What are the Surface Area and Volume Formulas for Cylinder, Cone, and Sphere with derivations?

Direct Answer

In 3D geometry and mensuration, curved surface area (CSA / lateral surface area), total surface area (TSA), and volume are defined by the following foundational formulas:

| 3D Geometric Solid | Curved Surface Area (CSA) | Total Surface Area (TSA) | Volume ($V$) |

| :--- | :--- | :--- | :--- |

| Right Circular Cylinder | $2\pi rh$ | $2\pi r(h + r)$ | $\pi r^2 h$ |

| Right Circular Cone | $\pi rl$ | $\pi r(l + r)$ | $\frac{1}{3}\pi r^2 h$ |

| Solid Sphere | $4\pi r^2$ *(same as TSA)* | $4\pi r^2$ | $\frac{4}{3}\pi r^3$ |

| Solid Hemisphere | $2\pi r^2$ | $3\pi r^2$ | $\frac{2}{3}\pi r^3$ |

| Hollow Cylinder | $2\pi h(R + r)$ | $2\pi(R + r)(h + R - r)$ | $\pi h(R^2 - r^2)$ |

*(Where $r$ = base radius, $R$ = outer radius, $h$ = vertical height, $l$ = slant height of cone).*

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Important Mathematical Relationships

#### 1. Slant Height of a Cone ($l$)

Using the Pythagorean theorem in the cross-sectional right triangle:

$$l = \sqrt{r^2 + h^2}$$

#### 2. Relationship Between Cylinder and Cone Volume

A cylinder and a cone sharing the same base radius $r$ and height $h$ satisfy:

$$\text{Volume of Cylinder} = 3 \times \text{Volume of Cone}$$

#### 3. Sphere Surface Area and Great Circle

The total surface area of a sphere ($4\pi r^2$) is exactly four times the area of its great circle ($\pi r^2$).