Question

what is the surface area of the shape?

Explanation:

Step1: Analyze the shape components

The shape appears to be composed of 4 congruent triangles (2 types? Wait, no, looking at the diagram: there are 4 triangles? Wait, actually, the central dashed square? Wait, no, let's re-examine. Wait, the figure has a top triangle, bottom triangle, left triangle, right triangle? Wait, no, the given lengths: the height of the top (and bottom) triangle is 2 in, and the distance from center to side (like the radius for the square's half-side?) Wait, maybe the shape is made of 4 triangles: two with base related to 4 in? Wait, no, let's see: the diagram shows a quadrilateral (a rhombus-like shape) made of 4 triangles? Wait, actually, the figure has 4 triangular faces? Wait, no, maybe it's a square-based pyramid's net? Wait, no, the net of a square pyramid has a square base and 4 triangular faces. Wait, but in the diagram, the central part is a dashed square? Wait, no, the given lengths: the height of the triangle (the slant height?) is 2 in? Wait, no, the vertical line is 2 in, and the horizontal line (from center to side) is 4 in? Wait, maybe the shape is composed of 4 triangles, each with base 8 in (since from center to side is 4 in, so total side is 8 in?) Wait, no, maybe I misinterpret. Wait, let's look again: the figure has a top triangle with height 2 in, and the distance from the center to the side (like the apothem or half-side) is 4 in? Wait, no, maybe the shape is made of 4 triangles: two with base 8 in (since 4 in from center to side, so total length 8 in) and height 2 in? Wait, no, that doesn't make sense. Wait, maybe the shape is a square with side 8 in (since 4 in from center to side, so 4*2=8 in) and four triangles? Wait, no, the diagram shows a green shape with a central dashed square and four triangles (top, bottom, left, right). Wait, maybe the surface area is the sum of the area of the square and the four triangles? Wait, no, the problem says "surface area of the shape" – maybe it's a 3D shape? Wait, no, the diagram is 2D? Wait, no, maybe it's a net of a square pyramid. Wait, the net of a square pyramid has a square base and four triangular faces. Let's assume that: the square base has side length 8 in (since the distance from center to side is 4 in, so side length is 8 in), and each triangular face has a base of 8 in and height of 2 in? Wait, no, that can't be. Wait, maybe the triangles have base 4 in? No, the 4 in is the distance from center to side. Wait, maybe the shape is composed of 4 congruent triangles, each with base 8 in (4*2) and height 2 in? Wait, no, let's calculate:

Wait, maybe the shape is a square with side 8 in (4*2) and four triangles, each with base 8 in and height 2 in? No, that would be the net of a square pyramid. Wait, but the surface area of a square pyramid's net is the area of the square base plus the area of the four triangular faces. Wait, but in the diagram, the central part is dashed, maybe the square is not included? No, the problem says "surface area of the shape" – maybe it's a 2D shape? Wait, the diagram shows a green shape with four triangular parts and a central square? No, the green area is the entire shape. Wait, maybe the shape is made of four triangles, each with base 8 in (since 4 in from center to side, so total length 8 in) and height 2 in? Wait, no, let's look at the numbers: 2 in and 4 in. Wait, maybe the shape is composed of 4 triangles, each with base 4 in and height 2 in? No, that seems too small. Wait, maybe the shape is a rhombus divided into four triangles, with the diagonals? Wait, no, the diagonals would be 8 in (4*2) and 4 in (2*2)? Wait, no, the area of a rhombus is (d1*d2)/2. But the diagram shows a shape with a central square? Wait, I think I made a mistake. Let's re-express:

Wait, the figure has four triangular faces: top, bottom, left, right. Each triangle has a base of 8 in (since the distance from center to side is 4 in, so total length is 8 in) and height of 2 in? Wait, no, the height of each triangle is 2 in. Wait, no, the 2 in is the height of the top triangle, and the 4 in is the distance from the center to the side (like the radius). Wait, maybe the shape is a square pyramid's net, with the square base having side length 8 in (4*2) and four triangular faces, each with base 8 in and height 2 in. Then the surface area would be the area of the square plus the area of the four triangles. Wait, but the square's area is 8*8=64, and each triangle is (8*2)/2=8, four triangles: 4*8=32, total 64+32=96? No, that can't be. Wait, maybe the 4 in is the side length of the square? Wait, the dashed square has side length 4 in? Then the square's area is 4*4=16. Then the four triangles: each has base 4 in and height 2 in? No, the height is 2 in. Wait, each triangle: area is (base*height)/2. If the square has side 4 in, then each triangle has base 4 in and height 2 in. There are four triangles (top, bottom, left, right). So area of square: 4*4=16. Area of four triangles: 4*( (4*2)/2 )=4*4=16. Total surface area: 16+16=32? Wait, but that doesn't match. Wait, maybe the shape is composed of four triangles, each with base 8 in (4*2) and height 2 in. Wait, 4 in is the distance from center to side, so the total length is 8 in (diameter). Then each triangle has base 8 in and height 2 in. There are four triangles. So area of each triangle: (8*2)/2=8. Four triangles: 4*8=32. Wait, but that's the same as before. Wait, maybe the shape is a square with side 8 in and four triangles, but the square is not included? No, the diagram shows the entire green shape. Wait, maybe I misinterpret the diagram. Let's look again: the figure has a top triangle with height 2 in, and the horizontal line (from center to side) is 4 in. So the base of the top triangle is 8 in (4*2). Similarly, the bottom triangle has base 8 in and height 2 in. The left and right triangles: wait, maybe the shape is a square with side 8 in, and four triangles attached to each side, each with base 8 in and height 2 in. But that would be a prism? No, the surface area would be the area of the square plus the area of the four rectangles? No, the triangles are triangular. Wait, I think the correct approach is: the shape is made of 4 congruent triangles, each with base 8 in (4*2) and height 2 in? No, 4 in is the distance from center to side, so the base of each triangle is 8 in (since from center to left is 4 in, center to right is 4 in, so total 8 in). The height of each triangle is 2 in. So area of one triangle: (8*2)/2=8. Four triangles: 4*8=32. Wait, but maybe the shape is a square with side 4 in, and four triangles, each with base 4 in and height 2 in. Then area of square: 4*4=16. Area of four triangles: 4*( (4*2)/2 )=16. Total: 32. Yes, that makes sense. So the surface area is 32 square inches.

Step1: Identify the components

The shape consists of a square (dashed) with side length \( 4 \, \text{in} \) and 4 triangular faces (top, bottom, left, right) each with base \( 4 \, \text{in} \) and height \( 2 \, \text{in} \).

Step2: Calculate area of the square

Area of square: \( \text{side}^2 = 4 \times 4 = 16 \, \text{in}^2 \).

Step3: Calculate area of one triangle

Area of a triangle: \( \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 4 \times 2 = 4 \, \text{in}^2 \).

Step4: Calculate area of 4 triangles

Total area of 4 triangles: \( 4 \times 4 = 16 \, \text{in}^2 \).

Step5: Sum the areas

Total surface area: \( 16 + 16 = 32 \, \text{in}^2 \).

Answer:

\( 32 \)