Question

quick check
a net of an equilateral triangular prism and some of its dimensions are shown in the diagram. the height of each triangular base is approximately 5.2 cm. which measurement is closest to the total surface area of the prism in square centimeters?
a. 280.8 cm²
b. 162 cm²
c. 193.2 cm²
d. 157.2 cm²

Explanation:

Step1: Find area of triangular bases

The formula for the area of a triangle is $A = \frac{1}{2} \times base \times height$. The base of the equilateral triangle is 8 cm (from the diagram) and height is 5.2 cm. So area of one triangle is $\frac{1}{2} \times 8 \times 5.2 = 20.8$ $cm^2$. Two triangular bases: $2 \times 20.8 = 41.6$ $cm^2$.

Step2: Find area of rectangular faces

The rectangular faces: the prism has three rectangles. The sides of the rectangles: one side is 7 cm (from diagram), and the other sides are 8 cm (the base of the triangle) for each rectangle? Wait, no, the equilateral triangle has side 8 cm, so the rectangles have dimensions 8 cm (length) and 7 cm (width)? Wait, no, the net: the three rectangles, each with length 7 cm (the distance between the triangles) and width equal to the side of the triangle (8 cm). Wait, no, looking at the diagram, the rectangles: maybe the three rectangles have dimensions 8 cm, 7 cm? Wait, no, the triangular prism's lateral faces: the perimeter of the base times the height of the prism. Wait, the base triangle has sides 8 cm (equilateral, so all sides 8 cm). The height of the prism (the distance between the two triangular bases) is 7 cm? Wait, the diagram shows 7 cm. So lateral surface area is perimeter of base (3*8) times height of prism (7). So lateral surface area: $3 \times 8 \times 7 = 168$? Wait, no, that can't be. Wait, maybe I misread. Wait, the diagram: the rectangles, maybe the length is 7 cm, and the width is 8 cm? Wait, no, let's re-express.

Wait, the triangular base: base 8 cm, height 5.2 cm. The prism's height (the distance between the two triangles) is 7 cm (from the diagram, the 7 cm is the length of the rectangles). So the three rectangular faces: each has area 8 cm (side of triangle) * 7 cm (height of prism). So three rectangles: 3 * 8 * 7 = 168? No, that's too big. Wait, no, maybe the rectangles are 8 cm, 7 cm? Wait, no, the problem: the net of an equilateral triangular prism. So the two triangular bases (area calculated) and three rectangular lateral faces. The lateral faces: each has a side equal to the side of the triangle (8 cm) and the length of the prism (7 cm). Wait, but then lateral surface area is 3 * 8 * 7 = 168. Then total surface area is lateral + 2*triangular. But 168 + 41.6 = 209.6, which is not in the options. So I must have misread the diagram. Wait, maybe the rectangles have different dimensions. Wait, the diagram has 8 cm, 7 cm, and maybe the other side? Wait, maybe the rectangles are 8 cm, 7 cm? No, maybe the base of the triangle is 8 cm, but the rectangles have length 7 cm and width 8 cm? Wait, no, the options are 162, 193, etc. Wait, maybe the rectangles are 8 cm, 7 cm? Wait, no, let's check again.

Wait, maybe the three rectangles: two with 8 cm and 7 cm, and one with 8 cm and 7 cm? No, no, triangular prism has three lateral faces, each with area equal to side of triangle times height of prism. Wait, maybe the height of the prism (the distance between the triangles) is 7 cm, and the sides of the triangle are 8 cm. But then lateral surface area is 3*8*7=168, plus two triangles 41.6, total 209.6, not matching options. So maybe the diagram has different numbers. Wait, the problem says "some of its dimensions are shown", maybe the rectangles have length 7 cm and width 8 cm? No, maybe the triangle's base is 6 cm? Wait, no, the diagram shows 8 cm. Wait, maybe I made a mistake. Wait, the options are A:280, B:162, C:193, D:157. Let's recalculate.

Wait, maybe the three rectangles: each with length 7 cm and width 8 cm? No, maybe the height of the prism is 7 cm, and the sides of the triangle are 8 cm, but maybe the rectangles are 8 cm, 7 cm, but there are three rectangles. Wait, no, maybe the triangle's base is 8 cm, height 5.2 cm, two triangles: 2*(0.5*8*5.2)=41.6. Then the lateral faces: three rectangles, each with area 8*7=56, so three of them: 3*56=168. Then total surface area: 41.6 + 168 = 209.6, which is not in options. So maybe the rectangles are not 8*7. Wait, maybe the three rectangles: two with 8 cm and 7 cm, and one with 8 cm and 7 cm? No, maybe the diagram has the rectangles with length 7 cm and width 8 cm, but maybe the side of the triangle is 6 cm? Wait, no, the diagram shows 8 cm. Wait, maybe the height of the prism is 7 cm, and the sides of the triangle are 8 cm, but the lateral faces are 8*7, 8*7, 8*7? No, that's three. Wait, maybe the triangle's base is 6 cm? Let's try: triangle area: 0.5*6*5.2=15.6, two triangles: 31.2. Lateral surface area: 3*6*7=126. Total: 31.2+126=157.2, which is option D. Ah! Maybe the base of the triangle is 6 cm, not 8 cm. Maybe the diagram has 6 cm, not 8 cm. Let's check the diagram again: the left triangle has a side labeled 8 cm? Wait, maybe the diagram's triangle has base 6 cm. Let's assume that. So:

Step1: Area of one triangle: 0.5*6*5.2=15.6. Two triangles: 31.2.

Step2: Lateral surface area: three rectangles, each with area 6*7=42. Three of them: 3*42=126.

Total surface area: 31.2 + 126 = 157.2, which is option D.

Ah, so maybe the base of the triangle is 6 cm, not 8 cm. So that's the mistake. So:

Step1: Area of triangular bases

Base of triangle = 6 cm (from diagram, maybe I misread 8 as 6), height = 5.2 cm. Area of one triangle: $\frac{1}{2} \times 6 \times 5.2 = 15.6$ $cm^2$. Two triangles: $2 \times 15.6 = 31.2$ $cm^2$.

Step2: Area of rectangular faces

The prism has three rectangular faces. Each rectangle has length 7 cm (from diagram) and width 6 cm (side of triangle). Area of one rectangle: $6 \times 7 = 42$ $cm^2$. Three rectangles: $3 \times 42 = 126$ $cm^2$.

Step3: Total surface area

Add lateral surface area and area of two bases: $126 + 31.2 = 157.2$ $cm^2$.

Answer:

D. $157.2\ cm^2$