TEAS Ready📏 Math · Measurement and Data · Lesson 6 of 7 · 6 min
Volume and surface area
🧊 How much a container holds is volume; how much material wraps it is surface area. Both show up on the TEAS.
💡 Key idea
Volume = base area × height in cubic units; surface area = sum of all faces in square units.
🧠 The idea
VOLUME is the space inside a solid, measured in cubic units (cm³, ft³). Rectangular prism (box): V = l × w × h. Cube: V = s³. Cylinder: V = πr²h, the area of the circular base times the height. Use π ≈ 3.14. SURFACE AREA is the total area of all outside faces, in square units. A box has six faces in three matching pairs: SA = 2lw + 2lh + 2wh. A cube has six equal squares: SA = 6s². Unit sense helps: 1 cm³ of liquid equals 1 mL, so a 10 cm × 10 cm × 10 cm box holds 1,000 cm³ = 1 L. If one dimension of a box doubles, its volume doubles; if every dimension doubles, the volume becomes 2 × 2 × 2 = 8 times as large.
✏️ Worked example
A cylindrical water tank has a radius of 2 m and a height of 5 m. How much water can it hold?
Step 1: Area of the circular base = πr² = 3.14 × 2² = 3.14 × 4 = 12.56 m².
Step 2: Volume = base area × height = 12.56 × 5 = 62.8 m³.
Answer: about 62.8 cubic meters.
📌 Takeaways
- Box V = l × w × h; cube V = s³; cylinder V = πr²h.
- Box SA = 2lw + 2lh + 2wh; cube SA = 6s².
- 1 cm³ = 1 mL; doubling every dimension multiplies volume by 8.
📖 Terms in this lesson
Volume: The amount of space a solid takes up, in cubic units; for example, a 2 × 3 × 4 cm box has a volume of 24 cm³.
✅ Check yourself
Question 1 of 3
A box is 4 cm long, 3 cm wide and 2 cm tall. What is its volume?
- 52 cm³
- 12 cm³
- 9 cm³
- 24 cm³
Show the answer and rationale
Answer: D
V = l × w × h = 4 × 3 × 2 = 24 cm³. Nine adds the dimensions, 52 is the surface area (2·12 + 2·8 + 2·6), and 12 multiplies only length × width.
Question 2 of 3
What is the volume of a cylinder with a radius of 2 in and a height of 5 in? (Use π ≈ 3.14.)
- 125.6 in³
- 31.4 in³
- 20 in³
- 62.8 in³
Show the answer and rationale
Answer: D
V = πr²h = 3.14 × 4 × 5 = 62.8 in³. The value 31.4 uses r instead of r², 125.6 doubles the answer, and 20 leaves out π.
Question 3 of 3
What is the surface area of a cube with edges of 3 ft?
- 27 ft²
- 36 ft²
- 162 ft²
- 54 ft²
Show the answer and rationale
Answer: D
Each face is 3 × 3 = 9 ft², and a cube has 6 faces: 6 × 9 = 54 ft². Twenty-seven is the volume (3³), 36 counts only 4 faces, and 162 cubes the edge instead of squaring it (6 × 3³).
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