Question

The drawing shows a heart-shaped figure. The heart is composed of a square and two semicircles. The semicircles have the center in the middle of two of the sides.

a) Set up a formula to calculate the area of ​​the heart.
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Answer 1

The radius of each semicircle is 5 cm. The area of the heart-shaped decoration is calculated using the formula
`\(A_{\text{heart}} = s^2 + 2 \cdot (1)/(2)\pi r^2\)`.

Let's denote the side length of the square as
`\(s\)` and the radius of each semicircle as
`\(r\)`.

The area of the heart-shaped decoration is given by the formula:

`\[A_{\text{heart}} = A_{\text{square}} + 2 \cdot A_{\text{semicircle}}.\]`

1. From the first statement, the diagonal of the square is
`\(10√(2)\)`.
Using the Pythagorean theorem
`(\(d^2 = s^2 + s^2\))`, we find
`\(s = 10\)` cm.

2. The area condition is
`\(s^2 - 2 \cdot (1)/(2)\pi r^2 = 100 - 25\pi\)`.
Substituting
`\(s = 10\)`, we get
`\(100 - \pi r^2 = 100 - 25\pi\)`. Solving for
`\(r\)`,we find
`\(r = 5\)`.

Therefore, the radius of each semicircle is 5 centimeters, consistent with the calculations from the given information.

In summary, the formula to calculate the area of the heart-shaped decoration is
`\(A_{\text{heart}} = s^2 + 2 \cdot (1)/(2)\pi r^2\)`, where
`\(s\)` is the side length of the square and
`\(r\)` is the radius of each semicircle.

Answer 2

Area = πx² + x² or x²(π + 1)

Explanation:

The area of the heart = area of a circle + area of the square

Note:

1. The 2 semicircles makes 1 full circle

2. The diameter of the circle = the side length of the square

Thus:

Area of the heart = πr² + s²

Where,

r = x

s = x

Formula for area of the heart = π(x²) + x² = πx² + x²

Area = x²(π + 1)