Solved Unit: Surface Area Homework 4 Nome Al Date Pd ob | Chegg.com - Free Printable

Let’s solve each problem step by step. We’ll calculate lateral surface area (area of the triangular faces only) and total surface area (lateral + base area).

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Problem 1: Square Pyramid

Base is a square with side = 4 cm
Slant height of each triangle = 12 cm

→ There are 4 identical triangular faces.

Lateral Surface Area = 4 × (½ × base × slant height)
= 4 × (½ × 4 × 12)
= 4 × (24)
= 96 cm²

Base Area = side² = 4 × 4 = 16 cm²

Total Surface Area = Lateral + Base = 96 + 16 = 112 cm²

✔ Check: All steps correct. Triangles use slant height, not vertical height — given as 12 cm, so we’re good.

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Problem 2: Regular Triangular Pyramid (Tetrahedron)

All faces are equilateral triangles? Wait — diagram shows base triangle with side 15 cm, and inner triangle labeled “height” 13 cm? Actually, looking at the diagram: it's a regular triangular pyramid meaning all faces are congruent equilateral triangles? But wait — the label says “15 cm” on one side of the base triangle, and inside there’s a smaller triangle with height 13 cm? That doesn’t make sense for an equilateral triangle.

Wait — re-examining: The diagram likely shows a regular triangular pyramid, which means the base is an equilateral triangle, and the three lateral faces are also congruent isosceles triangles. But in this case, since it’s called “regular”, sometimes it implies all faces are equilateral — but then the height wouldn’t be 13 cm for side 15.

Actually, let’s read carefully: It says “regular triangular pyramid”. In geometry, a *regular* triangular pyramid usually means the base is an equilateral triangle and the apex is directly above the centroid — but the lateral faces may not be equilateral unless specified.

But here, the diagram labels one side of the base as 15 cm, and inside the base triangle, there’s a line labeled “13 cm” — that must be the height of the base triangle, not the slant height of the lateral face.

Wait — no! Look again: The 13 cm is drawn from the center to a vertex? Or is it the height of the lateral face?

Actually, in many textbooks, when they show a regular triangular pyramid like this, and label a length inside the base triangle as “height”, it’s often the altitude of the base triangle.

But for lateral surface area, we need the slant height of the lateral faces — which is NOT given directly.

Hold on — perhaps I misread. Let me think differently.

In problem 2, the diagram shows a large triangle (the base?) with side 15 cm, and inside it, a smaller inverted triangle with height 13 cm? That doesn't help.

Alternatively — maybe the 13 cm is the slant height of the lateral faces? Because in pyramids, the slant height is usually marked on the lateral face.

Looking at standard problems: If it’s a regular triangular pyramid with base side 15 cm, and if the lateral faces are isosceles triangles with base 15 cm and height (slant height) 13 cm, then:

That makes sense — because otherwise we can’t compute without more info.

So assuming:
- Base is equilateral triangle with side 15 cm
- Each lateral face is a triangle with base 15 cm and height (slant height) = 13 cm

Then:

Lateral Surface Area = 3 × (½ × base × slant height)
= 3 × (½ × 15 × 13)
= 3 × (97.5)
= 292.5 cm²

Now, Base Area: equilateral triangle with side 15 cm
Area = (√3 / 4) × side² ≈ (1.732 / 4) × 225 ≈ 0.433 × 225 ≈ 97.425 cm²

But wait — do we need exact or decimal? Since other problems use whole numbers, maybe we should keep it symbolic? But the problem doesn’t specify.

Alternatively — perhaps the 13 cm is NOT the slant height? Maybe it’s the height of the base triangle?

If 13 cm is the height of the base triangle (equilateral), then for equilateral triangle, height h = (√3/2) × side → so side = h × 2 / √3 = 13 × 2 / 1.732 ≈ 15.01 — close to 15. So yes, 13 cm is likely the height of the base triangle.

But then what is the slant height of the lateral faces? Not given!

This is ambiguous.

Wait — look back at the diagram description: In problem 2, it says “regular triangular pyramid” and shows a triangle with side 15 cm, and inside it, a line labeled “13 cm” going from center to midpoint of a side? Or from apex to base?

Actually, in many such diagrams, when they draw a pyramid net or front view, and put a number inside the base triangle, it’s often the apothem or height of the base.

But for lateral surface area, we need the slant height — which is the height of the lateral triangular face.

Since it’s not given, perhaps in this context, the 13 cm IS the slant height? Because otherwise we can’t solve.

Let me check similar problems online or standard curriculum.

Alternatively — perhaps “regular triangular pyramid” here means all edges equal? Then it would be a tetrahedron with all sides 15 cm. Then each face is equilateral triangle with side 15.

Then area of one face = (√3/4)*15² = (√3/4)*225 ≈ 97.425 cm²

Lateral surface area = 3 faces = 3 * 97.425 ≈ 292.275 cm²

Total surface area = 4 faces = 4 * 97.425 ≈ 389.7 cm²

But the diagram has a label “13 cm” inside — which doesn’t match the height of equilateral triangle with side 15: actual height = (√3/2)*15 ≈ 12.99 ≈ 13 cm. Oh! So 13 cm is the height of the equilateral triangle face.

So yes — each face is equilateral triangle with side 15 cm, height approximately 13 cm (exactly (√3/2)*15 = (15√3)/2 ≈ 12.99, rounded to 13).

So for calculation purposes, we can use the formula for equilateral triangle area.

But the problem might expect us to use the given 13 cm as the height for area calculation.

Let’s do both ways.

Option A: Use given 13 cm as height of each triangular face (since it’s labeled inside, and matches approximate height).

Then area of one face = ½ * base * height = ½ * 15 * 13 = 97.5 cm²

Then:

Lateral Surface Area = 3 * 97.5 = 292.5 cm²

Base Area = same as one face? No — in a triangular pyramid, the base is one face, lateral are three others. But if it’s regular and all faces are congruent, then base is also 97.5 cm².

Total Surface Area = 4 * 97.5 = 390 cm²

And 13 cm is approximately the height, so using it is fine for school level.

Moreover, in the diagram, it’s labeled as “13 cm” inside the triangle, likely indicating the height of that triangle.

So I’ll go with that.

Thus:

Problem 2:
Lateral SA = 3 × (½ × 15 × 13) = 3 × 97.5 = 292.5 cm²
Total SA = 4 × 97.5 = 390 cm² (since all four faces are identical in a regular tetrahedron)

But is a "regular triangular pyramid" always a tetrahedron with all faces equilateral? Yes, typically.

So final answer for #2: Lateral 292.5, Total 390

But let’s confirm with exact value: if side=15, area of equilateral triangle = (√3/4)*225 = (225√3)/4

Lateral SA = 3 * (225√3)/4 = (675√3)/4 ≈ (675*1.732)/4 ≈ 1169.1/4 ≈ 292.275 — close to 292.5

Using 13 cm gives 97.5 per face, which is very close (actual is ~97.425), so for homework, using 13 is acceptable.

I think the problem intends for us to use 13 cm as the height of each triangular face.

So proceeding.

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Problem 3: Square Pyramid

Base side = 6 cm
Slant height = 16 cm (given on the lateral face)

→ 4 triangular faces

Lateral Surface Area = 4 × (½ × base × slant height)
= 4 × (½ × 6 × 16)
= 4 × (48)
= 192 cm²

Base Area = 6 × 6 = 36 cm²

Total Surface Area = 192 + 36 = 228 cm²

✔ Straightforward.

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Problem 4: Rectangular Pyramid

Base is rectangle: 8 cm by 6 cm
Two pairs of triangular faces:

- Two triangles with base 8 cm, slant height 6 cm (given on those faces)
- Two triangles with base 6 cm, slant height 6.5 cm (given)

Note: The diagram shows:
- For the 8 cm side, the slant height is 6 cm
- For the 6 cm side, the slant height is 6.5 cm

So:

Lateral Surface Area = [2 × (½ × 8 × 6)] + [2 × (½ × 6 × 6.5)]

Calculate each part:

First pair: 2 × (½ × 8 × 6) = 2 × 24 = 48 cm²

Second pair: 2 × (½ × 6 × 6.5) = 2 × (19.5) = 39 cm²

Total Lateral SA = 48 + 39 = 87 cm²

Base Area = 8 × 6 = 48 cm²

Total Surface Area = 87 + 48 = 135 cm²

✔ Correct.

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Problem 5: Dana has total surface area of rectangular pyramid, wants lateral surface area.

How? She subtracts the area of the base from the total surface area.

Because:
Total Surface Area = Lateral Surface Area + Base Area
So,
Lateral Surface Area = Total Surface Area - Base Area

She needs to know the dimensions of the base to find its area, or if the base area is given or can be calculated.

The question says: “Describe how Dana can use the given information...”

Assuming she knows the total surface area and the dimensions of the base (or the base area), she can subtract.

Answer: Dana can subtract the area of the rectangular base from the total surface area to get the lateral surface area.

More precisely:
Lateral Surface Area = Total Surface Area – (length × width of base)

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Now, compiling all answers.

Final Answers:

1. Lateral: 96 cm², Total: 112 cm²
2. Lateral: 292.5 cm², Total: 390 cm²
3. Lateral: 192 cm², Total: 228 cm²
4. Lateral: 87 cm², Total: 135 cm²
5. Subtract the area of the base from the total surface area.

For problem 2, to be precise, since 13 cm is given and likely intended as the height of each triangular face, we'll use that.

Some might argue that in a regular triangular pyramid, if base is equilateral with side 15, and it's regular, then lateral faces are also equilateral, so height is (√3/2)*15≈12.99, so using 13 is fine.

I think it's acceptable.

So final boxed answers.

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Final Answer:
1. Lateral Surface Area: 96 cm², Total Surface Area: 112 cm²
2. Lateral Surface Area: 292.5 cm², Total Surface Area: 390 cm²
3. Lateral Surface Area: 192 cm², Total Surface Area: 228 cm²
4. Lateral Surface Area: 87 cm², Total Surface Area: 135 cm²
5. Dana can subtract the area of the rectangular base from the total surface area to find the lateral surface area. Specifically, Lateral Surface Area = Total Surface Area − (length × width of base).