Question

1. here is a shape made of a semicircle and a square. what is the perimeter of this shape?

Explanation:

Step1: Determine the side length of the square

From the grid, the vertical (or horizontal) side of the square (and the diameter of the semicircle) is 6 units (since it spans 6 grid squares, each 1 unit). So the radius \( r \) of the semicircle is \( \frac{6}{2}=3 \) units.

Step2: Calculate the length of the semicircular arc

The formula for the circumference of a full circle is \( C = 2\pi r \), so the length of a semicircular arc is \( \frac{1}{2} \times 2\pi r=\pi r \). Substituting \( r = 3 \), we get \( \pi\times3 = 3\pi\approx 3\times3.14 = 9.42 \) units.

Step3: Calculate the length of the square's sides contributing to the perimeter

The shape's perimeter consists of the semicircular arc, two vertical sides of the square, and one horizontal side? Wait, no. Wait, looking at the diagram: the square has a side where the semicircle is attached (dashed line, not part of the perimeter). So the perimeter is: semicircular arc + top side of square + right side of square + bottom side of square + left side? Wait no, let's re-examine. The square: let's say the square has side length 6 (vertical) and horizontal length, let's see the horizontal side: from the dashed line to the right, how many units? Let's count the grid. The horizontal side (top and bottom) from the dashed line to the right is 6 units? Wait, no, the square: the vertical side (diameter of semicircle) is 6 units (since from top to bottom of the semicircle is 6 grid squares). Then the horizontal side (length of the square) is, let's see, from the dashed line to the right end: 6 units? Wait, no, looking at the grid, the horizontal side (the top and bottom of the square part) is 6 units? Wait, no, maybe the square has side length 6 (vertical) and horizontal length 6? Wait, no, the diagram: the semicircle is on the left, attached to the square. The square's vertical side (the one attached to the semicircle) is 6 units (diameter). Then the square's horizontal sides: top, bottom, and right side. Wait, the perimeter: semicircular arc (left) + top side (rightward) + right side (downward) + bottom side (leftward) +? Wait, no, the dashed line is the left side of the square, which is internal (not part of the perimeter). So the perimeter is:

- Semicircular arc (length \( \pi r \), \( r = 3 \), so \( 3\pi \approx 9.42 \))
- Top side of the square: length 6 (since the horizontal length from dashed line to right end is 6 units? Wait, no, let's count the grid. The horizontal side (top and bottom) of the square: from the dashed line (x=0) to the right end (x=6), so length 6. Then the vertical sides: right side (from y=0 to y=6, length 6) and bottom side? Wait, no, the shape is a square with a semicircle on the left. So the perimeter is:

Semicircle (left) + top of square (right) + right of square (down) + bottom of square (left) +? Wait, no, the bottom of the square: from right end to the start of the semicircle? No, the semicircle is on the left, so the bottom side is from the right end (bottom) to the bottom of the semicircle. Wait, maybe I made a mistake. Let's re-express:

The shape is composed of a semicircle (diameter 6, so radius 3) and a square with side length 6 (since the diameter of the semicircle is equal to the side length of the square). Wait, no, the square: looking at the grid, the horizontal length (from the dashed line to the right) is 6 units (since it's 6 grid squares, each 1 unit). The vertical length (from top to bottom) is 6 units (same as the diameter of the semicircle). So the square has side length 6 (vertical) and horizontal length 6? Wait, no, the square: if the vertical side (diameter) is 6, then the square's vertical side is 6, and horizontal side is 6. So the perimeter of the shape:

- Semicircular arc: length \( \frac{1}{2} \times \pi \times d \), where \( d = 6 \), so \( \frac{1}{2} \times \pi \times 6 = 3\pi \approx 9.42 \)
- Top side of square: length 6 (horizontal, from left dashed line to right end)
- Right side of square: length 6 (vertical, from top to bottom)
- Bottom side of square: length 6 (horizontal, from right end to left dashed line)
Wait, no, that can't be, because then the left side is the semicircle. Wait, no, the dashed line is the left side of the square, which is attached to the semicircle, so the perimeter is:

Semicircle (left) + top (right) + right (down) + bottom (left) +? Wait, no, the bottom side: from the bottom of the semicircle to the bottom right corner, then up the right side, then across the top, then left along the top to the top of the semicircle, then down the semicircle. Wait, maybe the correct breakdown is:

Perimeter = length of semicircular arc + 3 sides of the square (top, right, bottom). Wait, the square has 4 sides, but one side (left) is attached to the semicircle (dashed line, not part of perimeter). So the square contributes 3 sides: top, right, bottom. Each of these sides: let's check the length. The vertical side (right side of square) is 6 units (since the diameter of the semicircle is 6 units, so the square's vertical side is 6). The horizontal sides (top and bottom) are each 6 units? Wait, no, looking at the grid, the horizontal side (from the dashed line to the right end) is 6 units? Let's count the grid squares. The horizontal side (top) from the dashed line (x=0) to the right end (x=6) is 6 units. The vertical side (right) from y=0 (bottom) to y=6 (top) is 6 units. The bottom side from x=6 (right) to x=0 (dashed line) is 6 units. Then the semicircle is from y=0 (bottom) to y=6 (top), left side. So the perimeter is:

Semicircular arc (length \( 3\pi \approx 9.42 \)) + top side (6) + right side (6) + bottom side (6) = \( 3\pi + 18 \). Substituting \( \pi \approx 3.14 \), we get \( 9.42 + 18 = 27.42 \)? Wait, no, that can't be. Wait, maybe the horizontal side (top and bottom) is 6 units, and the vertical side (right) is 6 units, but the semicircle is on the left, so the perimeter is:

Semicircle (length \( 3\pi \)) + top (6) + right (6) + bottom (6) = \( 3\pi + 18 \approx 9.42 + 18 = 27.42 \). But wait, maybe the square's horizontal length is 6, and vertical length is 6, but the semicircle's diameter is 6, so radius 3. Wait, maybe I made a mistake in the square's side length. Let's re-express:

From the grid, the diameter of the semicircle (vertical) is 6 units (since it spans 6 grid squares, each 1 unit). So the radius \( r = 3 \). The length of the semicircular arc is \( \pi r = 3\pi \approx 9.42 \).

Now, the square: the horizontal side (length) is 6 units (since from the dashed line to the right end, it's 6 grid squares). The vertical side (height) is 6 units (same as the diameter). So the perimeter of the shape is:

- Semicircular arc: \( 3\pi \approx 9.42 \)
- Top side of square: 6 units
- Right side of square: 6 units
- Bottom side of square: 6 units

Wait, but that's three sides of the square. So total perimeter: \( 3\pi + 6 + 6 + 6 = 3\pi + 18 \). Substituting \( \pi \approx 3.14 \), we get \( 9.42 + 18 = 27.42 \). But wait, maybe the horizontal side is 6, and the vertical side is 6, but the square's horizontal length is 6, so the top and bottom sides are 6 each, and the right side is 6, and the semicircle is 3π. So total perimeter: 3π + 6 + 6 + 6 = 3π + 18 ≈ 27.42. Alternatively, maybe the square has side length 6, so three sides: 3*6=18, plus semicircle 3π≈9.42, total≈27.42.

Wait, maybe I miscounted the horizontal length. Let's look at the grid again. The square: from the dashed line (left) to the right end, how many grid squares? Let's count: the horizontal side (top) has 6 grid squares (from x=1 to x=7? Wait, maybe the grid is 1 unit per square, so the horizontal length is 6 units (since the number of squares from the dashed line to the right is 6). The vertical length (from top to bottom of the square) is 6 units (from y=1 to y=7, 6 squares). So the square's side length (horizontal and vertical) is 6 units.

Thus, the perimeter is:

Semicircular arc (length \( \frac{1}{2} \times \pi \times 6 = 3\pi \)) + top side (6) + right side (6) + bottom side (6) = 3π + 18.

Calculating numerically: 3*3.14 + 18 = 9.42 + 18 = 27.42.

Wait, but maybe the horizontal length is 6, and the vertical length is 6, but the semicircle is on the left, so the perimeter is semicircle + top + right + bottom. So that's correct.

Alternatively, maybe the square's horizontal length is 6, and vertical length is 6, so three sides: 3*6=18, plus semicircle 3π≈9.42, total≈27.42.

Answer:

The perimeter of the shape is approximately \( 27.42 \) units (or exactly \( 18 + 3\pi \) units). If we use \( \pi \approx 3.14 \), the perimeter is \( 18 + 9.42 = 27.42 \) units.