Perimeters and areas of plane shapes

Areas drive a large share of every construction estimate. Floors, walls, ceilings, roofs, paving, formwork contact surfaces, insulation, membranes and paint are all measured and priced by area, and perimeters measure footings, curbs, baseboards, fences, edge forms and flashing. The shapes on drawings are rarely simple rectangles: buildings have notches and bays, sites have curved and irregular boundaries, and walls are interrupted by doors and windows. This lesson builds a toolkit of area and perimeter formulas, shows how to break complex shapes into simple ones, and introduces three methods for irregular shapes: the trapezoidal rule, Simpson's rule and Heron's formula for triangles measured only by their sides.

Throughout the lesson, keep two habits. First, sketch the shape and label every dimension before calculating; most area errors come from using the wrong dimension, not from bad arithmetic. Second, write units on every line, because area units are squared and a missing square sign is the beginning of a conversion error.

Perimeter

The perimeter of a shape is the total length around its edge. For a rectangle it is twice the sum of the length and width. For a polygon it is the sum of all sides. For a circle the perimeter is called the circumference.

A useful shortcut applies to any shape made only of horizontal and vertical edges, such as an L-shaped or stepped building outline: its perimeter equals the perimeter of the smallest rectangle that encloses it, as long as the notches are on the outside corners and do not cut deep inlets into the middle of a side. The horizontal steps add up to the overall length and the vertical steps add up to the overall width. Inlets cut into a side, however, add twice their depth.

Perimeters in takeoffs usually need a decision about which line to measure. A strip footing is often measured along its centre line, because using the outside perimeter overstates corners and using the inside understates them. For a rectangular building, the centre-line perimeter equals the outside perimeter minus four times the footing width, or the inside perimeter plus four times the width. Lesson 4 applies this rule to footing volumes.

Areas of basic shapes

The table collects the area formulas used most often in construction. Each assumes consistent units: multiply metres by metres to get square metres, feet by feet to get square feet.

The perpendicular height is the most frequently misused dimension. The sloping side of a triangle or parallelogram is not its height; using it overstates the area. Gable ends, sloping sites and trapezoidal roof faces should be drawn with the perpendicular height shown explicitly before calculating.

Composite shapes

Most real outlines are composite shapes made of rectangles, triangles and curves. There are two strategies, and it is good practice to use one to calculate and the other to check.

• Addition: divide the shape into non-overlapping simple shapes, calculate each area and add them.
• Subtraction: take an enclosing simple shape, then subtract the missing pieces, such as a notch, a courtyard or an opening.

Label each piece on a sketch and tabulate its dimensions and area. A table makes the calculation easy to check, easy to revise when drawings change and easy for someone else to follow. Estimating software does the same thing behind the scenes, and understanding the hand method lets you check what the software reports.

Circles and curves

Circular and curved elements appear in tanks, columns, manholes, planters, curved walls and the radii at the corners of curbs and driveways. Use π to at least four figures (3.1416) or the calculator key. When a drawing gives a diameter, remember to halve it before squaring, or use the diameter form of the formula, A = π d² ÷ 4. The arc length of a sector, needed for curved edge forms and curbs, is a fraction of the circumference.

Wall areas and openings

Walls are measured as gross area, the perimeter times the height, less openings. Painters, drywall installers and cladding contractors each have rules about which openings to deduct: many estimators do not deduct small openings, because cutting around them takes as much time and wastes as much material as covering them, and deduct only openings above a stated size. The estimate should state the rule used so that quantities can be compared fairly.

Irregular shapes: the trapezoidal rule

Sites, ponds, cut slopes and curved boundaries often have no simple formula. A practical method is to draw a straight baseline through or along the shape and measure perpendicular offsets from the baseline to the boundary at equal intervals. Each strip between two offsets is then treated as a trapezoid. Adding the trapezoids gives the trapezoidal rule.

Here h is the interval between offsets and y₀ to yₙ are the offsets. The first and last offsets count half; all the others count fully. The rule treats the boundary between offsets as straight lines, so it slightly underestimates areas bounded by outward-curving edges and overestimates those bounded by inward-curving edges. Closer offsets make it more accurate.

Irregular shapes: Simpson's rule

Simpson's rule fits curves rather than straight lines through each group of three offsets, so it is usually more accurate for smoothly curving boundaries. It needs an even number of intervals, which means an odd number of offsets.

Odd-numbered offsets are multiplied by 4, even-numbered interior offsets by 2, and the first and last by 1. If a survey has an odd number of intervals, apply Simpson's rule to all but the last interval and the trapezoidal rule to the last one.

Triangles from three sides: Heron's formula

Survey plans and site sketches sometimes give the three sides of a triangle but not its height, for example when a lot is measured with a tape along its boundaries. Heron's formula gives the area directly from the three sides a, b and c using the semi-perimeter s.

Any polygon can be divided into triangles by drawing diagonals and measuring them, which is how sites were measured with chains and tapes for centuries, and how a quick check can be made today when a survey instrument is not available. Lesson 8 shows the coordinate method, which calculates the area of any polygon directly from the coordinates of its corners.

Units of area on site

Canadian projects measure building and site areas in square metres and larger land areas in hectares, although residential real estate is still often advertised in square feet. Converting with 1 m² = 10.7639 ft² is common when comparing a metric drawing with a product priced per square foot.

Building areas are also defined in different ways: gross floor area measured to the outside of walls, net or usable area measured to the inside, and rentable areas defined by standards such as those published by BOMA. The definition matters when areas are used for pricing, rent or code limits, and a drawing's area schedule should state which definition applies.