Level 2 · Enrichment

Reveal Math · Unit 5 · Supplemental

Geometry & Measurement

Grade 6 · Standards 6.G.A.1–4 — composite area, nets, volume & design reasoning

Name: Date:

Challenge Problems

Directions: Solve and show your strategy with correct units. For each "Explain" prompt, justify your reasoning in a complete sentence.
  1. An L-shaped room is made of a 10 ft × 6 ft rectangle joined to a 4 ft × 3 ft rectangle. Find the total floor area. Composite
    Explain: how does splitting the shape help you?
  2. A triangle has area 36 cm² and base 9 cm. Find its height. Reasoning
    Explain how you reversed the area formula.
  3. A rectangular prism has volume 120 cm³, length 5 cm, and width 4 cm. Find its height. Multi-step
  4. Find the surface area of a box 4 cm × 3 cm × 2 cm. Multi-step
    Explain why opposite faces come in matching pairs.
  5. A trapezoid is split into a rectangle (6 × 4) and a triangle (base 3, height 4). Find the total area. Composite
  6. Two boxes have the same volume of 48 cm³. Box A is 2 × 4 × 6; Box B is 1 × 6 × 8. Which uses less material (less surface area)? Reasoning
    Explain what this tells a packaging designer.
  7. A fish tank is 30 cm × 20 cm × 25 cm. How many liters of water does it hold? (1 liter = 1,000 cm³) Real-world
  8. A parallelogram and a triangle have the same base and height. How do their areas compare? Reasoning
    Explain using the formulas.
  9. Design two different rectangular prisms that each have a volume of exactly 64 cm³, using whole-number edges. Open-ended
  10. A wall is 12 ft × 8 ft. Paint covers 40 ft² per can. How many full cans must you buy to paint it? Real-world
    Explain why you must round up.

Stretch Investigation

Real-world application: Design a storage box that must hold exactly 1,000 cm³. Choose whole-number length, width, and height. Then compute the surface area of your design and one other design with the same volume. Build a net sketch of your box, and write a recommendation explaining which design would cost less cardboard to manufacture and why.

Answer Key

  1. (10 × 6) + (4 × 3) = 60 + 12 = 72 ft².
  2. Area = ½ × b × h → 36 = ½ × 9 × h → h = 72 ÷ 9 = 8 cm.
  3. Volume = l × w × h → 120 = 5 × 4 × h → h = 120 ÷ 20 = 6 cm.
  4. 2(4×3) + 2(4×2) + 2(3×2) = 24 + 16 + 12 = 52 cm². Opposite faces are congruent rectangles.
  5. (6 × 4) + (½ × 3 × 4) = 24 + 6 = 30 square units.
  6. Box A surface = 2(8+12+24) = 88 cm²; Box B = 2(6+8+48) = 124 cm². Box A uses less material; compact shapes save packaging.
  7. 30 × 20 × 25 = 15,000 cm³ ÷ 1,000 = 15 liters.
  8. The triangle's area is exactly half the parallelogram's (½bh vs bh).
  9. Answers vary, e.g., 4 × 4 × 4 and 2 × 4 × 8 (both 64 cm³).
  10. Wall area 12 × 8 = 96 ft²; 96 ÷ 40 = 2.4 → buy 3 cans (can't buy a partial can).
  11. Stretch: answers vary; full credit needs two valid 1,000 cm³ designs, correct surface areas, a net sketch, and a justified recommendation.