Question

1. determine the exact perimeter of this shape made from parts of circles.

show or explain your thinking

Explanation:

Step1: Analyze the shape's arcs

The shape has a large semicircle and three small semicircles. Let's find the radii (or diameters) of the circles. The large semicircle: from the grid, the diameter seems to be \( 3\times2 = 6 \) units (since each small semicircle's diameter is 2 units, and there are 3 small ones, so total length for large diameter is \( 3\times2 = 6 \)). The small semicircles: each has a diameter of 2 units.

Step2: Calculate the length of the large semicircle

The formula for the circumference of a full circle is \( C = \pi d \), so a semicircle is \( \frac{1}{2}\pi d \). For the large semicircle, \( d = 6 \), so length is \( \frac{1}{2}\pi\times6 = 3\pi \).

Step3: Calculate the length of the three small semicircles

Each small semicircle has \( d = 2 \), so length of one small semicircle is \( \frac{1}{2}\pi\times2=\pi \). Three of them: \( 3\times\pi = 3\pi \).

Step4: Total perimeter

Add the lengths of the large semicircle and the three small semicircles: \( 3\pi + 3\pi = 6\pi \). Wait, alternatively, notice that the sum of the diameters of the small semicircles equals the diameter of the large semicircle. So the total length of the curved parts: the large semicircle (d=6) and three small semicircles (each d=2, total d=6). So the length of the large semicircle is \( \frac{1}{2}\pi\times6 = 3\pi \), and the three small semicircles: each is \( \frac{1}{2}\pi\times2=\pi \), three of them is \( 3\pi \). So total perimeter is \( 3\pi + 3\pi = 6\pi \). Wait, actually, another way: the perimeter is the sum of the outer curved part (large semicircle) and the inner three curved parts (small semicircles). Wait, looking at the grid, the large semicircle's diameter is 6 (from x=1 to x=7, for example, 6 units), and each small semicircle has diameter 2 (from x=1 - 3, 3 - 5, 5 - 7, each 2 units). So the length of the large semicircle: \( \frac{1}{2}\times\pi\times6 = 3\pi \). The three small semicircles: each is \( \frac{1}{2}\times\pi\times2=\pi \), so three of them: \( 3\pi \). So total perimeter is \( 3\pi + 3\pi = 6\pi \). Wait, but let's check the diameters again. The large semicircle: from the leftmost point to the rightmost point, the horizontal distance is 6 units (since each small semicircle spans 2 units, 3 of them: 3*2=6). So diameter of large circle is 6, radius 3. The small semicircles: each has diameter 2, radius 1. So the length of the large semicircle: \( \pi r = \pi\times3 = 3\pi \) (since semicircle circumference is \( \pi r \) when radius is r, or \( \frac{\pi d}{2} \)). The small semicircles: each has radius 1, so semicircle length is \( \pi\times1=\pi \), three of them: \( 3\pi \). So total perimeter: \( 3\pi + 3\pi = 6\pi \).

Answer:

The exact perimeter is \( \boldsymbol{6\pi} \) units.