Rise over run is the whole language

A pitch is not an angle. It's a ratio a carpenter can hold in one hand: rise in inches, per 12 inches of run. Say "six twelve" on any job site on the continent and everyone knows the roof — and once you know the ratio, the rafter length is one multiplication away.

One foot of run, once

Take a single foot of run and stand the pitch's rise on the end of it. The sloped side of that little triangle is the only number you'll ever need to memorize about a pitch:

$\text{factor} = \frac{\sqrt{12^2 + \text{rise}^2}}{12}$

At 6/12 that's √180 ÷ 12 = 1.118. Every horizontal foot the rafter covers costs 1.118 feet of lumber. Pythagoras runs once, on one foot, and then you multiply — which is exactly what the old framing square printed on its blade decades before anyone carried a calculator.

Notice how gently the factor grows. Doubling the pitch from 6/12 to 12/12 doesn't double the rafter — it adds about 26%. Steep roofs cost their money in staging, safety and waste, not in linear feet.

The run is not half the span

Here's the step that eats beginners. The rafter doesn't run to the center of the building — it stops at the face of the ridge board. So:

$\text{run} = \frac{\text{span}}{2} - \frac{\text{ridge thickness}}{2}$

On a 24 ft span with a 1½" ridge: 12 ft − ¾" = 11.9375 ft of run. Multiply by 1.118 and the body of the rafter is 13 ft 4 3/16 in.

And note where the deduction happens: it comes off the horizontal run, before the factor. Take it off the sloped length instead and you'll be short — by a little at 4/12, by a lot at 12/12.

The tail is the same trick again

An overhang is quoted level — "a 12 inch overhang" means 12 inches of horizontal projection, not 12 inches of rafter. So the tail is just another run through the same factor: 1 ft × 1.118 = 1.118 ft.

Body plus tail = 14 ft 5 9/16 in, which is why you're buying 16 ft stock and cutting off about 18 inches. That offcut is the real argument for checking the number before the lumber shows up.

Run it

Open the rafter calculator with this exact roof loaded — 6/12, 24 ft span, 12 in overhang:

1. Read the run, the body, the tail and the stock length. Confirm the 13 ft 4 3/16 in above.
2. Change the ridge from 1½" to 2" and watch the run — not the factor — move.
3. Push the pitch to 12/12. The rafter grows, the ridge climbs 12 ft above the plate, and the stock length jumps a size. Same span, same building, different order.