Geometry: Practice Questions & Explanations

Area of a rectangle = length × width = 12 × 8 = 96 square units. Memorize basic geometry formulas: (1) Rectangle area = l × w, perimeter = 2(l + w); (2) Square area = s², perimeter = 4s; (3) Triangle area = ½ × base × height; (4) Circle area = πr², circumference = 2πr; (5) Parallelogram area = base × height; (6) Trapezoid area = ½(b₁ + b₂) × h. Perimeter sums the lengths around the boundary; area measures the surface inside. Volume formulas: rectangular prism = l × w × h; cylinder = πr²h; sphere = (4/3)πr³; cone = (1/3)πr²h. Units: area is in square units (sq ft, sq m, etc.); volume in cubic units; perimeter in linear units. The ASVAB MK includes several geometry questions; formula recognition is essential.

Area of a circle = πr², where r is the radius. With r = 5: A = π(5)² = 25π square units. Numerically: 25π ≈ 25 × 3.14159 ≈ 78.5 square units. Common ASVAB answer choices keep π symbolic to avoid arithmetic ambiguity. Related circle formulas: (1) Diameter = 2r (twice the radius); (2) Circumference = 2πr or πd (length around the circle); (3) Area = πr² (surface inside); (4) Arc length = (θ/360) × 2πr where θ is the central angle in degrees; (5) Sector area = (θ/360) × πr². Common error: confusing radius and diameter — read the problem carefully. The constant π ≈ 3.14 ≈ 22/7 (older approximation); for precise work, calculator π. Squaring the radius first, then multiplying by π, avoids confusion.

Pythagorean theorem: in a right triangle, a² + b² = c², where a and b are the legs and c is the hypotenuse (the longest side, opposite the right angle). Here: 3² + 4² = 9 + 16 = 25, so c² = 25, and c = √25 = 5. The 3-4-5 right triangle is a famous Pythagorean triple — memorize it and its multiples (6-8-10, 9-12-15, etc.). Other common Pythagorean triples to memorize: 5-12-13, 8-15-17, 7-24-25. Note: the hypotenuse is always the longest side. The Pythagorean theorem only applies to RIGHT triangles. Common applications: finding distances, ladder problems, navigation, construction. The theorem extends to coordinate geometry as the distance formula: distance = √((x₂-x₁)² + (y₂-y₁)²), which is just the Pythagorean theorem in disguise.

Volume of a rectangular prism (box) = length × width × height = 5 × 4 × 3 = 60 cubic units. Multiply in any order: 5 × 4 = 20, then 20 × 3 = 60. Volume formulas to know: (1) Rectangular prism: V = lwh; (2) Cube: V = s³ (side cubed); (3) Cylinder: V = πr²h (area of circular base times height); (4) Cone: V = (1/3)πr²h (one-third the cylinder); (5) Sphere: V = (4/3)πr³; (6) Pyramid: V = (1/3) × base area × height. Surface area is different — sum of all face areas. For a rectangular box, surface area = 2(lw + lh + wh). Units: volume is always cubic. Common error: confusing volume and surface area — read the question. Real-world applications: container capacity, concrete volume, water flow, gas volumes.

The sum of interior angles of any triangle is always 180°. This is a fundamental geometric fact. Other polygon angle sums: (1) Quadrilateral: 360°; (2) Pentagon: 540°; (3) Hexagon: 720°; (4) n-gon: (n − 2) × 180°. For regular polygons (all sides and angles equal): each interior angle = (n − 2) × 180° / n. Example: regular hexagon, each angle = 720°/6 = 120°. Triangle types by angles: (1) Acute — all angles < 90°; (2) Right — one angle = 90°; (3) Obtuse — one angle > 90°. Triangle types by sides: (1) Equilateral — all sides equal, all angles 60°; (2) Isosceles — two sides equal, two angles equal; (3) Scalene — no sides equal, no angles equal. Used to find missing angles: if two angles in a triangle are 50° and 70°, the third is 180° − 50° − 70° = 60°. Exterior angle of a triangle = sum of two non-adjacent interior angles.

Supplementary angles sum to 180°. So the other angle = 180° − 75° = 105°. Angle relationships to know: (1) Complementary — sum to 90° (think 'right angle'); (2) Supplementary — sum to 180° (think 'straight angle'); (3) Vertical (vertically opposite) — formed by intersecting lines; vertical angles are equal; (4) Adjacent — share a vertex and side, don't overlap; (5) Linear pair — adjacent angles forming a straight line; sum to 180° (always supplementary). Parallel lines cut by a transversal: (1) Corresponding angles — equal; (2) Alternate interior angles — equal; (3) Alternate exterior angles — equal; (4) Co-interior (same-side interior) angles — supplementary. Angle measurement: degrees (most common; circle = 360°), radians (advanced; 2π radians = 360°). Acute < 90°; right = 90°; obtuse 90°-180°; straight = 180°; reflex 180°-360°.

Circumference = πd = (22/7) × 14 = (22 × 14)/7 = 22 × 2 = 44 units. Or use 2πr with radius = 7: C = 2π(7) = 14π = 14 × (22/7) = 44. Circle formulas: (1) Diameter = 2 × radius, so radius = diameter/2; (2) Circumference C = πd = 2πr (length around the circle); (3) Area = πr² (surface inside); (4) The number π ≈ 3.14159 ≈ 22/7 (rough approximation). The 22/7 approximation works exactly when the radius or diameter is a multiple of 7. The ASVAB sometimes uses 22/7 to enable clean arithmetic without a calculator. For circles where π must remain symbolic, express answers in terms of π: a circle with radius 5 has C = 10π and A = 25π. The relationship C = πd is the geometric definition of π: π is the ratio of any circle's circumference to its diameter — always constant regardless of circle size.

Square: all four sides equal. Perimeter = 4 × side, so 36 = 4s, giving s = 9. Area = s² = 9² = 81 square units. Square properties to know: (1) All sides equal; (2) All angles 90°; (3) Diagonals are equal in length and bisect each other at 90°; (4) Diagonal length = s√2 (Pythagorean theorem with two sides of length s); (5) Perimeter = 4s; (6) Area = s²; (7) A square is a special rectangle, rhombus, and parallelogram. Don't confuse perimeter and area: perimeter is linear (boundary length), area is square (surface). Common error: dividing perimeter by 2 instead of 4 (forgetting all four sides), or finding side then forgetting to square it. Always identify what the problem asks for. Two-step problems are common on ASVAB MK: extract one quantity, use it to find another.

Sum of angles in a triangle = 180°. Third angle = 180° - 65° - 75° = 40°. The three interior angles of any triangle always total 180°, so once two are known the third is forced. Add the known angles first (65 + 75 = 140), then subtract from 180. A sanity check: 65°, 75° and 40° are all less than 90°, so this is an acute triangle, which is consistent. The distractors are simply other round numbers near the answer.

Perimeter of square = 4 × side = 4 × 9 = 36. Perimeter is the distance around a shape — for a square, four equal sides, so 4 × 9. The distractor 81 is the area (9 × 9), the most common mix-up; 18 is two sides (9 + 9), and 27 is three sides. Units confirm which you have: perimeter is in plain units (inches, feet), area in square units.

Pythagorean theorem: c² = a² + b² = 25 + 144 = 169. c = √169 = 13. 5-12-13 is one of the Pythagorean triples worth memorising, alongside 3-4-5 and 8-15-17 — the ASVAB uses them often because the square roots come out whole. The hypotenuse is always the longest side, opposite the right angle, so an answer smaller than 12 can be ruled out immediately. The distractors 14, 15 and 17 are not the square root of 169.

A quadrilateral (4 sides) has interior angles summing to 360°. Triangles sum to 180°; each additional side adds 180°. You can see why by splitting any quadrilateral along a diagonal into two triangles, each contributing 180°. The general formula is (n − 2) × 180° for an n-sided polygon: triangle 180°, quadrilateral 360°, pentagon 540° (the last distractor), hexagon 720°. The distractor 270° is three right angles, which is not a closed shape's total.

AREA of a triangle = ½ × base × height = ½ × 10 × 6 = ½ × 60 = 30 cm². ASVAB Mathematics Knowledge tests area formulas. Strategy: memorize the triangle area formula (½ × base × height) — a common error is forgetting the ½ and getting 60 (which is the area of a rectangle with those dimensions). The triangle is half of that rectangle. Units for area are squared (cm²). Key formulas to know: rectangle = l × w; triangle = ½bh; circle = πr².

PERIMETER of a rectangle = 2 × (length + width) = 2 × (12 + 5) = 2 × 17 = 34 inches. ASVAB Mathematics Knowledge tests perimeter vs area. Strategy: PERIMETER is the distance around (add all sides; for a rectangle, 2L + 2W); AREA is the space inside (L × W = 60 sq in here).

Common error: confusing perimeter with area — perimeter uses addition of sides and is in linear units (inches); area uses multiplication and squared units. The perimeter here is 34 inches (12+5+12+5).

AREA of a circle = πr² = 3.14 × 4² = 3.14 × 16 = 50.24 square units. ASVAB Mathematics Knowledge tests circle formulas. Strategy: square the radius FIRST (4² = 16), then multiply by π.

Common error: multiplying π × r first then squaring, or confusing area (πr²) with circumference (2πr = 2 × 3.14 × 4 = 25.12). Remember: AREA = πr² (squared, uses radius²); CIRCUMFERENCE = 2πr (the distance around). The radius is squared, not doubled, for area.

Use the PYTHAGOREAN THEOREM: a² + b² = c², where c is the hypotenuse. Here: 3² + 4² = c² → 9 + 16 = 25 = c². So c = √25 = 5. ASVAB Mathematics Knowledge tests the Pythagorean theorem. The 3-4-5 triangle is the most common 'Pythagorean triple' and appears frequently — worth memorizing (along with 5-12-13 and 6-8-10). Strategy: square the two legs, add them, then take the square root to find the hypotenuse. The hypotenuse is always the longest side, opposite the right angle.

Use the Pythagorean theorem: a² + b² = c², where c is the hypotenuse. Here 6² + 8² = 36 + 64 = 100, so c = √100 = 10. The 6-8-10 triangle is a multiple of the common 3-4-5 right triangle, so recognizing that pattern gives the answer instantly. Be careful: 100 is c², not c — you must take the square root. The hypotenuse is 10. Memorizing common Pythagorean triples (3-4-5, 5-12-13, 8-15-17) speeds up these geometry problems considerably.

The perimeter of a rectangle is 2 × (length + width), or 2L + 2W. Here perimeter = 2 × (9 + 4) = 2 × 13 = 26. The distractor 36 is the area (9 × 4), so don't confuse perimeter (distance around) with area (space inside). The perimeter is 26 units. A reliable approach is to add the length and width, then double, since a rectangle has two of each side. Always check whether a question asks for perimeter or area.

Circumference is π × diameter (C = πd), or equivalently 2πr. With a diameter of 10: C = 3.14 × 10 = 31.4. Be careful not to use the area formula (πr²); the area here would be 3.14 × 5² = 78.5, offered as a distractor. Since the diameter is given directly, C = πd is the fastest route. The circumference is 31.4. Remember that circumference uses the diameter (or 2× radius) to the first power, while area uses the radius squared.