Question

here is a triangular pyramid and its net. the lateral faces are congruent triangles. the base (shaded) is an equilateral triangle. (all lengths are in yards.) (a) find the area of the base of the pyramid. 35.1 yd² (b) find the area of one lateral face of the pyramid. 27 yd² (c) use the net to find the lateral surface area of the pyramid. the base is not included. yd² (d) use the net to find the total surface area of the pyramid. yd²

Explanation:

Step1: Identify the base triangle

The base is an equilateral triangle with side length 9 yd (from the net diagram). The formula for the area of an equilateral triangle is $\frac{\sqrt{3}}{4}s^2$, but wait, looking at the diagram, there's a height of 7.8 yd for the base? Wait, no, the base is an equilateral triangle? Wait, maybe the base triangle has base 9 and height 7.8? Wait, the problem says the base (shaded) is an equilateral triangle? Wait, no, maybe the base is a triangle with base 9 and height 7.8? Wait, the area of a triangle is $\frac{1}{2} \times base \times height$. Let's check the given values. The base triangle: if the side is 9, and the height (altitude) is 7.8? Wait, the net shows the base triangle with base 9 and height 7.8? Wait, no, the lateral faces are congruent triangles with base 9 and height 6? Wait, the problem (a) is to find the area of the base. Let's re-examine.

Wait, the base is a triangle (shaded) which is equilateral? Wait, no, the diagram: the base triangle has sides? Wait, the net has a triangle with base 9, and the height (altitude) of the base triangle is 7.8? Wait, the formula for the area of a triangle is $\frac{1}{2} \times base \times height$. So base = 9 yd, height = 7.8 yd. Then area is $\frac{1}{2} \times 9 \times 7.8$.

Step2: Calculate the area

$\frac{1}{2} \times 9 \times 7.8 = 4.5 \times 7.8 = 35.1$ square yards. Which matches the given (a) answer 35.1 yd².

For (b): One lateral face is a triangle with base 9 and height 6 (from the diagram: the lateral face has base 9 and height 6). Area of one lateral face: $\frac{1}{2} \times 9 \times 6 = 27$ yd², which matches (b) 27 yd².

For (c): Lateral surface area is 3 times the area of one lateral face (since it's a triangular pyramid, 3 lateral faces). So 3 × 27 = 81? Wait, no, wait the net: the triangular pyramid has 3 lateral faces, each congruent. So lateral surface area = 3 × 27 = 81? Wait, but the option (c) says "use the net to find the lateral surface area (base not included)". So 3 × 27 = 81? Wait, but the diagram's lateral faces: each has base 9 and height 6. So area per face: 27, three faces: 81. Wait, but the given (c) has a box, maybe I misread. Wait, the problem (c) is to find lateral surface area: 3 × 27 = 81? Wait, no, the original problem's (c) is "Use the net to find the lateral surface area of the pyramid. The base is not included." So lateral surface area = number of lateral faces × area of one lateral face. There are 3 lateral faces (since it's a triangular pyramid, base is a triangle, so 3 lateral faces). Each lateral face area is 27 (from (b)), so 3 × 27 = 81? Wait, but the given (c) has a box, maybe the answer is 81? Wait, no, the original problem's (c) is a box to fill, but the user's image shows (c) with a box and (d) as total surface area.

For (d): Total surface area = lateral surface area + base area. Lateral surface area (from (c)): let's confirm. Wait, (b) is area of one lateral face: 27, so three lateral faces: 3 × 27 = 81. Base area is 35.1 (from (a)). So total surface area = 81 + 35.1 = 116.1? Wait, but let's check again.

Wait, (a) is base area: 35.1 (correct as per calculation). (b) is one lateral face: 27 (correct: 0.5×9×6=27). (c) lateral surface area: 3×27=81. (d) total surface area: 81 + 35.1=116.1.

But the user's problem: let's answer each sub-question.

(a) Area of base: 35.1 yd² (as calculated).

(b) Area of one lateral face: 27 yd² (as calculated).

(c) Lateral surface area: 3×27=81 yd².

(d) Total surface area: 81 + 35.1=116.1 yd².

But the user's image shows (a) as 35.1, (b) as 27, (c) and (d)…

Answer:

(a) The area of the base is $\boxed{35.1}$ square yards.
(b) The area of one lateral face is $\boxed{27}$ square yards.
(c) The lateral surface area is $\boxed{81}$ square yards.
(d) The total surface area is $\boxed{116.1}$ square yards.