# 3D Geometry — Volume and Surface Area of the Essential Shapes

Rectangular prisms, cylinders, cones, spheres — the four 3D shapes that appear most in exams. Master the formulas and quick-calculation tricks.

3D geometry is a topic where students often lose marks from formula mix-ups, but it's also easy to recover those marks once you learn it correctly. This post covers the most important formulas with memory tips.

## Why do students get confused?

Students often mix up **surface area** (the total outer area) and **volume** (the interior space). The symbols rrr, hhh, and lll also tend to get confused when multiple shapes appear in the same problem.

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## 4 essential 3D shapes

### 1. Rectangular prism

Given length aaa, width bbb, height ccc:

V=abc

Stotal=2(ab+bc+ca)

> **Memory tip:** Total surface area = 2 times the sum of the areas of the three pairs of opposite faces.

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### 2. Cylinder

Given base radius rrr and height hhh:

V=πr²h

Slateral=2πrh

Stotal=2πr(h+r)

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### 3. Cone

Given base radius rrr, height hhh, slant height l=r²+h²:

V=\frac{1}{3}πr²h

Slateral=πrl

Stotal=πr(l+r)

> **Note:** The volume of a cone is \frac{1}{3} the volume of a cylinder with the same base and height.

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### 4. Sphere

Given radius rrr:

V=\frac{4}{3}πr³

S=4πr²

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## Summary table

| Shape              | Volume               | Total surface area         | 
|--------------------|----------------------|-----------------------------| 
| Rectangular prism  | abc                  | 2(ab+bc+ca)               | 
| Cylinder           | πr²h                 | 2πr(h+r)                   | 
| Cone               | \frac{1}{3}πr²h     | πr(l+r)                    | 
| Sphere             | \frac{4}{3}πr³      | 4πr²                       |

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## Real-world example

**Problem:** A cylindrical water tank has base radius r=1.5 m and height h=2 m. Find its maximum water capacity.

**Solution:**

V=π×1.5²×2=π×2.25×2=4.5π≈14.14 m³

The tank holds a maximum of approximately **14.14 m³** of water.