56-60. Calculate the length of the line in the figures.

Solution:

We need to calculate the perimeter of each figure. We will assume each square in the grid has a side length of 1 cm, unless otherwise specified.

Figure 56:

The figure is composed of straight lines and a semicircle.

1. The top horizontal line is 4 units long (4 cm).
2. The two vertical lines on the sides are each 3 units long (3 cm each).
3. The bottom horizontal line is 4 units long (4 cm).
4. The curved part is a semicircle. The diameter of the semicircle is the same as the length of the top horizontal line, which is 4 cm. The radius is 2 cm. The length of the semicircle (arc length) is \( \frac{1}{2} \times 2 \pi r = \pi r = 2\pi \) cm.

Perimeter = 4 + 3 + 3 + 4 + 2\(\pi\) = 14 + 2\(\pi\) cm.

If we approximate \(\pi \approx 3.14\), then Perimeter \(\approx 14 + 2 \times 3.14 = 14 + 6.28 = 20.28\) cm.

Figure 57:

This figure is made of straight lines and a quarter circle.

1. The bottom horizontal line is 3 units long (3 cm).
2. The right vertical line is 2 units long (2 cm).
3. The top horizontal line is 1 unit long (1 cm).
4. The left vertical line is 1 unit long (1 cm).
5. The curved part is a quarter of a circle. The radius of this quarter circle is 1 unit (1 cm). The arc length is \( \frac{1}{4} \times 2 \pi r = \frac{1}{2} \pi r = \frac{1}{2} \pi \times 1 = \frac{\pi}{2} \) cm.

Perimeter = 3 + 2 + 1 + 1 + \(\frac{\pi}{2}\) = 7 + \(\frac{\pi}{2}\) cm.

If we approximate \(\pi \approx 3.14\), then Perimeter \(\approx 7 + \frac{3.14}{2} = 7 + 1.57 = 8.57\) cm.

Figure 58:

This figure is composed of straight lines and two semicircles.

1. The bottom horizontal line is 4 units long (4 cm).
2. The two vertical lines on the sides are each 1 unit long (1 cm each).
3. The top horizontal line connecting the two semicircles is 2 units long (2 cm).
4. There are two semicircles. Each semicircle has a diameter of 2 units (2 cm), so a radius of 1 unit (1 cm). The length of each semicircle is \( \frac{1}{2} \times 2 \pi r = \pi r = \pi \times 1 = \pi \) cm.

Perimeter = 4 + 1 + 1 + 2 + \(\pi\) + \(\pi\) = 8 + 2\(\pi\) cm.

If we approximate \(\pi \approx 3.14\), then Perimeter \(\approx 8 + 2 \times 3.14 = 8 + 6.28 = 14.28\) cm.

Figure 59:

This figure resembles a cloud. It is made of straight lines and two semicircles at the top and bottom, connected by two straight lines.

Let's assume the grid lines represent 1 cm.

1. The bottom straight line is 4 units long (4 cm).
2. The two vertical lines on the sides are each 2 units long (2 cm each).
3. The top horizontal line is also 4 units long (4 cm).
4. There are two semicircles. Each has a diameter of 4 units (4 cm) and a radius of 2 units (2 cm). The length of each semicircle is \( \frac{1}{2} \times 2 \pi r = \pi r = \pi \times 2 = 2\pi \) cm.

Perimeter = 4 + 2 + 2 + 4 + 2\(\pi\) + 2\(\pi\) = 12 + 4\(\pi\) cm.

If we approximate \(\pi \approx 3.14\), then Perimeter \(\approx 12 + 4 \times 3.14 = 12 + 12.56 = 24.56\) cm.

Figure 60:

This figure is a symmetrical shape with four lobes. It appears to be made of straight lines and four semicircles.

Let's assume the grid lines represent 1 cm.

1. The central part seems to be a square of 4x4 units.
2. At the middle of each side, a semicircle is attached. Each semicircle has a radius of 2 units (2 cm). The length of each semicircle is \( \frac{1}{2} \times 2 \pi r = \pi r = \pi \times 2 = 2\pi \) cm.
3. The overall shape is made of four semicircles and four straight lines connecting them. The straight segments appear to be 4 units long.

Let's count the grid units for the perimeter.

The figure is composed of four semicircles, each with a radius of 2 units. The arc length of each semicircle is \( \frac{1}{2} \times 2 \pi r = \pi r = 2\pi \) cm.

There are four straight segments connecting the semicircles. Each of these segments appears to be 4 units long.

Perimeter = 4 \(\times\) (arc length of semicircle) + 4 \(\times\) (length of straight segment)

Perimeter = 4 \(\times\) (2\(\pi\)) + 4 \(\times\) 4 = 8\(\pi\) + 16 cm.

If we approximate \(\pi \approx 3.14\), then Perimeter \(\approx 8 \times 3.14 + 16 = 25.12 + 16 = 41.12\) cm.

Answer: 56: 14 + 2\(\pi\) cm; 57: 7 + \(\frac{\pi}{2}\) cm; 58: 8 + 2\(\pi\) cm; 59: 12 + 4\(\pi\) cm; 60: 16 + 8\(\pi\) cm.