Question

what is the area of the composite figure?
(6π + 4) cm²
(6π + 16) cm²
(12π + 4) cm²
(12π + 16) cm²
2 cm

Explanation:

Step1: Analyze the composite figure

The composite figure consists of a square and three semicircles (which can be combined to form 1.5 circles, or equivalently, three semicircles make 1.5 full circles, but looking at the diagram, there are three semicircles: top, left, right. Wait, actually, each side of the square has a semicircle. Wait, the square has side length? Wait, the distance between two points is 2 cm, so the side length of the square is 4 cm? Wait, no, looking at the diagram, the square has sides with segments of 2 cm? Wait, no, let's re-examine. The square: each side has a semicircle. The radius of each semicircle is 2 cm? Wait, no, the distance between the dots is 2 cm, so the side length of the square is 4 cm? Wait, no, the square has four sides, each with a semicircle. Wait, actually, the three semicircles: top, left, right. Wait, no, the figure has a square in the middle, and three semicircles: top, left, right. Wait, no, maybe the square has side length 4 cm? Wait, no, the given length is 2 cm. Wait, let's look at the options. The options have \(12\pi\) or \(6\pi\), so let's calculate the area of the circular parts and the square.

First, the square: from the diagram, the square has side length 4 cm? Wait, no, the distance between two adjacent dots on the square is 2 cm? Wait, no, the bottom side of the square has a length of 4 cm? Wait, no, the diagram shows a square with side length 4 cm? Wait, no, let's check the radius of the semicircles. The radius \(r = 2\) cm? Wait, no, the length given is 2 cm, which is the diameter? Wait, no, the distance between two points is 2 cm, so the radius \(r = 2\) cm? Wait, no, if the diameter is 4 cm, radius is 2 cm. Wait, maybe the square has side length 4 cm, and each semicircle has radius 2 cm.

Wait, the composite figure: square + three semicircles. Wait, three semicircles: top, left, right. Each semicircle has radius \(r = 2\) cm. The area of a semicircle is \(\frac{1}{2}\pi r^2\). So three semicircles: \(3 \times \frac{1}{2}\pi r^2 = \frac{3}{2}\pi r^2\). Wait, but if \(r = 2\), then \(\frac{3}{2}\pi (2)^2 = \frac{3}{2}\pi \times 4 = 6\pi\)? No, that's not right. Wait, maybe there are two full circles and one semicircle? Wait, no, let's count the number of semicircles. The top is a semicircle, left is a semicircle, right is a semicircle. Wait, no, actually, the three semicircles can be combined into 1.5 circles. Wait, no, maybe the figure has a square and three semicircles, but actually, the three semicircles make 1.5 circles, but let's check the area.

Wait, the square: side length 4 cm? Wait, no, the options have \(16\) or \(4\) for the square area. If the square has side length 4 cm, area is \(4 \times 4 = 16\) \(cm^2\). Now the circular parts: each semicircle has radius 2 cm (since the diameter is 4 cm? Wait, no, the distance between the dots is 2 cm, so the radius is 2 cm? Wait, no, the length given is 2 cm, which is the radius? No, that can't be. Wait, let's re-express.

Wait, the composite figure: the square has side length 4 cm (since from the diagram, the square has four sides, each with a semicircle, and the length between two points is 2 cm, so the side length of the square is 4 cm). The three semicircles: top, left, right. Each semicircle has radius \(r = 2\) cm. The area of one semicircle is \(\frac{1}{2}\pi r^2 = \frac{1}{2}\pi (2)^2 = 2\pi\). Three semicircles: \(3 \times 2\pi = 6\pi\)? No, that's not matching. Wait, maybe the figure has a square and two full circles and one semicircle? Wait, no, let's look at the options. The options have \(12\pi\) or \(6\pi\), and \(16\) or \(4\) for the square.

Wait, maybe the square has side length 4 cm, so area \(4 \times 4 = 16\) \(cm^2\). The circular parts: there are three semicircles, but actually, the three semicircles can be combined into 1.5 circles, but wait, no, if we have three semicircles, each with radius 2 cm, then the total area of the circular parts is \(3 \times \frac{1}{2}\pi (2)^2 = 3 \times 2\pi = 6\pi\)? No, that's 6π. But the options have 12π. Wait, maybe the radius is 2 cm, but there are six semicircles? No, the diagram shows a square with three semicircles: top, left, right. Wait, no, maybe the square has side length 4 cm, and each side of the square has a semicircle, but the bottom side doesn't? Wait, the diagram shows a square with top, left, and right semicircles. Wait, no, the figure looks like a square with three semicircles: top, left, right. Wait, but the area of the circular parts: if each semicircle has radius 2 cm, then three semicircles: 3*(1/2)*π*r² = 3*(1/2)*π*4 = 6π. The square: side length 4 cm, area 16 cm². So total area would be 6π + 16? But that's option B. Wait, no, wait, maybe the radius is 2 cm, but there are two full circles and one semicircle? Wait, no, let's recalculate.

Wait, maybe the square has side length 4 cm (since the distance between the dots is 2 cm, so each side of the square is 4 cm). The three semicircles: top, left, right. Each semicircle has radius 2 cm. The area of a semicircle is (1/2)πr² = (1/2)π(2)² = 2π. Three semicircles: 3*2π = 6π. The square: side length 4 cm, area 4*4 = 16 cm². So total area is 6π + 16? But the options have (6π + 16) as option B. Wait, but let's check again.

Wait, maybe the square has side length 4 cm, and the circular parts: there are three semicircles, each with radius 2 cm. So 3*(1/2)*π*2² = 3*(1/2)*π*4 = 6π. The square: 4*4 = 16. So total area 6π + 16. But wait, the options have (12π + 16). Wait, maybe I made a mistake in the number of semicircles. Wait, the diagram: the square has four sides, but the bottom side doesn't have a semicircle? No, the figure looks like a square with top, left, and right semicircles, and the bottom is the square's side. Wait, no, maybe the figure has a square and two full circles and one semicircle? Wait, no, let's look at the radius again. Wait, the length given is 2 cm, which is the radius? No, the diameter? If the diameter is 4 cm, radius is 2 cm. Wait, maybe the square has side length 4 cm, and there are three semicircles, each with diameter 4 cm (radius 2 cm). Then the area of each semicircle is (1/2)π*(2)² = 2π. Three semicircles: 3*2π = 6π. The square: 4*4 = 16. So total area 6π + 16, which is option B: (6π + 16) cm². Wait, but let's check the options again. The options are:

A. (6π + 4)

B. (6π + 16)

C. (12π + 4)

D. (12π + 16)

So if the square has area 16, and the circular parts 6π, then B is correct. Wait, but why 12π? Maybe I miscounted the number of semicircles. Wait, maybe the figure has a square and six semicircles? No, the diagram shows a square with three semicircles. Wait, maybe the radius is 2 cm, but there are two full circles (each made of two semicircles) and one semicircle? Wait, no, three semicircles make 1.5 circles. Wait, 1.5 circles with radius 2 cm: area 1.5*π*r² = 1.5*π*4 = 6π. So that's 6π. The square: side length 4 cm, area 16. So total area 6π + 16, which is option B.

Wait, but let's re-express:

- Square: side length 4 cm (since the distance between the dots is 2 cm, so each side of the square is 4 cm? Wait, no, the diagram shows a square with segments of 2 cm? Wait, the bottom side of the square has a length of 4 cm (two segments of 2 cm each). So the square has side length 4 cm, area 4*4 = 16 cm².

- Circular parts: three semicircles, each with radius 2 cm. Area of one semicircle: (1/2)πr² = (1/2)π(2)² = 2π. Three semicircles: 3*2π = 6π.

- Total area: 6π + 16 cm².

Step2: Calculate the area of the square

The square has side length \(4\) cm (since each segment is \(2\) cm, and there are two segments per side). The area of the square is \(side^2 = 4^2 = 16\) \(cm^2\).

Step3: Calculate the area of the circular parts

There are three semicircles, each with radius \(r = 2\) cm. The area of one semicircle is \(\frac{1}{2}\pi r^2\). So the area of three semicircles is \(3 \times \frac{1}{2}\pi r^2\).

Substituting \(r = 2\):

\(3 \times \frac{1}{2}\pi (2)^2 = 3 \times \frac{1}{2}\pi \times 4 = 3 \times 2\pi = 6\pi\) \(cm^2\).

Step4: Total area of the composite figure

Add the area of the square and the area of the circular parts:

\(6\pi + 16\) \(cm^2\).

Answer:

\(\boxed{(6\pi + 16) \text{ cm}^2}\) (Option B: \((6\pi + 16) \text{ cm}^2\))