Question

what is the surface area of this triangular pyramid? square inches submit

Explanation:

Step1: Analyze the base triangle

The base is an equilateral triangle? Wait, no, the base has sides 1 in, 1 in, and the height of the base triangle is 0.9 in. Wait, actually, the base is a triangle with base 1 in and height 0.9 in? Wait, no, looking at the diagram, the base is a triangle with sides 1 in, 1 in, and the height of the base (the triangle's height) is 0.9 in? Wait, maybe the base is an isosceles triangle with two sides 1 in and base 1 in? Wait, no, the base is a triangle with base 1 in and height 0.9 in? Wait, maybe the base area: the formula for the area of a triangle is $\frac{1}{2} \times base \times height$. So base area: $\frac{1}{2} \times 1 \times 0.9 = 0.45$ square inches.

Step2: Analyze the lateral faces

There are three lateral faces? Wait, no, a triangular pyramid (tetrahedron) has a triangular base and three triangular lateral faces. Wait, looking at the diagram, the lateral faces: two of them have a base of 1 in and height of 2 in? Wait, no, the diagram shows a lateral face with height 2 in? Wait, the diagram has a dashed line of 2 in, maybe the slant height? Wait, no, the diagram has a 1 in, 1 in, 0.9 in for the base, and a 2 in for the height of the lateral face? Wait, maybe the lateral faces: three triangles? Wait, no, maybe the base is an equilateral triangle with side 1 in, and the lateral faces are three triangles with base 1 in and height 2 in? Wait, no, the diagram shows a triangular pyramid where the base is a triangle with base 1 in, height 0.9 in, and the three lateral faces: two of them are triangles with base 1 in and height 2 in? Wait, no, maybe the base is a triangle with sides 1,1,1? Wait, no, the base has a height of 0.9 in. Wait, let's re-examine:

Wait, the base is a triangle with base 1 in and height 0.9 in (so area $\frac{1}{2} \times 1 \times 0.9 = 0.45$). Then the lateral faces: there are three triangular faces? Wait, no, a triangular pyramid (tetrahedron) has four triangular faces: the base and three lateral faces. Wait, but in the diagram, the base is a triangle with sides 1,1,1? Wait, no, the base has a height of 0.9 in, so it's an isosceles triangle with two sides 1 in and base 1 in? Wait, no, if the base is a triangle with base 1 in and height 0.9 in, then the two equal sides (legs) can be calculated, but maybe the diagram is a triangular pyramid where the base is an equilateral triangle with side 1 in, and the lateral faces are three triangles with base 1 in and height 2 in? Wait, no, the diagram shows a 2 in as the height of the lateral face? Wait, maybe the lateral faces: three triangles, each with base 1 in and height 2 in? Wait, no, the diagram has a 1 in, 1 in, 0.9 in for the base, and a 2 in for the height of the lateral face. Wait, maybe the base is a triangle with area $\frac{1}{2} \times 1 \times 0.9 = 0.45$, and the three lateral faces: two of them are triangles with base 1 in and height 2 in, and one with base 1 in and height 1 in? Wait, no, the diagram has a 1 in arrow pointing to a lateral face? Wait, maybe the lateral faces: three triangles, each with base 1 in and height 2 in? Wait, no, let's look again.

Wait, the problem is a triangular pyramid (tetrahedron) with a triangular base. The base is a triangle with base 1 in and height 0.9 in (so area $\frac{1}{2} \times 1 \times 0.9 = 0.45$). Then the three lateral faces: each is a triangle. Wait, the diagram shows a 2 in as the height of one lateral face, and a 1 in as the height of another? Wait, maybe the lateral faces: three triangles, two with base 1 in and height 2 in, and one with base 1 in and height 1 in? Wait, no, the diagram has a 1 in, 1 in, 0.9 in for the base, and a 2 in for the height of the lateral face. Wait, maybe the correct approach is:

Surface area of a triangular pyramid (tetrahedron) is the sum of the area of the base and the areas of the three lateral faces.

Base: triangle with base 1 in, height 0.9 in. Area = $\frac{1}{2} \times 1 \times 0.9 = 0.45$ in².

Lateral faces: three triangles. Wait, looking at the diagram, the lateral faces: two of them have a base of 1 in and height of 2 in? Wait, no, the diagram has a 2 in as the height of the lateral face (the dashed line), and a 1 in as the height of another lateral face? Wait, maybe the three lateral faces: each has a base of 1 in. Let's check the areas:

First lateral face: base 1 in, height 2 in. Area = $\frac{1}{2} \times 1 \times 2 = 1$ in².

Second lateral face: same as first? Wait, maybe two lateral faces with area 1 in² each, and one with area $\frac{1}{2} \times 1 \times 1 = 0.5$ in²? Wait, no, the diagram has a 1 in arrow pointing to a lateral face. Wait, maybe the three lateral faces: two with height 2 in and one with height 1 in? Wait, no, let's re-express:

Wait, the base is a triangle with area 0.45. Then the three lateral faces:

- Two faces: each is a triangle with base 1 in and height 2 in. Area of each: $\frac{1}{2} \times 1 \times 2 = 1$. So two of them: 2 * 1 = 2.

- One face: a triangle with base 1 in and height 1 in. Area: $\frac{1}{2} \times 1 \times 1 = 0.5$.

Then total surface area: base (0.45) + 2 + 0.5 = 2.95? Wait, no, that doesn't seem right. Wait, maybe the base is an equilateral triangle with side 1 in, so area $\frac{\sqrt{3}}{4} \times 1^2 \approx 0.433$, but the diagram shows 0.9 in as the height, so $\frac{1}{2} \times 1 \times 0.9 = 0.45$ is correct.

Wait, maybe the lateral faces: three triangles, each with base 1 in and height 2 in? Then area of each lateral face: $\frac{1}{2} \times 1 \times 2 = 1$. Three of them: 3 * 1 = 3. Then total surface area: 0.45 + 3 = 3.45? No, that's not matching. Wait, maybe the diagram is a triangular pyramid where the base is a triangle with sides 1,1,1 (equilateral), and the lateral faces are three triangles with base 1 in and height 2 in. But the height of the base triangle would be $\frac{\sqrt{3}}{2} \approx 0.866$, close to 0.9, so maybe that's the case. So base area: $\frac{\sqrt{3}}{4} \times 1^2 \approx 0.433$, but the diagram says 0.9 in, maybe it's a typo, or maybe the base is a triangle with base 1 in and height 0.9 in.

Wait, let's check the problem again. The diagram shows a triangular pyramid with base edges 1 in, 1 in, 1 in (equilateral), height of base triangle 0.9 in (approx $\frac{\sqrt{3}}{2} \approx 0.866$), and the lateral faces have a height (slant height) of 2 in? Wait, no, the 2 in is the height of the pyramid? No, the dashed line is 2 in, maybe the slant height.

Wait, maybe the correct formula is:

Surface area = base area + 3 * (area of lateral face)

Base area: $\frac{1}{2} \times 1 \times 0.9 = 0.45$

Lateral face: each is a triangle with base 1 in and height 2 in? Wait, no, the 1 in is the height of the lateral face? Wait, the diagram has a 1 in arrow pointing to a lateral face. Maybe the lateral faces: three triangles, each with base 1 in and height 1 in? No, that would be too small.

Wait, maybe I'm overcomplicating. Let's look at the numbers:

Base: triangle with base 1, height 0.9. Area: 0.5*1*0.9=0.45.

Lateral faces: three triangles. Two of them have base 1, height 2. Area: 0.5*1*2=1 each. So two of them: 2*1=2.

One lateral face: base 1, height 1. Area: 0.5*1*1=0.5.

Total: 0.45 + 2 + 0.5 = 2.95? No, that's not right. Wait, maybe the lateral faces are three triangles, each with base 1 and height 2. Then 3*1=3. Total: 0.45+3=3.45. But the diagram shows a 1 in arrow, maybe that's the height of the lateral face. Wait, maybe the lateral faces have height 1 in. Then area of each lateral face: 0.5*1*1=0.5. Three of them: 1.5. Total: 0.45+1.5=1.95. No, that doesn't make sense.

Wait, maybe the base is a triangle with sides 1,1,1 (equilateral), so area ≈0.433, and the lateral faces are three triangles with base 1 and height 2. Then total surface area: 0.433 + 3*1=3.433, approx 3.43. But the diagram has 0.9 in, which is close to 0.866, so maybe the base area is 0.45 (using 0.9 as height). Then total surface area: 0.45 + 3*1=3.45. But maybe the lateral faces have height 1 in. Then 0.45 + 3*0.5=1.95. No, the diagram shows 2 in, so maybe 2 in is the height of the lateral face.

Wait, maybe the correct answer is 3.45, but let's check again.

Wait, the base is a triangle with base 1 in and height 0.9 in: area = 0.5 * 1 * 0.9 = 0.45.

The three lateral faces: each is a triangle with base 1 in and height 2 in: area of each = 0.5 * 1 * 2 = 1. So three of them: 3 * 1 = 3.

Total surface area: 0.45 + 3 = 3.45.

But maybe the lateral faces are two with height 2 and one with height 1. Then 2*1 + 0.5*1*1 = 2 + 0.5 = 2.5. Then total: 0.45 + 2.5 = 2.95.

Wait, the diagram has a 1 in arrow pointing to a lateral face, so maybe one lateral face has height 1 in, and the other two have height 2 in. So:

Base: 0.45

Lateral face 1: 0.5*1*2=1

Lateral face 2: 0.5*1*2=1

Lateral face 3: 0.5*1*1=0.5

Total: 0.45 + 1 + 1 + 0.5 = 2.95.

But I'm not sure. Wait, maybe the base is a triangle with area 0.5*1*0.9=0.45, and the three lateral faces are each 0.5*1*2=1, so total 0.45 + 3*1=3.45.

Alternatively, maybe the base is a triangle with sides 1,1,1, so area ≈0.433, and lateral faces each 0.5*1*2=1, total ≈3.433, which is approximately 3.43 or 3.45.

But the problem gives 0.9 in as the height of the base, so we should use that. So:

Base area: 0.5 * 1 * 0.9 = 0.45

Lateral faces: three triangles, each with base 1 in and height 2 in: 3 * (0.5 * 1 * 2) = 3 * 1 = 3

Total surface area: 0.45 + 3 = 3.45

So the answer is 3.45.

Answer:

3.45