Steradian

Overview

Engineering reference data for Steradian in miscellaneous.

Key Formulas

Unit Conversion

$y = x \cdot k$

Multiply by conversion factor.

Linear Interpolation

$y = y_1 + \frac{(x - x_1)(y_2 - y_1)}{x_2 - x_1}$

Estimate between two known points.

Percentage

$p = \frac{\text{part}}{\text{whole}} \times 100\%$

Part as fraction of whole.

Variables

Symbol	Description	Unit
$x$	Input value	—
$y$	Output value	—
$k$	Conversion factor	—

Definition and Properties

The steradian (symbol: sr) is the SI unit for measuring solid angles. It is derived from the radian, which measures plane angles, extended to three dimensions.

Geometric Definition: One steradian is the solid angle subtended at the center of a sphere by a surface area equal to the square of the sphere's radius.

$$1 \text{ sr} = \frac{\text{Area on sphere surface}}{r^2}$$

Total Solid Angle: The entire surface of a sphere subtends a solid angle of exactly:

$$\Omega_{\text{sphere}} = \frac{4\pi r^2}{r^2} = 4\pi \text{ sr} \approx 12.566 \text{ sr}$$

Key Relations:

• A hemisphere (half-sphere) subtends $2\pi$ sr at its center.
• A right circular cone with half-angle $\theta$ subtends $\Omega = 2\pi (1 - \cos\theta)$ sr.
• The steradian is a dimensionless quantity (sr = m²/m² = 1).

Practical Visualization:

• A sphere of radius 1 meter has a surface area of $4\pi$ m², so one steradian corresponds to an area of 1 m² on that sphere.
• On Earth (radius ≈ 6371 km), one steradian covers approximately 40.7 million km²—about 8% of the planet's surface.

References

• Original Source