Lesson 3 of 12 · Chapter 3
Trade Math for Solar Design: Pitch, Area, Orientation and Size
Trade math is the bridge between standing in the driveway looking up and a layout that is buildable, compliant and worth the customer’s money. Every bench in this lesson gives you new numbers each round: the slope factor of a pitch, the usable surface of a real roof plane, a compass reading corrected to true north, and a system sized from sun hours, including what happens when the roof will not fit it.
Choose how you want to learn
You can change this any time. Every level does the same hands-on steps. Press Listen on any part to hear it read aloud.
- 1 · Pitch and slope factor
- 2 · Usable surface area
- 3 · Azimuth and true north
- 4 · Orientation and the goal
- 5 · Sizing the system
- 6 · Try it at home
- 7 · Your card
- 8 · Check yourself
Part 1
Pitch, degrees and the slope factor
Pitch is rise over a 12-inch run: a 6-in-12 roof rises 6 inches for every 12 across.
Measure it with a level held 12 inches out and a tape, or with a proper inclinometer (phones are a convenience, not a primary tool).
The roof surface is bigger than its footprint. The slope factor is √(144 + rise²) ÷ 12.
At 6-in-12, √180 = 13.42 inches of sloped length per 12 of run, a slope factor of 1.118: about 11.8% more surface than footprint. At 12-in-12 it is 1.414, about 41% more. That difference decides whether an array fits and how many attachments, rails and feet of wire you need.
Carry both views. The ratio view is quick for slope-factor math; the angle view (6-in-12 is about 26.5°, 8-in-12 about 33.7°, 12-in-12 is 45°) matters for tilt. Pitch also changes how a crew moves: a 10-in-12 roof makes module handling a two-person job even in calm weather.
“Pitch measurement is not worth a slip.”
“That is where your pitch-to-degrees intuition starts paying rent.”
Read the whole chapter free in the book reader.
The Pitch Bench. Pick a pitch to see it, then work a new one yourself.
Part 2
Usable surface area
Measure the plane in plan view (from plans, the attic, the ground or aerial tools, then verify).
Break awkward shapes into rectangles and right triangles, and subtract chimneys and hip cuts.
Multiply what is left by the slope factor. That is usable roof surface, before setbacks.
A right triangle’s area is half the base times the height. Subtract in plan view first and apply the slope factor once, so errors do not compound. Eyeballing a hip or valley line almost always steals space you only notice when modules hit a setback.
Usable area is not total area. Fire pathways, ridge and eave setbacks, and room to land feet and route wire all come off. Attachments land on rafters or trusses, often 16 or 24 inches apart, but never assume: a layout pushed too close to an edge can put a rail foot where there is no framing.
“A roof is a shape, not a guess.”
“improvisation is where roofs leak.”
The Roof Area Bench. The first roof is the chapter’s own example; after that, every roof is new.
Part 3
Azimuth, tilt, and true north
Azimuth is the direction the array faces, in degrees clockwise from true north: east 90, south 180, west 270.
A compass points to magnetic north. The difference is declination, and it can be 10° or more.
Tilt is the angle above horizontal. On a flush-mounted roof, the tilt is the roof pitch in degrees.
Designers work in true bearings. If a plan set lists azimuth, treat it as true unless it says otherwise. If you measure with a compass, correct for declination, and watch for metal gutters, vehicles and equipment that pull the needle.
Cross-check: compare the plan-set azimuth with what you see, confirm ridges and roof lines match the software, and note any twist. Roofs are not always square to the street, and streets are not always square to the compass.
“The building will happily lie to your intuition.”
The True North Bench. Correct each compass reading to a true azimuth.
Part 4
Orientation and the customer’s goal
In the northern hemisphere, true south with tilt near latitude usually gives the most energy in a year.
East favours mornings, west favours afternoons, and an east-west split flattens the curve.
Shade can undo a perfect orientation, so it is modelled, not hoped away.
Many customers care when the power arrives, not just how much. On time-of-use rates, afternoon production from a west plane can be worth more than extra annual kilowatt-hours from a small south one. That is why designers talk about production profiles, not only totals.
Steeper is not always better: a steep tilt helps winter and snow shedding but loses summer sun, and a shallow tilt does the opposite. The same chimney casts a small summer shadow and a long winter diagonal. A roof plane that catches strong midday sun can also run hot, which lowers voltage, as in Lesson 1.
“shading is orientation’s quiet partner”
“you do not “fix” that with optimism.”
Six customers and roofs. What does the orientation decision look like?
Part 5
Sizing the system, and listening to the roof
Peak sun hours are the day’s sunlight counted as hours at 1,000 W/m².
Required kW DC = daily kWh ÷ (peak sun hours × performance ratio). Then divide by module size and round up.
If the roof fits fewer modules, work out what they will produce and tell the customer the real offset.
The performance ratio (often around 0.75 to 0.85) covers heat, inverter efficiency, soiling, mismatch, wiring loss, clipping, shade, snow and downtime. Break it into parts you can justify; if you are unsure, choose conservatively and write the assumption down.
Look at monthly sun and monthly use, not only the annual average: a system that offsets 100% a year may give 120% in summer and 80% in winter. Arrays are often sized larger than the inverter; the resulting clipping can be a good trade, as long as it is explained and within equipment and utility limits.
“Performance ratio is where experience and honesty show up.”
“the roof is telling you a truth the spreadsheet cannot negotiate.”
“A “perfect” energy size that cannot be built safely and legally is not a design; it is a wish.”
The System Sizer. The first customer is the chapter’s own example; after that, every customer is new.
Part 6
Try it at home
Do these at home and from the ground. Tick each one when it is done.
Part 7
Your “Roof and Sizing Sheet” card
One roof plane and one system, worked the chapter’s way. Nothing you type here leaves this device.
My Roof and Sizing Sheet
Global Sovereign University · Solar PV Installers, lesson 3
The math on one card
- Slope factor = √(144 + rise²) ÷ 12.
- Subtract in plan, then multiply by slope factor.
- True azimuth: add east declination, subtract west.
- South for total; east or west for timing.
- kW = daily kWh ÷ (PSH × PR); round modules up.
- If the roof fits fewer, tell the real offset.
A layout is only good when it fits, meets clearances, lands on framing and meets the goal.
Part 8
Check yourself
Six real situations. Pick what you would do. You will see why after each one.
Lesson 3 complete.
You can now turn a pitch into a slope factor, measure usable roof surface, correct a compass to true north, match orientation to a goal, and size a system honestly. Next: Lesson 4, tools of the trade.
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Solar Photovoltaic (PV) Installers by Dr. Gene A Constant. This lesson comes from chapter three. Read the whole book free in the reader, or keep a copy.
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