Question

what is the surface area of this triangular pyramid? image of triangular pyramid with edge labels 5 yd, 5 yd, 5 yd and base height label 4.3 yd blank square yards submit

Explanation:

Step1: Identify the type of pyramid

This is a regular triangular pyramid (tetrahedron) with a base that is an equilateral triangle (side length 5 yd) and three congruent triangular faces (also with side length 5 yd, and the slant height for the lateral faces can be calculated or we can use the given height of the base triangle's altitude? Wait, actually, looking at the diagram, the base is an equilateral triangle with side length 5 yd, and the height of the base triangle (altitude) is 4.3 yd. The lateral faces are three congruent isosceles triangles with side length 5 yd. Wait, no, actually, a triangular pyramid (tetrahedron) with all edges equal? Wait, the edges from the base vertices to the apex are 5 yd, and the base edges are 5 yd. So it's a regular tetrahedron? Wait, no, the base triangle has an altitude of 4.3 yd. Wait, maybe the base is an equilateral triangle with side 5 yd, so area of base is (base * height)/2 = (5 * 4.3)/2. Then the lateral faces: each lateral face is a triangle with base 5 yd and height? Wait, no, in a regular triangular pyramid (tetrahedron) with all edges equal, the slant height would be different, but here, maybe the lateral faces are equilateral triangles? Wait, the diagram shows the edges from the apex to each base vertex as 5 yd, and the base edges as 5 yd. So all edges are 5 yd? Wait, but the base triangle has an altitude of 4.3 yd. Let's check: for an equilateral triangle with side length \( s \), the altitude \( h = \frac{\sqrt{3}}{2}s \approx 0.866s \). For \( s = 5 \), \( h \approx 4.33 \), which is approximately 4.3 yd (maybe rounded). So the base is an equilateral triangle with side 5 yd, area \( A_{base} = \frac{1}{2} \times 5 \times 4.3 \). Then the lateral faces: there are three congruent equilateral triangles (since all edges are 5 yd) with side length 5 yd. Wait, no, the lateral faces are triangles with two sides 5 yd (from apex to base vertices) and base 5 yd, so they are also equilateral triangles. Wait, but the area of an equilateral triangle with side \( s \) is \( \frac{\sqrt{3}}{4}s^2 \approx 0.433s^2 \). For \( s = 5 \), that's \( 0.433 \times 25 \approx 10.825 \) square yards per lateral face. But maybe the problem is simpler: the triangular pyramid (tetrahedron) has a base that is an equilateral triangle with side 5 yd and height 4.3 yd, and three lateral faces that are congruent triangles with base 5 yd and height (slant height) equal to 5 yd? Wait, no, the diagram shows the slant height? Wait, maybe the lateral faces are triangles with base 5 yd and height 5 yd? Wait, no, the 5 yd is the edge length. Wait, maybe the surface area is the sum of the base area and the three lateral face areas.

Wait, let's re-express:

A triangular pyramid (tetrahedron) has 4 triangular faces: 1 base and 3 lateral faces.

If the base is an equilateral triangle with side length \( s = 5 \) yd and altitude \( h_{base} = 4.3 \) yd, then area of base \( A_{base} = \frac{1}{2} \times s \times h_{base} = \frac{1}{2} \times 5 \times 4.3 \).

Each lateral face: if the pyramid is regular (all lateral edges equal and base is equilateral), then each lateral face is an isosceles triangle with base \( s = 5 \) yd and height (slant height) \( l \). But in the diagram, the edges from apex to base vertices are 5 yd, so the lateral faces are triangles with two sides 5 yd and base 5 yd, so they are equilateral triangles. Wait, but the area of an equilateral triangle with side 5 is \( \frac{\sqrt{3}}{4} \times 5^2 \approx 10.825 \), but maybe the problem is using the given 4.3 yd as the height for the lateral faces? No, that doesn't make sense. Wait, maybe the diagram is a regular triangular pyramid where the base is an equilateral triangle with side 5 yd, and the lateral faces are also equilateral triangles with side 5 yd. Wait, but the base's altitude is 4.3 yd, which is approximately \( \frac{\sqrt{3}}{2} \times 5 \approx 4.33 \), so that's consistent. So the base area is \( \frac{1}{2} \times 5 \times 4.3 = 10.75 \) square yards. Each lateral face: since it's an equilateral triangle with side 5 yd, area is \( \frac{1}{2} \times 5 \times 4.3 \)? Wait, no, that would be the same as the base, but that's only if the lateral faces are also triangles with base 5 and height 4.3. Wait, maybe the pyramid is a regular tetrahedron, so all four faces are equilateral triangles. Then the surface area would be 4 times the area of one equilateral triangle.

Wait, let's calculate the area of one equilateral triangle with side 5 yd. The formula for the area of an equilateral triangle is \( A = \frac{\sqrt{3}}{4} s^2 \). Plugging in \( s = 5 \):

\( A = \frac{\sqrt{3}}{4} \times 25 \approx \frac{1.732}{4} \times 25 \approx 0.433 \times 25 \approx 10.825 \) square yards per face.

Then total surface area (4 faces) would be \( 4 \times 10.825 = 43.3 \) square yards.

Alternatively, if the base is a triangle with base 5 and height 4.3, and the three lateral faces are triangles with base 5 and height 5 (since the edge length is 5), then:

Base area: \( \frac{1}{2} \times 5 \times 4.3 = 10.75 \)

Each lateral face area: \( \frac{1}{2} \times 5 \times 5 = 12.5 \)

Three lateral faces: \( 3 \times 12.5 = 37.5 \)

Total surface area: \( 10.75 + 37.5 = 48.25 \)

But that doesn't match the equilateral triangle area. Wait, maybe the diagram is a regular triangular pyramid where the base is an equilateral triangle with side 5, and the lateral faces are also equilateral triangles with side 5, so all four faces are equilateral. Then the surface area is 4 times the area of one equilateral triangle.

Wait, the given height of the base triangle is 4.3, which is approximately \( \frac{\sqrt{3}}{2} \times 5 \approx 4.33 \), so that's a rounded value. So the area of one equilateral triangle is \( \frac{1}{2} \times 5 \times 4.3 = 10.75 \) (using the given 4.3 instead of the exact \( \frac{\sqrt{3}}{2} \times 5 \)). Then four faces would be \( 4 \times 10.75 = 43 \) square yards? Wait, no, 4 times 10.75 is 43. But let's check:

If the base is an equilateral triangle with side 5 and height 4.3, area is \( \frac{1}{2} \times 5 \times 4.3 = 10.75 \).

Each lateral face: since the pyramid is regular, the lateral faces are also equilateral triangles with side 5, so their area is also \( 10.75 \) (using the same height 4.3, assuming the slant height is equal to the base height, which might be the case here). Then total surface area is \( 10.75 + 3 \times 10.75 = 4 \times 10.75 = 43 \) square yards. Wait, but 4.3 times 5 is 21.5, divided by 2 is 10.75. Then 4 times that is 43. But maybe the problem is considering the lateral faces as triangles with base 5 and height 5? Let's see:

If lateral faces have base 5 and height 5, area is \( \frac{1}{2} \times 5 \times 5 = 12.5 \). Three of them: 37.5. Base area: 10.75. Total: 48.25. But which is correct?

Wait, the diagram shows the edges from the apex to the base vertices as 5 yd, and the base edges as 5 yd. So all edges are 5 yd, making it a regular tetrahedron. In a regular tetrahedron, all four faces are equilateral triangles. The height (altitude) of each equilateral triangle with side 5 is \( \frac{\sqrt{3}}{2} \times 5 \approx 4.33 \), which is approximately 4.3 yd (rounded). So the area of each face is \( \frac{1}{2} \times 5 \times 4.3 \approx 10.75 \) square yards. Therefore, total surface area is 4 times that: \( 4 \times 10.75 = 43 \) square yards. Wait, but 4.3 times 5 is 21.5, divided by 2 is 10.75. Then 4 times 10.75 is 43. But let's check with the exact value: \( \frac{\sqrt{3}}{4} \times 5^2 \times 4 = \sqrt{3} \times 25 \approx 43.30 \), which is approximately 43.3 square yards. The given 4.3 is a rounded value of \( \frac{\sqrt{3}}{2} \times 5 \approx 4.33 \), so using 4.3 gives \( \frac{1}{2} \times 5 \times 4.3 = 10.75 \) per face, times 4 is 43. But maybe the problem is considering the base as one triangle and three lateral faces, each with base 5 and height 5? No, that would be if the lateral faces are isosceles triangles with height 5, but the edge length is 5, so the slant height (height of the lateral face) would be different. Wait, maybe the diagram is a triangular pyramid with a base that is an equilateral triangle (side 5) and three lateral faces that are congruent triangles with base 5 and height 5. Let's calculate both ways:

1. All faces are equilateral triangles (regular tetrahedron):
- Area per face: \( \frac{1}{2} \times 5 \times 4.3 = 10.75 \)
- Total surface area: \( 4 \times 10.75 = 43 \)

2. Base is equilateral (area 10.75) and lateral faces are isosceles with base 5 and height 5:
- Lateral face area: \( \frac{1}{2} \times 5 \times 5 = 12.5 \)
- Three lateral faces: \( 3 \times 12.5 = 37.5 \)
- Total: \( 10.75 + 37.5 = 48.25 \)

But the diagram shows the height of the base triangle as 4.3, and the edges from apex to base vertices as 5. So the lateral faces are triangles with two sides 5 (the edges) and base 5, so they are equilateral, so their height should also be 4.3 (since all sides are equal). Therefore, the first calculation is correct: total surface area is 4 times the area of one equilateral triangle with side 5 and height 4.3.

So:

Step1: Calculate area of one triangular face

The base of each triangle is 5 yd, and the height is 4.3 yd. The area of a triangle is \( \frac{1}{2} \times \text{base} \times \text{height} \).
\( A_{\text{face}} = \frac{1}{2} \times 5 \times 4.3 = 10.75 \) square yards.

Step2: Calculate total surface area

A triangular pyramid (tetrahedron) has 4 triangular faces. So total surface area \( A_{\text{total}} = 4 \times A_{\text{face}} \).
\( A_{\text{total}} = 4 \times 10.75 = 43 \) square yards. Wait, but 4.3 is a rounded value, so maybe the exact value is 43.3. Let's check with the exact height: \( \frac{\sqrt{3}}{2} \times 5 \approx 4.330 \). Then \( A_{\text{face}} = \frac{1}{2} \times 5 \times 4.330 \approx 10.825 \). Then total surface area \( 4 \times 10.825 \approx 43.3 \) square yards. The given 4.3 is probably a rounded value, so the answer is approximately 43.3 or 43 (if using 4.3).

Wait, let's do the calculation with 4.3:

\( \frac{1}{2} \times 5 \times 4.3 = 10.75 \)

\( 4 \times 10.75 = 43 \)

But if we use the more accurate 4.33:

\( \frac{1}{2} \times 5 \times 4.33 \approx 10.825 \)

\( 4 \times 10.825 \approx 43.3 \)

The problem gives 4.3, so we should use that. Therefore, the total surface area is 43 square yards (or 43.3 if we consider the more accurate height).

Wait, maybe the pyramid is a triangular pyramid with a base that is an equilateral triangle and three lateral faces that are congruent triangles, but the base is one face and the lateral faces are three. Wait, no, a triangular pyramid (tetrahedron) has 4 faces: 1 base, 3 lateral. So if the base is an equilateral triangle with side 5 and height 4.3, and the three lateral faces are also equilateral triangles with side 5 and height 4.3, then total surface area is 4 * (1/2 * 5 * 4.3) = 4 * 10.75 = 43.

Yes, that makes sense. So the steps are:

1. Calculate area of one triangular face: \( \frac{1}{2} \times 5 \times 4.3 = 10.75 \)
2. Multiply by 4 (since there are 4 faces): \( 4 \times 10.75 = 43 \)

Answer:

43 (or 43.3 if using more precise height, but based on given 4.3, it's 43)