter of the shaded region, we first need to determine the side lengths of the squares.
Let's denote the side length of the smallest square as "a" and the side length of the middle square as "b". We know that the area of a square is given by the formula A = side length^2.
Given that the area of the smallest square is 5 cm^2, we have: a^2 = 5.
Similarly, the area of the middle square is 20 cm^2, so we have: b^2 = 20.
To find the side lengths a and b, we can take the square root of both sides of the equations:
a = sqrt(5) cm,
b = sqrt(20) cm.
Now, let's determine the side length of the large square that encloses both smaller squares. Since the side length of the large square equals the sum of the side lengths of the two smaller squares, we have:
Side length of large square = a + b = sqrt(5) + sqrt(20) cm.
To find the area of the shaded region, we need to subtract the area of the middle square from the area of the large square. The area of the large square is given by the formula A = side length^2:
Area of large square = (sqrt(5) + sqrt(20))^2 cm^2.
To calculate the area of the shaded region, we subtract the area of the middle square from the area of the large square:
Area of shaded region = Area of large square - Area of middle square.
Finally, to find the perimeter of the shaded region, we need to add the lengths of all four sides of the shaded region.
Perimeter of shaded region = Length of top side + Length of bottom side + Length of left side + Length of right side of the shaded region.