Triangular pyramid with a right triangle base, used in a math practice problem.

Let’s solve each problem one by one, step by step. We’ll calculate surface areas for different 3D shapes using the right formulas.

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Problem 1: Cone (wrapping paper = lateral surface area)
Given: slant height = 12 inches, radius = 6 inches
Formula for lateral surface area of a cone: π × r × l
→ π × 6 × 12 = 72π ≈ 72 × 3.14 = 226.08 square inches

✔ Final Answer for #1: 226.08 in²

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Problem 2: Cylinder (plastic wrap = lateral surface area only — since it’s a roll, we don’t cover top/bottom)
Given: height = 11 inches, diameter = 6 → radius = 3 inches
Formula for lateral surface area of cylinder: 2 × π × r × h
→ 2 × π × 3 × 11 = 66π ≈ 66 × 3.14 = 207.24 square inches

✔ Final Answer for #2: 207.24 in²

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Problem 3: Rectangular box (fabric on all sides = total surface area)
Dimensions: length = 14, width = 3.5, height = 2.5
Formula: 2(lw + lh + wh)
→ lw = 14 × 3.5 = 49
→ lh = 14 × 2.5 = 35
→ wh = 3.5 × 2.5 = 8.75
Sum = 49 + 35 + 8.75 = 92.75
Multiply by 2: 92.75 × 2 = 185.5 square inches

✔ Final Answer for #3: 185.5 in²

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Problem 4: Cylinder (wrapping paper = total surface area? Wait — “wrap a present packed in a cylinder” — usually means covering the whole thing, so include top and bottom unless specified otherwise.)
But let’s check context: In Problem 2, they said “wrap the roll” and meant lateral only. Here, it says “wrap a present packed in a cylinder” — likely full surface area. But sometimes wrapping paper doesn’t cover ends if it’s like a gift tube... Hmm. Let’s assume total surface area to be safe, but note: many textbooks treat “wrapping a cylinder-shaped present” as including bases.

Radius = 4, height = 7
Total SA = 2πr² + 2πrh
→ 2π(4)² = 2π×16 = 32π
→ 2π×4×7 = 56π
Total = 32π + 56π = 88π ≈ 88 × 3.14 = 276.32 square inches

Wait — actually, re-reading: “wrap a present packed in a cylinder” — if it’s a solid present inside, you’d wrap the outside completely. So yes, total surface area.

✔ Final Answer for #4: 276.32 in²

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Problem 5: Triangular prism (paint all sides = total surface area)
It has two triangular bases and three rectangular sides.

Triangle base: equilateral? No — it says “equal on all three sides” → so equilateral triangle with side 8 ft, and height of triangle = 6.9 ft.

Area of one triangle = (base × height)/2 = (8 × 6.9)/2 = 55.2 / 2 = 27.6 ft²
Two triangles: 27.6 × 2 = 55.2 ft²

Now rectangles: each rectangle is side of triangle × height of prism (which is 10 ft). Since all 3 sides are 8 ft, each rectangle is 8 × 10 = 80 ft²
Three rectangles: 80 × 3 = 240 ft²

Total paint needed = 55.2 + 240 = 295.2 ft²

✔ Final Answer for #5: 295.2 ft²

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Problem 6: Rectangular pyramid (paper to cover all faces = total surface area)
Base: 7 in long, 2 in wide → area = 7 × 2 = 14 in²

Lateral faces: 4 triangles. Two pairs:
- Two triangles with base = 7 in, slant height = 7 in → area each = (7×7)/2 = 24.5 → two of them = 49
- Two triangles with base = 2 in, slant height = 7 in → area each = (2×7)/2 = 7 → two of them = 14

Wait — is slant height the same for all faces? The problem says “the slant height is 7 inches” — probably meaning for all triangular faces. But in a rectangular pyramid, there are two different slant heights unless it’s a square base. However, the problem states “slant height is 7 inches” without specifying which face — so we must assume it applies to all four triangular faces? That would be unusual, but perhaps it’s simplified.

Actually, looking again: “Its base has a length of 7 inches and a width of 2 inches. The slant height is 7 inches.” It doesn’t specify which slant height, so maybe they mean the slant height for the faces along the length and width are both 7? That can’t be geometrically accurate, but for this level, perhaps they want us to use 7 for all.

Alternatively, maybe “slant height” refers to the height of each triangular face from base to apex — and since it’s not specified per face, we’ll assume all four triangular faces have slant height 7.

So:

Two triangles with base 7: area = 2 × (1/2 × 7 × 7) = 49
Two triangles with base 2: area = 2 × (1/2 × 2 × 7) = 14
Total lateral = 49 + 14 = 63
Plus base = 14
Total = 63 + 14 = 77 in²

But wait — is the base included? The problem says “cover all the faces” — yes, so include base.

✔ Final Answer for #6: 77 in²

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Problem 7: Triangular pyramid (tetrahedron?) — but it has a right triangle base and slant height given.

Base: right triangle with sides 5, 12, 13 → that’s a right triangle (5-12-13), so legs 5 and 12, hypotenuse 13.

Slant height = 14 feet — but slant height for which face? The hint says “add the given measurements to the below picture” — but no picture shown. However, typically in such problems, “slant height” refers to the height of each triangular face from the base edge to the apex.

But here’s the issue: a triangular pyramid has 4 faces: 1 base and 3 lateral faces. If the base is a right triangle, and the pyramid is not regular, the three lateral faces may have different areas.

However, the problem gives only ONE slant height: 14 feet. This suggests that perhaps all three lateral faces have the same slant height? Or maybe it’s the slant height corresponding to each base edge?

Looking at standard interpretation: Often in these problems, when they say “a triangular pyramid with slant height X”, and give a base, they mean that each lateral face has a height (from base edge to apex) of X — even if the base edges are different lengths.

So let’s assume: each of the three triangular lateral faces has a height (slant height) of 14 feet, and their bases are the three sides of the base triangle: 5, 12, and 13 feet.

Then:

Area of lateral face 1 (base 5): (1/2) × 5 × 14 = 35
Area of lateral face 2 (base 12): (1/2) × 12 × 14 = 84
Area of lateral face 3 (base 13): (1/2) × 13 × 14 = 91
Total lateral area = 35 + 84 + 91 = 210

Base area: right triangle with legs 5 and 12 → (1/2) × 5 × 12 = 30

Total surface area = 210 + 30 = 240 ft²

This makes sense.

✔ Final Answer for #7: 240 ft²

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Final Answers Summary:

1. 226.08 in²
2. 207.24 in²
3. 185.5 in²
4. 276.32 in²
5. 295.2 ft²
6. 77 in²
7. 240 ft²

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Final Answer:
1. 226.08
2. 207.24
3. 185.5
4. 276.32
5. 295.2
6. 77
7. 240