Question

quiz 1: surface areas
what is the surface area of the prism shown?
12
15
20
855 units²
828 units²
720 units²
900 units²

Explanation:

Step1: Identify the prism type

This is a right triangular prism (the triangular base has a right angle). The triangular base has legs 12 and 15? Wait, no, wait. Wait, the triangular base: let's see, the prism has a triangular base with sides? Wait, no, the dimensions: the triangular base is a right triangle? Wait, the height of the triangle: wait, the prism's length is 20, the triangular base has a leg 12, another leg? Wait, maybe the triangular base is a right triangle with legs 12 and 15? Wait, no, the diagram: the triangular face has a right angle, one side 12, the other leg? Wait, maybe the triangular base is a right triangle with legs 12 and 15? Wait, no, the prism's length is 20. Wait, the surface area of a triangular prism is given by \( SA = 2B + Ph \), where \( B \) is the area of the triangular base, \( P \) is the perimeter of the triangular base, and \( h \) is the length of the prism (the distance between the two triangular bases).

Wait, first, find the area of the triangular base. The triangular base is a right triangle? Wait, the diagram shows a right angle on the triangle. So the legs of the right triangle: let's see, one leg is 12, another leg? Wait, maybe the other leg is 15? Wait, no, the prism's length is 20. Wait, maybe the triangular base has legs 12 and 15? Wait, no, let's check the dimensions. Wait, the prism has a triangular base with sides: let's see, the triangle has a right angle, one leg 12, another leg? Wait, maybe the hypotenuse? Wait, no, the perimeter of the triangle: let's calculate the area of the triangle first. If it's a right triangle with legs 12 and 15, then area \( B = \frac{1}{2} \times 12 \times 15 = 90 \). Then the perimeter of the triangle: the hypotenuse is \( \sqrt{12^2 + 15^2} = \sqrt{144 + 225} = \sqrt{369} \approx 19.21 \), but that doesn't make sense. Wait, maybe I got the legs wrong. Wait, maybe the triangular base is a right triangle with legs 12 and 9? No, wait, the options are 828, 855, etc. Wait, maybe the triangular base has legs 12 and 9? No, let's re-examine. Wait, the prism's length is 20, and the triangular base has sides 12, 15, and the hypotenuse? Wait, no, maybe the triangular base is a right triangle with legs 12 and 9? Wait, no, let's use the formula for surface area of a triangular prism: \( SA = 2 \times (\text{area of triangle}) + (\text{perimeter of triangle}) \times \text{length of prism} \).

Wait, maybe the triangular base is a right triangle with legs 12 and 9? No, wait, the given dimensions: 12, 15, 20. Wait, maybe the triangle has legs 12 and 9? No, let's check the options. Let's suppose the triangular base is a right triangle with legs 12 and 9? No, wait, 12, 15, and the hypotenuse? Wait, 12-15-19.21, but that's messy. Wait, maybe the triangle is a right triangle with legs 12 and 15, and the length of the prism is 20. Wait, then area of triangle \( B = \frac{1}{2} \times 12 \times 15 = 90 \). Perimeter of triangle: 12 + 15 + hypotenuse. Hypotenuse: \( \sqrt{12^2 + 15^2} = \sqrt{144 + 225} = \sqrt{369} \approx 19.21 \). Then perimeter \( P \approx 12 + 15 + 19.21 = 46.21 \). Then surface area \( SA = 2 \times 90 + 46.21 \times 20 = 180 + 924.2 = 1104.2 \), which is not one of the options. So I must have misidentified the triangle.

Wait, maybe the triangular base is a right triangle with legs 12 and 9? No, wait, maybe the triangle has legs 12 and 9, and the length is 20? No, the options are 828, 855, 720, 900. Let's try another approach. Wait, maybe the triangular base is a right triangle with legs 12 and 9? No, wait, maybe the triangle has legs 12 and 9, and the length is 20? No, let's check the formula again. Wait, maybe the prism is a right triangular prism with the triangular base having legs 12 and 9, and the length 20? No, that's not matching. Wait, maybe the triangular base is a right triangle with legs 12 and 15, and the length is 18? No, the length is 20. Wait, maybe I made a mistake. Wait, let's look at the options. The options are 828, 855, 720, 900. Let's calculate \( 2B + Ph \). Let's suppose the triangle has legs 12 and 9, then area \( B = 54 \), perimeter \( 12 + 9 + 15 = 36 \) (since 12-9-15 is a right triangle: \( 9^2 + 12^2 = 81 + 144 = 225 = 15^2 \)). Ah! There we go. So the triangular base is a right triangle with legs 9 and 12, hypotenuse 15. So that's a 9-12-15 triangle (a multiple of 3-4-5: 3*3=9, 3*4=12, 3*5=15). Perfect. So the triangular base has legs 9 and 12, hypotenuse 15. Then the area of the triangle \( B = \frac{1}{2} \times 9 \times 12 = 54 \). The perimeter of the triangle \( P = 9 + 12 + 15 = 36 \). The length of the prism (the distance between the two triangular bases) is 20. Then the surface area \( SA = 2B + P \times \text{length} = 2 \times 54 + 36 \times 20 = 108 + 720 = 828 \).

Step2: Verify the calculation

- Area of triangular base: \( \frac{1}{2} \times 9 \times 12 = 54 \). Two bases: \( 2 \times 54 = 108 \).
- Perimeter of triangular base: \( 9 + 12 + 15 = 36 \).
- Lateral surface area: \( 36 \times 20 = 720 \).
- Total surface area: \( 108 + 720 = 828 \).

Answer:

828 units²