ASVAB Mathematics Knowledge — Geometry Formulas

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°.

Multiply coefficients and apply exponent rules to variables. Coefficients: 3 × 4 = 12. Variables: x² × x³ — use the product rule a^m × a^n = a^(m+n), so x² × x³ = x^(2+3) = x⁵. Combined: 12x⁵. Common error: multiplying exponents instead of adding (multiplication of like bases ADDS exponents; raising a power to a power MULTIPLIES exponents). Distinguish: (1) x² × x³ = x⁵ (add); (2) (x²)³ = x⁶ (multiply). Other key exponent operations: (1) (3x²y)(4xy³) = 12 x³y⁴ (multiply coefficients, add exponents of each variable separately); (2) 12x⁵ ÷ 3x² = 4x³ (divide coefficients, subtract exponents); (3) (2x³)² = 4x⁶ (raise both coefficient and variable to the outside exponent); (4) (3x²)³ = 27x⁶ (3³ = 27, and (x²)³ = x⁶). Master these patterns to handle polynomial arithmetic efficiently.

When multiplying powers with the same base, add the exponents: 2³ × 2⁴ = 2^(3+4) = 2⁷. This is the product rule for exponents. Do not multiply the exponents (that rule applies to a power raised to a power) and do not change the base. So 2⁷ = 128 if you compute it out, but in simplified exponential form the answer is 2⁷. The common mistakes are adding the bases or multiplying the exponents; remember: same base, multiply the terms, add the exponents.

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 square root of 144 is the number that, multiplied by itself, gives 144. Since 12 × 12 = 144, √144 = 12. Memorizing perfect squares up to at least 15² (1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225) makes these instant. The distractor 72 is 144 ÷ 2, a common error of halving instead of taking the root. The square root of 144 is 12.

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.

When raising a power to another power, multiply the exponents: (3²)³ = 3^(2×3) = 3⁶. This is the power-of-a-power rule, distinct from the product rule (where you add exponents for multiplication of like bases). So (3²)³ = 3⁶, which equals 729 if computed out. The common mistake is adding the exponents (giving 3⁵) instead of multiplying them. Remember: multiply exponents when a power is raised to a power; add exponents when multiplying like bases. The simplified form is 3⁶.