Volume & Surface Area of 3D Shapes

📋 GED Formula Sheet Connection

📋 The GED Gives You a Formula Sheet

You do not need to memorize every geometry formula. The GED provides a formula sheet during the test.

Your job is to: identify the shape → find the right formula → substitute correctly → use your calculator.

→ View the official GED Formula Sheet (PDF)

⚖️ Volume vs Surface Area

This is the most important distinction in 3D geometry. Every problem is either asking about the inside or the outside.

📦 Volume

Measures the space inside
Used for: filling, holding, containing
Units: cubic (cm³, ft³, in³)

Examples: water in a tank, cereal in a box, concrete in a mold

🎁 Surface Area

Measures the outside covering
Used for: wrapping, painting, covering
Units: square (cm², ft², in²)

Examples: wrapping paper, paint needed, tent fabric

Quick test: Is the problem about filling something or covering something? Filling = volume. Covering = surface area.

🔷 The Most Common GED Shapes

The GED loves circle-based geometry. Cylinders, cones, and spheres appear frequently. Become very comfortable with circular measurements — especially the difference between radius and diameter.

Rectangular Prism

V = l × w × h

Cereal box, shipping container, brick. Six rectangular faces.

Cylinder ⭐ Common

V = πr²h

Cans, tanks, pipes. Two circular ends + curved side.

Cone ⭐ Common

V = 13πr²h

Traffic cone, party hat, ice cream cone. One circular base, comes to a point at the top.

Sphere ⭐ Common

V = 43πr³

Ball, globe, tank. Perfectly round — all radius, no height.

Right Prism (Triangular)

V = (base area) × h

Tent shape, A-frame. Two triangular ends + rectangular sides. GED calls this a "right prism."

Pyramid

V = 13 × base area × h

Square base + 4 triangular sides meeting at a point. Use net strategy for surface area.

About the "Right Prism" on the GED: The GED formula sheet uses the term "right prism." This almost always means a triangular prism — a tent-shaped object with triangular ends. It may be shown lying on its side or standing up, but the shape is the same.

⭕ Radius vs Diameter — Most Common Mistake

This is the single most common GED geometry error. Every circular formula uses the radius. If a problem gives you the diameter, you must divide by 2 first.

Two circles — diameter vs radius

Always divide the diameter by 2 before using any formula. Every circular formula on the GED uses the radius, not the diameter. If the problem gives you d = 10, your radius is r = 5.

🔢 Worked Examples

Every example follows the same four-step structure: identify the shape → write the formula → list what you know → solve.

Example 1 — Rectangular Prism Surface Area

A box: length = 8 ft, width = 3 ft, height = 2 ft. Find the surface area.

1. Shape: Rectangular prism (box shape)

2. Formula: SA = 2lw + 2lh + 2wh

This means: add all 6 faces as 3 pairs — top/bottom, front/back, two sides.

3. What I know: l = 8 ft, w = 3 ft, h = 2 ft

4. No conversion needed

5. Solve — substitute into each pair:

2lw = 2×8×3 = 48 ft² (top & bottom)

2lh = 2×8×2 = 32 ft² (front & back)

2wh = 2×3×2 = 12 ft² (two sides)

Total: 48+32+12 = 92 ft²

Example 2 — Cylinder Volume

A cylindrical water tank has diameter = 10 ft, height = 15 ft. How much water can it hold?

1. Shape: Cylinder

2. Formula: V = πr²h

3. What I know: diameter = 10 ft, h = 15 ft

4. Convert diameter → radius: r = 10 ÷ 2 = 5 ft

5. Solve: V = π × (5)² × 15 = π × 25 × 15 = 375π

Calculator: 375 × 3.14 ≈ 1,178 ft³

⚠️ Using d=10 instead of r=5 gives 4× the right answer.

Example 3 — Square Pyramid Surface Area

A square pyramid: base side = 6 cm, slant height = 5 cm. Find the surface area.

1. Shape: Square pyramid

2. Formula: Add all faces — 4 triangles + 1 square base

3. What I know: base side = 6 cm, slant height = 5 cm

4. No conversion needed

5. Solve:

4 triangle faces: each = 12 × 6 × 5 = 15 cm² → 4×15 = 60 cm²

Square base: 6×6 = 36 cm²

Total: 60+36 = 96 cm²

Example 4 — Cone Volume

A cone: radius = 3 cm, height = 8 cm. Find the volume.

1. Shape: Cone

2. Formula: V = 13πr²h

3. What I know: r = 3 cm, h = 8 cm

4. No conversion needed (radius already given)

5. Solve: V = 13 × π × (3)² × 8

= 13 × π × 9 × 8 = 13 × 72π

Calculator: 13 × 3.14 × 72 ≈ 75 cm³

⚠️ Don't forget to divide by 3 at the end.

Example 5 — Sphere Volume

A sphere has diameter = 6 inches. Find the volume.

1. Shape: Sphere

2. Formula: V = 43πr³

3. What I know: diameter = 6 in

4. Convert diameter → radius: r = 6 ÷ 2 = 3 in

5. Solve: V = 43 × π × (3)³ = 43 × π × 27

Calculator: 43 × 3.14 × 27 ≈ 113 in³

🚫 Common Mistakes

Mistake 1 — Most Common

Using diameter instead of radius

If diameter = 10, then r = 5. Always divide by 2 before using any circular formula.

Mistake 2

Confusing volume and surface area

Filling/holding = volume (cubic units). Covering/wrapping = surface area (square units).

Mistake 3

Wrong units on the answer

Surface area → ft² or cm². Volume → ft³ or cm³. The GED uses unit errors as distractors.

Mistake 4

Forgetting to multiply by 1/3 for cones and pyramids

Cones and pyramids use V = 13 × base area × height. The ÷3 is easy to forget.

GED Test Strategy: Some geometry problems are intentionally difficult. If a shape looks unfamiliar or the problem feels overwhelming, skip it and answer easier questions first. Every question is worth the same — don't get stuck on one hard geometry problem.

Volume & Surface Areaof 3D Shapes