Area of Triangles, Gable Ends and Sloped Walls

Why this matters

Gable ends are where siding, paint and sheathing estimates go wrong, and they go wrong in both directions. One estimator measures the house as a box and forgets the triangle on top, so the crew runs out of siding at the last course on the peak. Another treats the whole gable as a rectangle as tall as the ridge and orders twice the material the triangle needs. On the gable worked below, the second mistake adds more than a quarter to that wall's siding.

The triangle is not hard. Its area is half the base times the height, and the height of a gable comes straight from the roof pitch. The skill is splitting a wall into shapes you can measure, getting the right height into the formula, and checking the triangle against the rectangle that contains it. This card covers triangles and the trapezoids that show up on shed-roof end walls and sloped knee walls. Breaking an irregular floor plan into pieces is its own card; see related: How to Find the Area of an Irregular Room by Decomposition.

The formula, and which height

Triangle area = 0.5 x base x height.

The height is the perpendicular distance from the base to the opposite point, measured square to the base. On a gable end, the base is the wall width at the eave line and the height is the vertical rise from that line to the peak. It is not the length of the sloped edge (the rake), which is always longer than the rise.

That one fact covers most of the errors. The rake is the edge you can see and tape from a ladder, and it is the wrong number for this formula. It is the right number for trim and drip edge along the rake, which is a length, not an area.

The peak does not have to be centered. For any triangle, 0.5 x base x height works wherever the peak sits along the base, as long as the height is measured straight up from the base line. An off-center gable, a saltbox end or a wall under a stair stringer all use the same formula. What changes for an off-center peak is how you find the height: you cannot get it from half the span and a single pitch, because each side has its own run.

Getting the height from the pitch

Roof pitch is written as rise per 12 units of horizontal run: 6/12 means 6 inches of rise for every 12 inches of run. On a symmetric gable, the run on each side is half the wall width. So:

Gable rise = (pitch rise / 12) x (wall width / 2).

For a 24 ft wide gable at 6/12: run = 24 / 2 = 12 ft, rise = (6 / 12) x 12 ft = 6 ft. See 'Roof Slope, Pitch, + Measurement Reference' for measuring pitch on an existing roof.

Where this goes wrong. Using the full width as the run doubles the rise: (6 / 12) x 24 ft = 12 ft, and the triangle comes out at twice its real area. The units do not catch it, because a width and a run are both feet. The geometry does: a 6/12 roof rises 6 inches per foot of run, and the run of one side of a gable is half the wall.

The worked example: siding a gable end

A siding crew is estimating one gable end of a house. The wall is 24 ft wide at the eave, the wall is 8 ft from the top of the foundation to the eave line, the roof is 6/12, and there is one window measuring 3 ft by 4 ft in the rectangular part of the wall.

GABLE END, SPLIT INTO TWO SHAPES

                  peak
                   /\
                  /  \   <- rake, 13.42 ft each
                 /    \
                /  T   \    T = triangle
               /        \   rise 6 ft
   eave line  +----------+  <- base 24 ft
              |          |
              |  R  [w]  |  R = rectangle 24 x 8 ft
              |          |  w = window 3 x 4 ft
              +----------+  <- foundation line
               24 ft wide

Step 1, the rectangle. R = 24 ft x 8 ft = 192 sq ft.

Step 2, the triangle. Rise = (6 / 12) x (24 ft / 2) = 6 ft. T = 0.5 x 24 ft x 6 ft = 72 sq ft.

Step 3, gross wall. 192 + 72 = 264 sq ft.

Step 4, openings. Window = 3 ft x 4 ft = 12 sq ft. Net = 264 - 12 = 252 sq ft.

Step 5, the check against the rectangle. The triangle sits inside a rectangle 24 ft wide and 6 ft tall, which is 144 sq ft. Every triangle is exactly half of the rectangle drawn around its base and height, and 72 is half of 144. If your triangle comes out at more than half of its bounding rectangle, the height is wrong. If it comes out equal to the rectangle, the 0.5 was dropped.

Step 6, the rake length, for trim. The rake is the hypotenuse of a right triangle with legs of 12 ft (run) and 6 ft (rise): square root of (144 + 36) = square root of 180 = 13.42 ft per side. As a check, 6/12 has a slope factor of 1.118, and 12 ft x 1.118 = 13.42 ft. Two rakes are 26.83 ft of rake trim for this gable. See related: Roof Pitch Multipliers for Any Sloped Surface for that factor.

What the estimate does with 252 sq ft. Siding is usually sold by the square (100 sq ft of coverage), so the net wall is 2.52 squares before waste. Paint is figured by dividing the area by the coverage rate on the can. Sheathing is figured in sheets: a 4 ft by 8 ft sheet covers 32 sq ft. The waste allowance comes after, and gable triangles carry more of it than rectangles do because every course is cut on an angle at both ends. See 'The Materials Waste Factor Worth Measuring'.

The two wrong answers, side by side. If the estimator uses the full width as the run, the rise becomes 12 ft and the triangle 144 sq ft, giving a gross wall of 336 sq ft, 72 sq ft more than real. If the estimator uses the rake as the height, the triangle becomes 0.5 x 24 x 13.42 = 161 sq ft, giving 353 sq ft gross, 89 sq ft more than real. Either error is most of a square (0.72 and 0.89 squares) of extra siding on every gable it touches.

Should you subtract openings?

Subtracting doors and windows is correct for the wall area. Whether you subtract them from the material order is a separate judgment that depends on the material:

• Paint is applied to the area actually painted, so small openings are often left in as a cushion and large ones (a garage door, a wall of windows) are taken out.
• Siding and sheathing are cut around openings, and the piece removed from a window opening is often too small to reuse. Many estimators subtract only openings larger than a set size and treat small ones as waste. Whatever rule your shop uses, write it on the takeoff so the next estimator does not apply a different one.

The geometry does not change; the order does. Keep the gross area, the openings and the net on separate lines so the decision is visible.

Trapezoids: shed ends and sloped knee walls

A trapezoid has two parallel sides of different lengths. It is the shape of a shed-roof end wall (short side at the low eave, tall side at the high eave) and of a knee wall that runs under a sloped ceiling and steps down.

Trapezoid area = (side 1 + side 2) / 2 x the distance between them.

In words: the average of the two parallel sides, times the width between them.

A shed end wall is 12 ft deep, 8 ft tall at the low side and 11 ft tall at the high side. Area = (8 ft + 11 ft) / 2 x 12 ft = 9.5 ft x 12 ft = 114 sq ft.

Check it by splitting. The same wall is a rectangle 12 ft x 8 ft = 96 sq ft with a triangle on top whose base is 12 ft and height is 11 - 8 = 3 ft: 0.5 x 12 x 3 = 18 sq ft. 96 + 18 = 114 sq ft, which matches. Splitting into a rectangle and a triangle is always available as the second method, and it is the most reliable rough check for a trapezoid.

The heights must be measured square to the base. On a shed end, both heights are vertical and the base is level, so a tape held plumb gives the right numbers. On a wall with an out-of-level foundation line, measure each height plumb from a level reference line and keep the base horizontal; using the sloped foundation as the base without correcting for it is a small error on most walls but not on a lot with a strong grade.

Triangles that are not vertical

Everything above applies to vertical walls, where the height you need is a true vertical rise. Two cases look similar and are not:

• A sloped ceiling or a sloped wall under a roof. The area you cover is the sloped surface, which is larger than its footprint. A room 14 ft by 20 ft with a cathedral ceiling has more ceiling than 280 sq ft. That correction is a pitch multiplier, not a triangle formula; see related: Roof Pitch Multipliers for Any Sloped Surface.
• A hip roof face. The triangular end of a hip roof lies on the slope. Its area is 0.5 x base x the slope height (eave to peak along the surface), not the vertical rise. Or work it in plan and apply the pitch multiplier, which gives the same answer.

The question that sorts them: is the surface you are covering vertical? If yes, the height is the plumb rise. If not, the height must be measured along the surface, or the plan area corrected for pitch.

When the pitch is unknown

On an existing house you often cannot read the pitch, and you cannot reach the peak. Three ways to get the rise without climbing:

• Count siding courses. If the lap siding exposure is known (measured on a lower course), count courses from the eave line to the peak and multiply.
• Measure from inside the attic. Measure the vertical distance from the top of the ceiling joists to the underside of the ridge at the gable wall; adjust for framing depth if you need the exterior line.
• Measure the pitch at the eave with a level and a tape, then compute the rise from half the width. See related: How to Measure Pitch With a Level and a Tape.

Whichever method you use, write it on the takeoff. A rise obtained by counting courses at an assumed exposure carries the assumption with it, and if the exposure was wrong, every gable on the job is wrong by the same ratio.

References

• Roof Slope, Pitch, + Measurement Reference (library card, for reading pitch on an existing roof)
• The Materials Waste Factor Worth Measuring (library card, for cut waste on angled courses)
• See related: How to Find the Area of an Irregular Room by Decomposition
• See related: Roof Pitch Multipliers for Any Sloped Surface
• See related: How to Measure Pitch With a Level and a Tape