4.1 Perimeter & area (rectangles, triangles, circles, L-shapes)

Measuring Flat Spaces on the Job

Almost every trade measures flat surfaces. Flooring installers price a room by its area, painters buy paint by wall area, and fence crews order material by perimeter. On the ACT WorkKeys Applied Math test you receive a formula sheet, so you do not have to memorize these relationships. What you must do is choose the right formula for the situation and combine formulas when a shape is not a simple rectangle. This section covers perimeter, the area of rectangles, triangles, and circles, and the composite (L-shaped) figures that show up at Levels 5 through 7.

The core formulas

Two habits prevent most mistakes. First, remember that perimeter is a length (feet, meters) while area is a surface (square feet, square meters). Check that your units match what the question asks. Second, in the circle formulas r is the radius, which is half the diameter. If a problem gives you a diameter, cut it in half before you square it.

Rectangles: flooring and fencing

Flooring example. A break room measures 12 ft by 15 ft. Vinyl plank flooring is sold in boxes that each cover 20 sq ft. How many boxes are needed?

Area = L x W = 12 x 15 = 180 sq ft. Boxes = 180 / 20 = 9 boxes. Because 180 divides evenly, no rounding is needed. If the store instead sold boxes covering 22 sq ft, you would compute 180 / 22 = 8.18 and round up to 9 boxes, because you can never buy a fraction of a box.

Fencing example. A rectangular storage yard is 40 ft long and 25 ft wide. Fencing comes in 8 ft panels. How many panels are required to enclose the yard?

Perimeter = 2(L + W) = 2(40 + 25) = 2 x 65 = 130 ft. Panels = 130 / 8 = 16.25, so round up to 17 panels. Notice this problem uses perimeter, not area, because a fence follows the outside edge of the yard rather than covering the ground inside it.

Triangles

Triangle example. A triangular flower bed has a base of 18 ft and a height of 12 ft. How much mulch area must be covered?

Area = 1/2 x b x h = 1/2 x 18 x 12 = 1/2 x 216 = 108 sq ft. The height must be the perpendicular distance from the base to the opposite point, not the slanted side. WorkKeys diagrams label the perpendicular height for you, so use the labeled value rather than a sloped edge.

Circles

Circle example. A circular patio has a radius of 7 ft. Using pi = 3.14, find its area and the length of edging needed to go around it.

Area = pi x r x r = 3.14 x 7 x 7 = 3.14 x 49 = 153.86 sq ft. Circumference (the edging distance) = 2 x pi x r = 2 x 3.14 x 7 = 43.96 ft. If edging is sold by the whole foot, you would round up and buy 44 ft.

Composite and L-shaped areas: the Level 6 to 7 skill

Real rooms are rarely simple rectangles. The reliable method is to decompose the shape into rectangles, find each area, and then add them together. A useful cross-check is to take the full outer rectangle and subtract the missing piece.

L-shaped example. An office floor is L-shaped. The full outer rectangle would be 20 ft by 14 ft, but an 8 ft by 6 ft corner is cut out because that alcove will not be floored. Find the carpet area.

Subtract method: Full rectangle = 20 x 14 = 280 sq ft. Cut-out = 8 x 6 = 48 sq ft. Carpet area = 280 - 48 = 232 sq ft.

Add method (as a check): Split the L into two rectangles. One piece is 20 ft x 8 ft = 160 sq ft, and the remaining strip is 12 ft x 6 ft = 72 sq ft. Total = 160 + 72 = 232 sq ft. Both methods give the same answer, which confirms the result.

Paint example combining steps. A wall is 24 ft long and 9 ft high, with a door 3 ft by 7 ft that will not be painted. One gallon of paint covers 350 sq ft. How many gallons are needed for two coats?

Wall area = 24 x 9 = 216 sq ft. Door = 3 x 7 = 21 sq ft. Paintable area = 216 - 21 = 195 sq ft. Two coats need 195 x 2 = 390 sq ft of coverage. Gallons = 390 / 350 = 1.11, so round up to 2 gallons.

Test tips

• Read whether the question wants perimeter (edging, fencing, trim, baseboard) or area (flooring, paint, sod, tile).
• For any "how many boxes, gallons, or panels" question, round up, because partial units cannot be purchased.
• Decompose composite shapes into rectangles, and verify with the add-and-subtract cross-check when time allows.
• Always halve a diameter to get the radius before you use either circle formula.