Surface Area & Volume
Grades 7–9 Geometry 3D
Flat shapes get perimeter and area; solid shapes get surface area (the wrapping paper) and volume (the space inside). Answers in units² and units³ respectively.
Boxes and cylinders
Box (l × w × h):
Volume V = l × w × h
Surface area SA = 2(lw + lh + wh) ← three pairs of faces
Cylinder (r, h):
Volume V = πr²h ← circle area × height
Box 5 × 3 × 2:
V = 5 × 3 × 2 = 30 cm³
SA = 2(15 + 10 + 6) = 2 × 31 = 62 cm²
The cube shortcut
Side s: V = s³ and SA = 6s². A 4 cm cube holds 64 cm³ of stuff
under 96 cm² of skin.
Watch out
Doubling a box's sides does not double the volume —
2× every dimension multiplies volume by 2³ = 8. Scaling bites.
Practice
Box 4 × 3 × 2 — volume = ___ cm³
Same box — surface area = ___ cm²
Cube with side 5 — volume = ___ cm³
Doubling every side of a cube multiplies volume by:
Next: the coordinate plane · Back: perimeter & area