Trade Math Advanced (angles, area, volume, geometry on the job)

Fractions and basic area get you through most days. The next level is the math that lets you solve a problem nobody has written down for you: how long a rafter needs to be, how to cut an eight-sided frame, how much pipe goes between two 45s, or how many amps a heater will pull. None of it is hard once you see the pattern. Most of it comes down to right triangles, ratios and keeping your units straight. Every section below has worked numbers, so grab a calculator and follow along.

What You Will Be Able to Do

• Use the Pythagorean theorem and basic trig to find lengths and angles.
• Set a miter saw for any corner or any number of sides.
• Calculate pipe offsets, rafter lengths and stair layouts.
• Estimate tile, paint, roofing, drywall and concrete.
• Use Ohm's law and the power formula for loads, and find voltage drop.
• Convert between slope, percent grade and ratio.

Right Triangles

The long side of a right triangle is the hypotenuse. If a and b are the other two sides:

c = square root of (a x a + b x b)

Worked example: a wall brace. A wall is 8 feet tall, and the brace lands 6 feet from the base. Brace = square root of (64 + 36) = square root of 100 = 10 feet.

Worked example: one stair step. The rise is 7-1/2 inches and the run is 10-1/2 inches. The slope length is the square root of (56.25 + 110.25) = square root of 166.5 = about 12.9 inches.

Trig Without the Fear

Trig ties angles to side lengths. For any angle in a right triangle, the opposite side faces the angle, and the adjacent side touches it.

To get the angle back, use the inverse key on your calculator, marked arctan, or tan with a small -1.

Roof Pitch

Pitch is the rise in inches for each 12 inches of run. To get degrees, take the arctan of rise divided by run.

Worked example: rafter length. A building is 24 feet wide with a 6/12 roof.

1. Run is half the span: 12 feet.
2. Rise: 12 x 6 / 12 = 6 feet.
3. Rafter: square root of (144 + 36) = square root of 180 = 13.42 feet, before any overhang.

Roof area. The pitch factor turns flat area into roof area. For 6/12, the factor is 13.42 / 12 = 1.118. A house that covers 1,500 square feet of flat plan has about 1,500 x 1.118 = 1,677 square feet of roof.

Miters and Angles

A miter saw reads 0 when it cuts square, so its scale shows how far you swing away from square.

Miter setting = (180 - corner angle) / 2

You may see 120-degree corners described as 60-degree cuts, or 135-degree corners as 67.5. Those numbers are measured from the edge of the board, not from square. Same cut, different reference.

For a frame with equal sides, use this:

Miter setting = 360 / number of sides / 2

Worked example: A six-sided frame. 360 / 6 = 60, and 60 / 2 = 30 degrees on each end of every piece.

A compound cut uses both a miter and a blade bevel. Crown molding is the classic case. For 38-degree spring crown on a 90-degree wall, cutting flat on the saw, use a 31.6-degree miter and a 33.9-degree bevel. Test on scrap, because real walls are rarely perfect.

Pipe Offsets

An offset moves a pipe sideways or up around something. The angled piece is the travel, the sideways distance is the offset, and the distance along the original line is the run.

Worked example: A pipe needs a 10-inch offset, using 45-degree fittings.

1. Travel = 10 x 1.414 = 14.14, so about 14-1/8 inches, center to center.
2. Run = 10 x 1.000 = 10 inches.
3. Subtract the fitting takeout for each end to get your cut length.

Stairs

Divide the total rise into equal steps. Codes set the limits. The IRC generally allows a maximum riser of 7-3/4 inches and a minimum tread of 10 inches, while commercial codes are often 7 and 11. Check your adopted code.

Worked example: The floor-to-floor rise is 108 inches, and you want risers near 7 inches.

1. 108 / 7 = 15.4, so try 15 risers.
2. Actual rise: 108 / 15 = 7.2 inches.
3. Treads: one fewer than risers, so 14.
4. Total run at 11 inches: 14 x 11 = 154 inches, or 12 feet 10 inches.
5. Length along the slope: square root of (108 x 108 + 154 x 154) = about 188 inches, or 15 feet 8 inches.

Add for the top and bottom cuts when you buy stock, so a 16-foot board may be too short.

Estimating Materials

Common Takeoffs

• Tile. Add about 10 percent for straight layouts, and 15 percent for diagonals or complex rooms.
• Paint. Coverage is on the can, often 300 to 400 square feet per gallon. A 12 x 14 room with 8-foot walls has (12 + 14 + 12 + 14) x 8 = 416 square feet of wall.
• Roofing. One square is 100 square feet. A 2,000 square foot roof is 20 squares, before waste for starters and ridges.
• Drywall. A 4 x 8 sheet is 32 square feet. A 14 x 16 room has 480 square feet of wall, which takes 480 / 32 = 15 sheets, plus 224 / 32 = 7 for the ceiling.

Concrete

Worked example: round footings. Six footings are 24 inches across and 36 inches deep.

1. Radius 1 foot, depth 3 feet.
2. Each: 3.14 x 1 x 1 x 3 = 9.42 cubic feet.
3. Six: 56.5 cubic feet.
4. Divide by 27: 2.09 cubic yards. Order with a waste allowance, such as 2.5.

Electrical Math

Two formulas do most of the work.

• Ohm's law: volts = amps x ohms, so amps = volts / ohms.
• Power: watts = volts x amps, so amps = watts / volts.

Worked example: A 24-volt supply across an 8-ohm load: 24 / 8 = 3 amps.

Worked example: A 4,500-watt water heater on 240 volts: 4,500 / 240 = 18.75 amps.

Continuous Loads

A continuous load is one that runs for 3 hours or more. The usual rule is to keep it to 80 percent of the breaker rating.

Worked example: A 1,500-watt heater pulls 12.5 amps at 120 volts. If it runs for hours on a 15-amp circuit, it is over the 12-amp limit.

Voltage Drop

Long runs lose voltage. For single-phase circuits:

Voltage drop = 2 x K x amps x one-way distance / circular mils

K is about 12.9 for copper and 21.2 for aluminum. The NEC recommends in its informational notes keeping branch circuits to about 3 percent, and feeders plus branch circuits to about 5 percent.

Worked example: A 240-volt circuit carries 20 amps over 100 feet of 10 AWG copper, which is 10,380 circular mils.

1. 2 x 12.9 x 20 x 100 = 51,600.
2. 51,600 / 10,380 = 4.97 volts.
3. 4.97 / 240 = 2.1 percent, which is under 3 percent.

HVAC Math

A common rule of thumb is 400 cubic feet per minute (cfm) of air for each ton of cooling. Duct area in square feet is cfm divided by air speed in feet per minute.

Worked example: A 2-ton system moves 800 cfm at 700 feet per minute. Area = 800 / 700 = 1.14 square feet, and 1.14 x 144 = 164 square inches. A 14-inch round duct has 154 square inches, so it is a bit small. A real design uses Manual D or your employer's method.

Efficiency. A furnace rated 96 percent AFUE turns about 96 percent of its fuel energy into heat over a season. Burning 80,000 BTU of fuel delivers about 76,800 BTU of heat.

Slope, Grade and Ramps

The same slope can be written three ways.

To get percent, divide rise by run and multiply by 100. For 1/4 inch per foot, 0.25 / 12 x 100 = about 2 percent.

• Drains. Under the IPC, the minimum slope is 1/4 inch per foot for pipe 2-1/2 inches or smaller, 1/8 inch for 3 to 6 inches, and 1/16 inch for 8 inches or larger.
• Accessible ramps. ADA standards limit most ramps to 1:12, so each inch of rise needs 12 inches of run. A 30-inch rise needs 360 inches, or 30 feet, plus level landings at least 5 feet long.

Worked example: A drain runs 40 feet at 1/4 inch per foot. Total fall = 40 x 0.25 = 10 inches.

More Conversions

Common Mistakes

Check Your Understanding

1. A deck frame is 12 by 16 feet. What should the diagonal read if it is square?

Square root of (144 + 256) = square root of 400 = 20 feet.

2. A 10-sided frame needs what miter setting?

360 / 10 = 36, and 36 / 2 = 18 degrees.

3. A 1,500-watt microwave and a 700-watt refrigerator share a 20-amp, 120-volt circuit. Is it a good idea?

1,500 / 120 = 12.5 amps, and 700 / 120 = 5.8 amps, so 18.3 amps total. That is close to the breaker limit, so the microwave should have its own circuit.

4. A 22.5-degree offset must move a pipe 8 inches. What is the travel?

8 x 2.613 = about 20.9 inches, center to center.

5. A rise of 126 inches needs a stair with risers near 7.25 inches. How many risers, and what is the actual rise?

126 / 7.25 = 17.4, so 17 risers at 126 / 17 = 7.41 inches, or 18 risers at 7 inches.

Key Takeaways

• Most layout problems are right triangles.
• Arctan of rise over run gives the angle, so 6/12 is 26.6 degrees.
• Miter setting = (180 - corner angle) / 2, or 360 / sides / 2.
• Pipe travel = offset x multiplier, and 45-degree fittings use 1.414.
• Amps = watts / volts, and continuous loads stay at 80 percent.
• 1/4 inch per foot is about 2 percent, and ramps run 1:12.