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Difficulty: MediumArea of Two-Dimensional Shapes

An L-shaped region is created by removing a smaller square from the corner of a larger square. The perimeter of the L-shaped region is 4848 inches, and the area of the smaller square that was removed is 1616 square inches. What is the area, in square inches, of the L-shaped region?

Answer: 128 square inches

Answer

128
The area of the L-shaped region is the difference between the area of the original larger square and the area of the removed corner square. The perimeter of an L-shaped region formed by removing a corner square is identical to the perimeter of the original square, which is 4848 inches. This means the side length of the larger square is 1212 inches, and its area is 144144 square units. Subtracting the area of the removed square (1616 square units) from the area of the larger square gives 128128 square units.

Step-by-Step Solution

1
Calculate the side length of the smaller square from its area.
The side length of the smaller square is 44 inches.
The area of a square is the square of its side length (A=s2A = s^2). Since the area is 1616, we have s=16=4s = \sqrt{16} = 4.
2
Find the side length of the larger square using the perimeter of the L-shaped region.
The side length of the larger square is 1212 inches.
When a corner square is removed from a larger square, the perimeter remains unchanged because the two cut-out edges going inward have the same lengths as the two outer edges that were removed. Thus, the perimeter of the L-shaped region is equal to 4S4S. With a perimeter of 4848, the side length SS is 48÷4=1248 \div 4 = 12.
3
Compute the area of the L-shaped region.
The area of the L-shaped region is 128128 square inches.
The area of the L-shaped region is the area of the larger square minus the area of the removed smaller square: 122−16=144−16=12812^2 - 16 = 144 - 16 = 128.

Key Concept

The area of a composite shape can be calculated by subtracting the area of a removed sub-region from the area of the outer boundary. The perimeter of a rectangle or square remains unchanged when a corner square is removed.
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Area of Two-Dimensional Shapes

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