Geometric Solids — Surface Area and Volume

Surface area and volume are different measures. A cube of side 3 has SA 54, V 27. Different units, different meaning. The formulas are the control. Indirect aim: preparation for calculus, practical measurement, biology (cell SA/V ratio).

• Framework: Montessori
• Unit: Sensorial
• Learning area: Geometric Cabinet
• Level: Ages 9-11

Objectives

• Compute surface area of each geometric solid
• Compute volume of each solid
• Apply formulas to real objects
• Understand SA to V ratio

Materials

• Geometric solids (cube, cylinder, sphere, cone, pyramid)
• Rulers, calculators
• Formula reference chart
• Recording paper

Instructions

1. Take cube side 3: SA = 6×9 = 54; V = 27. 2. Cylinder r=2 h=5: SA = 2πr² + 2πrh = 8π + 20π = 28π; V = πr²h = 20π. 3. Sphere r=3: SA = 4πr² = 36π; V = (4/3)πr³ = 36π. 4. Cone r=3 h=4: SA = πr² + πrl (l=5) = 9π + 15π = 24π; V = (1/3)πr²h = 12π. 5. Square pyramid base 4 h=6: SA = 16 + ½×4×4×l (l=~6.3) ≈ 16+50.4 = 66.4; V = (1/3)×16×6 = 32. 6. Apply to a real can: compute SA and V. 7. Discuss SA:V ratio. 8. Research why small animals lose heat faster.

Educator tip

SA:V ratio explains why cells are small and why mice eat constantly. Connect to biology.

Assessment

Observe formula application, correct units, real-world transfer.

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