Question

the area of the shaded part of the figure below can be determined in different ways. use the drop - down menus to explain how to find the area by decomposing the figure into rectangles or trapezoids.
(there is a square with side length 5 in, and a smaller square with side length 2 in inside it, the area between them is shaded.)

Explanation:

Step1: Calculate area of large square

$A_{large} = 5 \times 5 = 25$ square inches

Step2: Calculate area of small square

$A_{small} = 2 \times 2 = 4$ square inches

Step3: Subtract small from large area

$A_{shaded} = 25 - 4 = 21$ square inches

Decomposition Method (Rectangles):

Step1: Split shaded area into 4 rectangles

• 2 outer rectangles: $5 \times \frac{5-2}{2} = 5 \times 1.5 = 7.5$ sq in each
• 2 inner rectangles: $2 \times \frac{5-2}{2} = 2 \times 1.5 = 3$ sq in each

Step2: Sum all 4 rectangle areas

$A_{shaded} = (2 \times 7.5) + (2 \times 3) = 15 + 6 = 21$ square inches

Answer:

The area of the shaded part is 21 square inches. It can be found by either subtracting the area of the inner small square from the area of the outer large square, or by decomposing the shaded region into 4 smaller rectangles, calculating each of their areas, and summing them together.