7.2 Basic Geometry and Measurement
Key Takeaways
- OPM names basic geometry as Occupational Math content, and its preparation tips advise drawing diagrams to visualise scenarios.
- Perimeter is a length, area is a length squared, and volume is a length cubed, so the answer units tell you which is wanted.
- Circle formulas take the radius, but questions supply the diameter; using the diameter in the area formula makes the answer four times too large.
- When the answer counts physical items such as cartons or gallons, round up regardless of the decimal, because 2.16 gallons means buying 3.
- Doubling both sides of a rectangle doubles the perimeter but quadruples the area, which is why a square yard is 9 square feet and not 3.
7.2 Basic Geometry and Measurement
Quick Summary: OPM lists basic geometry as Occupational Math content, and its preparation tip — draw diagrams whenever possible to help visualise different scenarios — is aimed squarely at these items. What is tested is practical measurement: perimeter, area, volume, and unit conversion in ordinary settings such as floor space, storage, and materials.
Perimeter, area, and volume
The first decision on any geometry item is which of the three you need. The units in the answer options tell you.
| Quantity | What it measures | Units |
|---|---|---|
| Perimeter | Distance around a shape | feet, metres — linear |
| Area | Surface covered | square feet, square metres |
| Volume | Space filled | cubic feet, cubic metres |
If the options are in square feet, you need an area. If they are in plain feet, a perimeter. This single check resolves a surprising number of items before any calculation.
Core formulas
| Shape | Formula |
|---|---|
| Rectangle perimeter | 2 × (length + width) |
| Rectangle area | length × width |
| Square area | side × side |
| Triangle area | (base × height) ÷ 2 |
| Circle circumference | pi × diameter |
| Circle area | pi × radius × radius |
| Rectangular box volume | length × width × height |
Use 3.14 for pi unless the question specifies otherwise.
A room measures 18 feet by 14 feet.
Perimeter = 2 × (18 + 14) = 2 × 32 = 64 feet. Area = 18 × 14 = 252 square feet.
A question about baseboard trim needs the perimeter; a question about carpet needs the area. The word in the question — around, border, edging versus cover, flooring, surface — points to which.
Circles
Circles appear less often than rectangles, but they carry the single most reliable trap in this content area: the formulas take the radius, and questions almost always give the diameter.
A circular conference table has a diameter of 6 feet. Protective edging is sold by the foot. How much edging is needed?
Circumference = pi × diameter = 3.14 × 6 = 18.84 feet.
The same table needs a fitted cover. What area must the cover have?
The radius is half the diameter, so 6 ÷ 2 = 3 feet. Area = 3.14 × 3 × 3 = 28.26 square feet.
Feeding the diameter straight into the area formula gives 3.14 × 6 × 6 = 113.04, which is exactly four times too large — and that value is a dependable distractor. Circumference is the one circle formula that uses the diameter directly; area does not. Halve first, then square.
Working with cost per unit area
Most practical geometry items pair a measurement with a rate.
Carpet costs $4.25 per square foot. How much for the 18-by-14-foot room?
Area = 252 square feet. Cost = 252 × 4.25 = $1,071.
Storage boxes measure 2 feet by 1.5 feet by 1 foot. How many fit in 90 cubic feet of shelving?
Box volume = 2 × 1.5 × 1 = 3 cubic feet. Boxes = 90 ÷ 3 = 30.
Materials come in whole units
Practical measurement items usually end in a purchase, and you cannot buy 12.6 cartons of tile or 2.16 gallons of paint. When the answer is a count of physical items, round up regardless of the decimal.
Floor tile is sold in cartons that cover 20 square feet each. How many cartons cover the 18-by-14-foot room?
Area = 252 square feet. 252 ÷ 20 = 12.6, so order 13 cartons. Twelve cartons leave 12 square feet bare.
One gallon of paint covers 350 square feet, and a 42-by-9-foot wall needs two coats.
Wall area = 42 × 9 = 378 square feet. Two coats = 756 square feet. 756 ÷ 350 = 2.16, so buy 3 gallons.
Ordinary rounding happens to give the right carton answer, because 12.6 rounds up anyway. It gives the wrong paint answer, because 2.16 rounds down to 2 and two gallons cover only 700 of the 756 square feet needed. The decision to round up is made by the situation, never by the digit after the decimal point. Read the question for the word that signals it — how many are needed, how many must be ordered, at minimum — and separate it from questions asking for the bare measurement, which stay exact.
Composite shapes
An L-shaped area is handled by splitting or subtracting. Draw it — this is exactly the case OPM's diagram tip addresses.
An office floor is an L-shape: a 20-by-12-foot rectangle with a 6-by-5-foot rectangular alcove removed from one corner. What is the floor area?
Subtraction method: 20 × 12 = 240, minus 6 × 5 = 30, giving 210 square feet.
Addition method: split the L into a 20-by-7 rectangle (140) and a 14-by-5 rectangle (70), giving 140 + 70 = 210 square feet. ✓
The two methods agreeing is a strong check. When a composite item allows both, doing both takes little time and is decisive.
Units and conversion
Unit errors cause more wrong answers here than formula errors. Convert everything to one unit before calculating.
| Conversion | Factor |
|---|---|
| 1 foot | 12 inches |
| 1 yard | 3 feet |
| 1 square yard | 9 square feet |
| 1 cubic yard | 27 cubic feet |
| 1 metre | 100 centimetres |
| 1 square metre | 10,000 square centimetres |
| 1 pound | 16 ounces |
| 1 gallon | 4 quarts |
The rows worth memorising are the square and cubic ones, because they trip people up. A square yard is 9 square feet, not 3 — because both dimensions convert. A cubic yard is 27 cubic feet, because all three do.
A room is 18 by 14 feet. Flooring is sold by the square yard. How many square yards are needed?
Area = 252 square feet. 252 ÷ 9 = 28 square yards.
Dividing by 3 instead gives 84, and 84 is reliably offered as an option.
A shelf is 4 feet 6 inches long. Express in feet.
6 inches = 6 ÷ 12 = 0.5 feet, so 4.5 feet. Writing 4.6 feet is a standard error — inches are twelfths, not tenths.
When a dimension changes
Some items ask about the effect of a change rather than a total, and intuition is unreliable here because perimeter and area do not scale the same way.
Doubling one side doubles the area but adds far less than double to the perimeter. Doubling both sides doubles the perimeter and quadruples the area, because area multiplies two lengths and each one has doubled. Tripling both sides multiplies the area by nine.
A 2-by-3-foot sign has a perimeter of 10 feet and an area of 6 square feet. Enlarged to 4 by 6 feet, its perimeter is 20 feet — doubled — but its area is 24 square feet, four times the original.
Testing a claim with a small concrete rectangle like this settles the question in seconds and is far safer than reasoning about it in the abstract. The same logic explains the square-yard conversion above: the factor is squared because two dimensions convert, and it is cubed for volume because three do.
A working procedure
- Draw the shape and label every measurement given.
- Read the answer options' units to determine whether perimeter, area, or volume is wanted.
- Convert all measurements to one unit before calculating, and halve any diameter you were given.
- Split or subtract if the shape is composite.
- Apply any rate given in the problem.
- Round up if the answer counts physical items such as cartons, gallons, or rolls.
- Check the units of your answer match the options.
The last step catches the most common failure. If you computed 252 and the options are in square yards, you are not finished.
A room measures 18 feet by 14 feet, and flooring is sold by the square yard. How many square yards are required?
An office floor is a 20-by-12-foot rectangle with a 6-by-5-foot alcove removed from one corner. What is the floor area?
A shelf measures 4 feet 6 inches. Expressed in feet, what is its length?
One gallon of paint covers 350 square feet. A wall measuring 42 feet by 9 feet needs two coats. How many gallons must be purchased?