# Solid Angles

**Truly Understanding Luminous Intensity**

Ian Ashdown, P. Eng., FIES

Chief Scientist, Lighting Analysts Inc.

[ Please send comments to allthingslighting@gmail.com. ]Do you suffer from *math anxiety*? A surprising number of us do (e.g., Wigfield 1988). I would tell you the exact numbers, but you would need to understand statistical analysis …

Fortunately, we can mostly muddle through our lives without having to deal with statistics, vector calculus, differential geometry, algebraic topology and all that. As an electrical engineer in the 1980s for example, I never needed anything more than a four-function calculator to do my work designing billion-dollar transportation systems.

Our fear (note the implicit “we”) can, however, disadvantage us in subtle ways. In studiously ignoring the mathematics of a topic, we all too often overlook the underlying concepts that help us better understand what we are interested in.

An example from lighting design: *luminous intensity*. We measure the luminous intensity of a light source in *candela*, which is defined as “one lumen per steradian” (IES 2010). A lumen is easy enough to understand, but what the blazes is a “steradian”?

The all-knowing Wikipedia has an answer: it is the measure of a “solid angle.” Going to the Wikipedia definition of this phrase, we see:

Anxiety? What anxiety?

But now for a trade secret: most mathematicians do not think in terms of equations like these double integrals. Instead, they *visualize*. Just as lighting designers can look at architectural drawings and imagine lighting designs, mathematicians can look at a set of equations – which are really nothing more than an arcane written language – and visualize new mathematical concepts and proofs.

I learned this from a professor of mine whose specialty was hyperspace geometry – he could “easily imagine” four- and five-dimensional objects by mentally projecting them into three-dimensional shapes and imagining how their shadows changed as he rotated the objects in his mind. Some people …

So, we start by visualizing a circle (FIG. 1):

FIG. 1 – Circle with radius *r*

If you remember anything at all from mathematics in school, it is that the circumference *C *of a circle with radius *r *is equal to two times *pi* times its radius, or:

*C *= 2 * *pi ** *r*

where pi is approximately 3.14159. (Remember that 1980s-era four-function calculator – it is all you will need for this.)

What this means is that if we take a piece of string with length *r*, we will need to stretch it by a factor of two *pi *(6.28328 …) to wrap around the circumference of the circle.

But suppose we wrap the string with length *r *part way around the circle (FIG. 2). The resultant angle is precisely one *radian*, which is abbreviated *rad*.

FIG. 2 – One radian

Most of us are used to thinking of angles in terms of degrees – there are 360 degrees in a circle. (The reason for the magic number 360 is lost in history, according to Wikipedia.) This means that one radian is equal to 360 / (2 * *pi*) = 180 / *pi* degrees, which is approximately 57.3 degrees. Radians are more useful simply because they are related to the geometry of the circle rather than some magic number – they are easier to visualize and so understand.

Now, imagine a sphere with radius *r*, and with a cone-shaped section whose base has a surface area of *r ** *r*, or *r*^{2} (FIG. 3):

FIG. 3 – Solid angle

This cone has a *solid angle *of precisely one *steradian* (or one “solid radian”), which is abbreviated *sr*.

No mathematics required – easy.

(To be precise, a solid angle does not need to be a circular cone-shaped section as shown in FIG. 3. The top of the cone can be any shape; all that matters is the ratio of the surface area of the base to the radius *r*.)

How many “square degrees” in a steradian? That’s also easy: if one radian is equal to 180 / *pi* degrees, then one steradian is equal to (180 / *pi*) * (180 / *pi*), or approximately 3282.8, square degrees.

To be honest, I also suffer from math anxiety when first reading a set of equations. I do not really understand them until I can visualize what they mean. Mathematical equations are just the formal written language we use to express what we have visualized.

… now if only I could understand batting averages in baseball and cricket …

**References**

IES. 2010. IES Lighting Handbook, Tenth Edition. New York, NY: Illuminating Engineering Society of North America.

Wigfield, A., and J. L. Meece. “Math Anxiety in Elementary and Secondary School Students,” Journal of Educational Psychology 80(2):210-216.