Perimeter and area of floors, walls and roofs

Perimeter and area are the first measurements of quantity. Perimeter, the distance around a shape, tells you how much baseboard, edge form, firestop sealant around a room or cable along a wall you need. Area, the amount of surface, tells you how much flooring, drywall, insulation, roofing, membrane or paint. Every trade uses both, and every estimate begins with them.

On Kingsway Tower, the drywall, flooring and painting contractors all start from the same suite drawings. Maya's mechanical crew uses area too: floor area drives the radiant heating layout and the number of sprinkler heads, and wall area sets the amount of fire-rated sealant and insulation around pipe penetrations. In this lesson the apprentices work through a typical one-bedroom suite on the seventh floor.

Perimeter

The perimeter of any shape made of straight sides is the sum of the side lengths. For a rectangle with length L and width W, that is twice the length plus twice the width.

For irregular rooms, add every side, going around in order so that no side is missed or counted twice. A useful shortcut for L-shaped and stepped rooms whose walls all meet at right angles: the perimeter equals the perimeter of the smallest rectangle that encloses the room. The steps in and out add and subtract the same lengths.

Perimeter is a length, so it is measured in metres, millimetres, feet or inches, never in square units. When you order trim or cable by perimeter, subtract the widths of doorways where the material does not run, then add an allowance for joints and waste.

Area of rectangles and composite shapes

The area of a rectangle is length times width. The answer is in square units: square metres (m²) if the lengths were in metres, square feet if they were in feet. Both lengths must be in the same unit before you multiply. Multiplying metres by millimetres, or feet by inches, is one of the most common estimating errors.

A composite shape is any area that can be split into rectangles, triangles and circles. Split it, find each area, then add them. If the shape has a piece missing, such as a column or a shaft, find the whole area and subtract the missing piece.

Wall area

The area of the walls in a room is the perimeter times the wall height, less the openings. For drywall and paint, subtract doors and windows. For some materials, such as vapour barrier or fire-rated board, the specification may say not to subtract small openings because the offcuts are not usable; follow the specification.

Experienced drywall estimators often count sheets wall by wall instead, laying out where each sheet goes, because whole sheets and offcuts do not always fit together the way areas suggest. The area method gives a quick check on that count.

Triangles, trapezoids and roofs

A triangle's area is half its base times its height, where the height is measured at right angles to the base. A gable end wall, the triangular wall under a pitched roof, is the classic example. A trapezoid, a four-sided shape with two parallel sides, has an area equal to the average of the parallel sides times the distance between them. Hip roof planes and the side walls of a sloped ceiling are often trapezoids.

Roof areas need care, because a sloped roof has more surface than the floor area it covers. Lesson 10 shows how to find the true length of a sloped roof from its pitch; multiply the plan area by the same factor to get the roof surface.

Circles

Pipes, round ducts, core holes, columns and tanks bring circles into the trades. The distance across a circle through its centre is the diameter d; half of it is the radius r. The distance around the circle, the circumference, is \(\pi d\), and the area is \(\pi r^2\). The number \(\pi\) is about 3.1416.

The same circumference formula tells you how wide a sheet of jacketing must be to wrap insulated pipe: the circumference of the outside of the insulation plus an overlap.

Converting area units

Because area is length times length, area conversion factors are length factors squared. One metre is 3.2808 ft, so one square metre is 3.2808 squared, about 10.764 square feet. One foot is 12 inches, so one square foot is 144 square inches.

The million in the metric row surprises people: a square metre is 1,000 mm by 1,000 mm, so it contains a million square millimetres. Wire and rebar cross-sections are given in mm², which is why the numbers are small: a common 2.5 mm² conductor is tiny compared with a square metre of wall.

Coverage rates

Many materials are sold by coverage: a can of paint, a bag of thinset, a pail of fire-stop sealant or a roll of membrane covers a stated area. Divide the area by the coverage per unit, round up, and add any second coat or waste allowance.

Paint, for example, commonly covers about 9 m² per litre per coat on smooth drywall, which is roughly 350 to 400 square feet per US gallon, but the label on the actual product is what counts. Rough or porous surfaces take more.

Area by coordinates

For an irregular shape drawn on a grid, such as a site boundary or an odd-shaped mechanical room, there is a neat method that needs only the coordinates of the corners, listed in order around the shape. It is sometimes called the shoelace formula, after the criss-cross pattern of the multiplication. Survey and estimating software use it.

def shoelace(points):
    """Area of a polygon from its corners listed in order around the shape."""
    s = 0.0
    for (x1, y1), (x2, y2) in zip(points, points[1:] + points[:1]):
        s += x1 * y2 - x2 * y1
    return abs(s) / 2

# The L-shaped suite, corners in metres, going around counterclockwise
suite = [(0, 0), (6.20, 0), (6.20, 4.80), (3.10, 4.80), (3.10, 7.20), (0, 7.20)]
perimeter = sum(((x2 - x1) ** 2 + (y2 - y1) ** 2) ** 0.5
                for (x1, y1), (x2, y2) in zip(suite, suite[1:] + suite[:1]))
print(f"Area = {shoelace(suite):.2f} m2, perimeter = {perimeter:.2f} m")

Area = 37.20 m2, perimeter = 26.80 m

The program confirms the hand calculation for the suite. It works for any straight-sided shape, including ones with angled walls that are awkward to split into rectangles.

Irregular areas: the average-width method

Not every area is made of neat rectangles and triangles. A curved planting bed beside the entrance, a pond liner, or a patch of damaged slab may have an irregular edge. A practical way to estimate such an area is to divide it into equal strips, measure the width of the shape at regular stations along a straight baseline, average the widths, and multiply by the length.

The more stations you measure, the closer the estimate. For ordering material, an estimate within a few percent is enough, and the waste allowance from lesson 12 covers the rest.

When to deduct openings

For wall finishes, the question is whether to subtract doors, windows and other openings. Trades handle it differently, and it is worth knowing the custom before you estimate.

• Paint and wall coverings: subtract large openings such as garage doors and storefront glazing; many painters ignore openings smaller than about 1 square metre or 10 square feet, because cutting in around them uses extra time and material anyway.
• Drywall: estimators often do not deduct ordinary doors and windows, because the cut-out pieces are usually waste. They do deduct very large openings.
• Insulation and vapour barrier: deduct openings, but add overlaps at seams.
• Masonry: deduct openings, then add for the lintels, sills and extra cutting around them.

Whatever the custom, write down what you did. A takeoff that says "openings not deducted" can be checked by anyone; one that does not say leaves the next person guessing.