To solve this problem, we can use the fact that each unit in the grid is equal to 5 cm, as stated in the problem. Here's how to calculate the areas of shapes $A$ and $B$ in Picture 16:

1. Area A (triangular sections):

From the user's calculations in the image, Area A is made up of two triangular regions. Here's how the user approached this:

• First triangle: $\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 20 \, \text{cm} \times 10 \, \text{cm} = 100 \, \text{cm}^2$

• Second triangle: $\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 30 \, \text{cm} \times 5 \, \text{cm} = 75 \, \text{cm}^2$

• Total area for A: $100 \, \text{cm}^2 + 75 \, \text{cm}^2 = 175 \, \text{cm}^2$

2. Area B (irregular shape):

The area of shape B requires calculating the number of grid units it covers, and multiplying that by the area of a single grid unit, which is:

$\text{Area of 1 grid unit} = 5 \, \text{cm} \times 5 \, \text{cm} = 25 \, \text{cm}^2$

The shape B is not a regular polygon, but we can approximate its area by breaking it into triangles or trapezoids based on the grid. It seems the user has calculated that the total area for B is approximately 86.6 cm², as seen in the notes.

Final Results:

• Area A: $175 \, \text{cm}^2$
• Area B: $86.6 \, \text{cm}^2$

Would you like to confirm or break down the steps for Area B further?

Here are 5 related questions to explore:

1. How can you find the area of an irregular polygon more precisely?
2. How does multiplying by the scale of 5 cm affect the overall calculation?
3. What other geometric methods could be used to approximate the area of irregular shapes like B?
4. How would the total area change if the grid unit was larger or smaller?
5. Can symmetry be used to simplify the calculation of shape B?

Tip: When dealing with irregular shapes on a grid, breaking them down into simple geometric shapes (like triangles or rectangles) can greatly simplify your calculations!