ASVAB Mathematics Knowledge — Algebra, Geometry, and Formulas

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.

2⁵ means 2 multiplied by itself 5 times: 2 × 2 × 2 × 2 × 2 = 32. ASVAB Mathematics Knowledge tests exponents. Work step by step: 2×2=4, ×2=8, ×2=16, ×2=32. COMMON ERROR: confusing 2⁵ with 2 × 5 = 10 — an exponent means repeated MULTIPLICATION, not multiplying the base by the exponent. Powers of 2 to memorize: 2¹=2, 2²=4, 2³=8, 2⁴=16, 2⁵=32, 2⁶=64. Recognizing common powers speeds up the test.

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).

Use the DISTRIBUTIVE property: 4(x + 3) = 4x + 12. Then subtract 2x: 4x + 12 − 2x = 2x + 12. ASVAB Mathematics Knowledge tests simplifying expressions. Strategy: (1) distribute the 4 across the parentheses: 4·x + 4·3 = 4x + 12; (2) combine like terms: 4x − 2x = 2x, and keep the +12. Result: 2x + 12. COMMON ERROR: forgetting to distribute the 4 to BOTH terms in the parentheses, or trying to combine 12 with the x-terms (you can't combine unlike terms).

Use ORDER OF OPERATIONS (PEMDAS): multiplication before addition/subtraction. 3 × 4 = 12 first. Then: 6 + 12 − 2 = 18 − 2 = 16. ASVAB Mathematics Knowledge tests order of operations. PEMDAS: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right). COMMON ERROR: working strictly left to right (6 + 3 = 9, × 4 = 36, − 2 = 34) is WRONG — multiplication must come before addition. The correct answer is 16. Always do multiplication/division before addition/subtraction.

If x² = 49, then x = ±7 (both 7 AND −7), because 7² = 49 AND (−7)² = 49. ASVAB Mathematics Knowledge tests square roots and equations. KEY CONCEPT: when solving x² = a number, there are usually TWO solutions — a positive and a negative root — because squaring a negative number gives a positive result. COMMON ERROR: giving only the positive root (7). The square root symbol √ alone denotes the positive root, but solving the EQUATION x² = 49 yields both +7 and −7. Check: (−7)² = (−7)(−7) = 49. ✓

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.

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.