Surface Area and Volume Formulas for All 3D Shapes

Every 3D solid in standard geometry — cube, rectangular prism (box), cylinder, sphere, cone, square pyramid — has a one-line volume formula and a one-line surface area formula. Memorize these and you’ve covered 95% of school 3D problems. This guide collects them all in one place with worked examples.

Quick Reference Table

Cube

The simplest 3D shape — all 12 edges equal, 6 square faces.

• Volume: V = s³  (side cubed)
• Surface Area: SA = 6 × s²  (6 faces, each s × s)
• Space Diagonal: d = s√3  (3D Pythagorean for cube)

Example: a cube with side 4 cm: V = 64 cm³, SA = 96 cm², diagonal = 4√3 ≈ 6.93 cm.

Rectangular Prism (Box)

Length l, width w, height h. Most common 3D solid in real life (rooms, boxes, swimming pools).

• Volume: V = l × w × h
• Surface Area: SA = 2(lw + lh + wh)  (3 pairs of opposite faces)
• Space Diagonal: d = √(l² + w² + h²)

Example: a box 8 × 5 × 3: V = 120, SA = 2(40 + 24 + 15) = 158, d = √(64+25+9) = √98 ≈ 9.90.

Cylinder

Two circular bases (radius r) connected by a curved lateral surface, height h.

• Volume: V = π × r² × h  (base area × height)
• Lateral Surface Area: LSA = 2πrh  (unrolled rectangle: width = circumference, height = h)
• Total Surface Area: SA = 2πr² + 2πrh = 2πr(r + h)

Example: cylinder r = 3, h = 10: V = π × 9 × 10 = 90π ≈ 282.74, SA = 2π × 9 + 2π × 30 = 78π ≈ 245.04.

Sphere

The simplest 3D shape — defined by radius alone. Volume and surface area both depend only on r.

• Volume: V = (4/3) × π × r³
• Surface Area: SA = 4 × π × r²  (equivalent to 4 great circles)

Example: a basketball with diameter 24 cm has r = 12. V = (4/3)π × 1728 = 2304π ≈ 7238 cm³, SA = 4π × 144 = 576π ≈ 1810 cm².

Cone

One circular base (radius r) tapering to a single point, height h. The trickiest of the bunch because surface area uses the SLANT height (not the vertical height).

• Volume: V = (1/3) × π × r² × h  (exactly 1/3 of the equivalent cylinder)
• Slant Height: l = √(r² + h²)  (Pythagorean — connects rim to apex along the surface)
• Lateral Surface Area: LSA = π × r × l
• Total Surface Area: SA = πr² + πrl = πr(r + l)

Example: cone r = 6, h = 8. Slant l = √(36 + 64) = √100 = 10. V = (1/3)π × 36 × 8 = 96π ≈ 301.59, SA = π × 6 × (6 + 10) = 96π ≈ 301.59 cm².

Square Pyramid

Square base (side b) tapering to a point, vertical height h.

• Volume: V = (1/3) × b² × h
• Slant Height (face): l = √(h² + (b/2)²)
• Surface Area: SA = b² + 4 × (½ × b × l) = b² + 2bl

Why Volume of Cone = (1/3) × Cylinder?

This is one of the cooler facts in geometry: a cone and a cylinder with the same base and same height — the cone has exactly 1/3 the volume of the cylinder. You can verify experimentally with water: fill a cone-shaped container, pour into a cylinder of same dimensions — it takes exactly 3 cone-fulls. Same applies to pyramid vs prism with same base + height.

Why Volume of Sphere = (4/3)πr³?

Two ways to derive: (1) calculus integration of disk slices, (2) Cavalieri’s principle comparing to a cylinder with cones removed. The “(4/3)” comes from the integral ∫(πr² − πx²) dx evaluated from −r to r.

Real-World Applications

• Swimming pool volume: rectangular prism formula → l × w × average depth
• Tank capacity: cylinder volume → πr²h
• Ice cream cone: cone for the cone, half-sphere for the scoop on top
• Box surface area for packaging: 2(lw + lh + wh) gives total cardboard needed
• Painting a sphere (e.g., a globe): 4πr² gives exact paint coverage area

Common Mistakes

• Confusing slant height and vertical height in cone problems. Slant l is along the surface; vertical h is the perpendicular drop from apex to base center. Use l = √(r² + h²) to convert.
• Forgetting the (1/3) for cone/pyramid volume. The base area × height gives the cylinder/prism volume; cone/pyramid is exactly 1/3 of that.
• Using diameter instead of radius. All these formulas use radius, NOT diameter. d = 2r — divide by 2 first.
• Forgetting one of the surface area pieces. Cylinder SA has TWO circular bases + ONE curved side. Cone SA has ONE base + ONE curved lateral. Don’t drop a term.

For instant calculation, our Sphere/Cylinder/Cone Calculator handles all three, and Cube/Box Calculator handles boxes. For the full set of 2D and 3D formulas in one place, see our Complete Geometry Formulas Reference.

FAQ

How do I find volume of an irregular 3D shape? Three approaches: (1) decompose into standard solids and sum, (2) water displacement (immerse in water and measure displaced volume), (3) integration (calculus) for shapes defined by curves.

What units does volume use? Cubic units (cm³, m³, in³, ft³). 1 m³ = 1,000 liters = 1,000,000 cm³.

What units does surface area use? Squared units (cm², m², in²). Same units as 2D area.

Which formula does Google use for “volume of a cylinder”? The standard V = πr²h, exactly the same as we list. There’s no other formula — this one is universal.