To find the area of this shape, which is a combination of a rectangle and a half-circle, we'll use a few steps and employ the Pythagorean theorem to verify dimensions if necessary. Let's proceed as follows:

Step 1: Find the Radius of the Half-Circle

The given dimensions indicate that the length from the left side to the right edge of the rectangle is $7 \, \text{cm}$. Since the rectangle has a width of $4 \, \text{cm}$, this leaves a radius of the half-circle of: $r = 7 \, \text{cm} - 4 \, \text{cm} = 3 \, \text{cm}$

Step 2: Calculate the Area of the Rectangle

The area $A_{\text{rectangle}}$ of the rectangle is given by: $A_{\text{rectangle}} = \text{width} \times \text{height} = 4 \, \text{cm} \times 6 \, \text{cm} = 24 \, \text{cm}^2$

Step 3: Calculate the Area of the Half-Circle

The area of a full circle with radius $r = 3 \, \text{cm}$ is: $A_{\text{circle}} = \pi r^2 = 3.14 \times (3)^2 = 3.14 \times 9 = 28.26 \, \text{cm}^2$ Since we only have a half-circle, we divide this by 2: $A_{\text{half-circle}} = \frac{28.26}{2} = 14.13 \, \text{cm}^2$

Step 4: Add the Areas

Now, we add the area of the rectangle and the half-circle to find the total area of the figure: $A_{\text{total}} = A_{\text{rectangle}} + A_{\text{half-circle}} = 24 \, \text{cm}^2 + 14.13 \, \text{cm}^2 = 38.13 \, \text{cm}^2$

Answer

The total area enclosed by the figure is: $\boxed{38.13 \, \text{cm}^2}$

Would you like further details or have any other questions?

Relative Questions:

1. How would the area change if the radius of the half-circle were different?
2. How is the Pythagorean theorem used in verifying the dimensions of composite shapes?
3. What if the shape was a full circle instead of a half-circle?
4. How do different values of $\pi$ affect area calculations?
5. Can this method be generalized for other composite shapes?

Tip:

When dealing with composite shapes, break down each part, calculate their areas individually, and then sum them to find the total area.