Applied Geometry: Practice Questions & Explanations

Area of a rectangle = length × width = 8 × 5 = 40 square feet. Common geometry formulas tested on ASVAB: Rectangle area = L × W; Square area = side²; Triangle area = (1/2) × base × height; Circle area = π × radius² (use π ≈ 3.14 or 22/7). Rectangle perimeter = 2L + 2W; Square perimeter = 4 × side; Circle circumference = 2 × π × radius (or π × diameter). Watch units: linear measurements like perimeter are in regular units (feet, meters); area measurements are in square units (sq ft, sq m); volume is in cubic units (cu ft, cu m). Common mistake: confusing perimeter (length around) with area (space inside). For a rectangle: 8 ft long, 5 ft wide — perimeter = 2(8) + 2(5) = 26 ft; area = 40 sq ft. Real-world applications: flooring (area), fencing (perimeter), paint (area), trim (perimeter).

Volume of a rectangular prism (or box) = length × width × height = 20 × 10 × 5 = 1,000 cubic feet. Volume formulas commonly tested: Rectangular prism = L × W × H; Cube = side³; Cylinder = π × r² × h; Sphere = (4/3) × π × r³. Volume units: cubic feet (cu ft or ft³), cubic meters, gallons (1 cu ft ≈ 7.48 gallons; useful for converting pool volume to gallons of water needed). Applications: pool capacity, container volume, concrete needed, water/gas volume in a tank. Note the relationship: doubling all dimensions multiplies volume by 8 (2³); doubling just one dimension doubles the volume. For irregular shapes, find the volume by computing the volume of regular sections and summing. Practice converting between cubic feet and gallons: 1 cubic foot ≈ 7.48 gallons.

Circumference = π × diameter. Using π ≈ 22/7: C = (22/7) × 14 = 22 × 2 = 44 feet. Alternative: C = 2 × π × radius; radius = diameter/2 = 7; C = 2 × (22/7) × 7 = 2 × 22 = 44 feet. Both forms of the formula give the same answer. When diameter is a multiple of 7, using π = 22/7 makes the arithmetic clean. Use π ≈ 3.14 for other diameters. Don't confuse circumference (distance around) with area (space inside): area of this fountain = π × r² = (22/7) × 49 = 154 square feet. Many ASVAB problems test the difference. Real-world applications: edging needed (circumference), surface area for paint (area), water volume (volume formula involving radius squared). Memorize: C = π × d = 2π × r; A = π × r². The square in the area formula makes area grow much faster than circumference as radius increases.

Multi-step problem: first find the area, then multiply by the unit cost. Area = 12 × 15 = 180 square feet. Cost = 180 × $4 = $720. Common mistake: computing the perimeter (2 × 12 + 2 × 15 = 54 feet) instead of the area — but carpet covers the floor area, not just the perimeter. The perimeter would be relevant for baseboards or molding. Always identify whether the problem asks for: linear measurement (perimeter, length, distance) — uses regular units; area measurement (floor coverage, paint coverage, land area) — uses square units; volume measurement (water, air, soil) — uses cubic units. Real-world applications: flooring/carpet (area × cost per sq ft); paint (area × cost per sq ft, with coverage adjustments for second coats); concrete (volume × cost per cubic yard); gravel (volume × cost per ton). Step-by-step approach to multi-step problems: identify all required calculations, solve each in turn, combine.

Area = length × width = 15 × 12 = 180 square feet. Carpet covers a flat surface, so you need area, not perimeter. The distractor 54 sq ft is the perimeter (15 + 12 + 15 + 12 = 54) mislabelled as area, and 27 is just 15 + 12. Always ask whether the question wants a length (perimeter), a surface (area) or a space (volume) before picking the formula.

Area = π r² = 3.14 × 7² = 3.14 × 49 ≈ 153.86 square inches. Square the radius before multiplying by π — the most common error is doubling it instead (3.14 × 14 = 43.96, the first distractor, which is actually the circumference). 87.92 is 2 × 43.96 and 219.8 is 3.14 × 70. If the question gives a diameter instead, halve it first to get the radius.

Volume = length × width × height = 4 × 3 × 2 = 24 cubic feet. Volume is a three-dimensional measure, so all three dimensions multiply. The distractor 9 is 4 + 3 + 2 (adding instead of multiplying), 18 and 36 come from multiplying the wrong pair twice (3 × 2 × 3 and 4 × 3 × 3). Units matter too: feet × feet × feet gives cubic feet, which is what the answer choices show.

The area of a rectangle is length × width. Here area = 12 ft × 9 ft = 108 square feet. Carpeting a floor is an area problem, so you multiply the two dimensions. Be careful not to confuse area with perimeter: the perimeter (distance around) would be 2 × (12 + 9) = 42 feet, which is offered as a distractor. Square footage of a floor is always length times width. The room needs 108 square feet of carpet.

The area of a circle is π × r². Here r = 7, so r² = 49, and area = (22/7) × 49 = 22 × 7 = 154 square feet. Using 22/7 for π is convenient when the radius is a multiple of 7 because the 7 cancels neatly. Don't confuse area (πr²) with circumference (2πr); the circumference here would be 2 × (22/7) × 7 = 44 feet, offered as a distractor. The garden's area is 154 square feet. Remember area uses the radius squared, while circumference uses the radius to the first power.