Question

Two identical squares with sides of length 10cm overlap to form a shaded region as shown. A corner of one square lies at the intersection of the diagonals of the other square. Find the area of the shaded region in square centimetres.

Answer 1

The area of the shaded region formed by the overlap of two identical squares, each with a side length of 10cm, is 100 cm².

Step-by-step explanation:

To calculate the area of the shaded region where two identical squares overlap, we need to understand that the corner of one square lying at the intersection of the diagonals of the other square creates an octagon within the square. Since each side of the square is 10 cm, the diagonal of the square can be determined by using the Pythagorean theorem: the diagonal (d) is the square root of (side ² + side ²), leading to:

d = \(√{10² + 10²}\) = 10\(√{2}\) cm

Each triangle that has been cut off to create the octagon is an isosceles right triangle with legs measuring 5 cm (half the side of the square). The area of each triangle is then:

\(\frac{1}{2} × 5 × 5\) = 12.5 cm²

There are four such triangles, so the total area of the triangles is 4 × 12.5 cm² = 50 cm². Since the area of the entire square is 100 cm², the area of the octagon is then:

100 cm² - 50 cm² = 50 cm²

Two overlapping octagons form the shaded area in question, so we sum the areas of the two octagons to get the area of the shaded region:

Area of shaded region = 50 cm² + 50 cm² = 100 cm²

Answer 2

After subtracting the area of the four curved triangles from the total area of one square, the area of the shaded region is approximately
`\(68.54 \, \text{cm}^2\)`.

The shaded area is the area of one square minus the area of the four curved triangles formed by the quarter-circles in each corner.

The area of one square with side length
`\(10 \, \text{cm}\) is \(10 \, \text{cm} * 10 \, \text{cm} = 100 \, \text{cm}^2\).`

Each curved triangle is a quarter-circle with a radius equal to the side length of the square
`(\(10 \, \text{cm}\))`.

The area of one curved triangle (one-quarter of a circle) is:

`\[ \text{Area of one curved triangle} = (1)/(4) * \pi * \text{radius}^2 \]`

`\[ \text{Area of one curved triangle} = (1)/(4) * \pi * 10^2 \, \text{cm}^2 \]`

`\[ \text{Area of one curved triangle} = (1)/(4) * \pi * 100 \, \text{cm}^2 = 25 \, \text{cm}^2 * \pi \]`

Since there are four such curved triangles, their combined area is
`\(4 * 25 \, \text{cm}^2 * \pi = 100 \, \text{cm}^2 * \pi\)`.

Now, subtract this total area of the curved triangles from the total area of one square:

`\[ \text{Shaded area} = \text{Area of one square} - \text{Area of curved triangles} \]`

`\[ \text{Shaded area} = 100 \, \text{cm}^2 - 100 \, \text{cm}^2 * \pi \]`

Let's compute the numerical value:

`\[ \text{Shaded area} = 100 \, \text{cm}^2 - 100 \, \text{cm}^2 * \pi \approx 68.54 \, \text{cm}^2 \]`