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°.

Perimeter of square = 4 × side = 4 × 9 = 36.

Pythagorean theorem: c² = a² + b² = 25 + 144 = 169. c = √169 = 13.

A quadrilateral (4 sides) has interior angles summing to 360°. Triangles sum to 180°; each additional side adds 180°.

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.