IT-8

Chapter: PERIMETER AND AREA
Class: 6 (Class 6)
Category: in_text

Question

Find the area of the following figures.

Question diagram(s):

Figure 1: The purple 'F' shape. We need to find its area. Let us count the full squares inside the figure. Let us count the half squares inside the figure. We will count each full square as 1 unit. We will count each half square as 1/2 unit.

Step 1 — Count full squares for Figure 1 Let us look at the purple figure. We can see 3 full squares in the top part. We can see 1 full square in the middle part. We can see 1 full square in the bottom part. So, the number of full squares is $3 + 1 + 1$. $\text{Number of full squares} = 5$

Step 2 — Count half squares for Figure 1 Let us look at the purple figure again. There are no half squares in this figure. All boundary lines are either horizontal, vertical, or pass through grid points such that they enclose full squares or parts that are not half squares. The diagonal line at the bottom encloses parts of squares, but these are not exactly half squares. Let us use a different method for this shape, by dividing it into rectangles and triangles.

Let's rethink the counting method for Figure 1, as it has a diagonal line. For shapes with diagonal lines, we can try to break them into simpler shapes like rectangles and right-angled triangles. Or, we can use the "count full squares, count half squares, count more than half squares as 1, count less than half squares as 0" method.

Let's try to decompose Figure 1:

1. A rectangle on the left side: 1 unit wide, 2 units high. Area = $1 \times 2 = 2$ square units. This rectangle covers the points (0,0), (0,1), (0,2), (1,0), (1,1), (1,2).
2. A rectangle on the top right: 1 unit wide, 1 unit high. Area = $1 \times 1 = 1$ square unit. This rectangle covers points (2,2), (2,3), (3,2), (3,3).
3. A rectangle in the middle: 1 unit wide, 1 unit high. Area = $1 \times 1 = 1$ square unit. This rectangle covers points (1,1), (1,2), (2,1), (2,2).

This decomposition is also tricky because of overlapping areas or missing parts.

Let's use the standard method for grid figures: Count full squares (F). Count squares that are more than half (M). Count squares that are exactly half (H). Area = F + M + H/2.

Let's re-examine Figure 1 (purple): Full squares (F):

• The column from (0,0) to (1,2) has 2 full squares.
• The column from (1,1) to (2,1) has 1 full square.
• The column from (2,2) to (3,3) has 1 full square. This is still not straightforward.

Let's use the "bounding box and subtract" method for Figure 1. Enclose the figure in a rectangle. The figure fits in a rectangle of width 3 units and height 4 units (from bottom-most dot to top-most dot). Let's use the coordinates from the original image, assuming the bottom-left dot of the whole grid is (0,0). The purple figure's vertices are at: (1,1), (1,3), (2,3), (2,2), (3,2), (3,4), (4,4), (4,1). The bounding box is from x=1 to x=4 and y=1 to y=4. Width = $4 - 1 = 3$ units. Height = $4 - 1 = 3$ units. Area of bounding box = $3 \times 3 = 9$ square units.

Now, let's subtract the empty spaces (white areas) from the bounding box. The empty spaces are:

1. A rectangle at the top-left: (1,3)-(2,4) is part of the bounding box, but the figure only goes up to (1,3) on the left. The actual figure goes up to (3,4) and (4,4). Let's use the simpler method of counting full squares and combining partial squares.

Let's draw Figure 1 on a grid and count carefully: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .