Geometry involves understanding shapes, angles, measurements, and spatial reasoning. This study guide covers the essential concepts you need to know for the ASVAB Mathematics Knowledge subtest, including formulas and definitions.
1. Area & Volume
Area and volume measure the space within a shape or a solid object.
Area of a Rectangle:
A=Length×Width
Volume of a Rectangular Prism:
V=Length×Width×Height
2. Angles
Angles measure the space between two intersecting lines.
Right Angle: A 90° angle (symbol: ⊾).
Complementary Angles: Two angles whose sum is 90°.
Example: 30° + 60° = 90°
Supplementary Angles: Two angles whose sum is 180°.
Example: 110° + 70° = 180°
Angles in a Triangle:
The sum of the interior angles of a triangle is always 180°.
Types of Angles:
| Angle Type | Measurement |
| Acute | Less than 90° |
| Right | Exactly 90° |
| Obtuse | Greater than 90° but less than 180° |
| Straight | Exactly 180° |
| Reflex | Greater than 180° |
Vertical Angles:
- Opposite angles formed by two intersecting lines.
- Always equal in measure.
3. Triangles
Triangles are three-sided polygons with specific angle and side properties.
Types of Triangles:
| Type | Properties |
| Equilateral | All sides equal, all angles 60°. |
| Isosceles | Two equal sides, two equal base angles. |
| Scalene | No equal sides, no equal angles. |
| Right Triangle | One 90° angle. |
Right Triangle Properties:
Pythagorean Theorem: Applies to right triangles:
a^2 + b^2 = c^2
where c is the hypotenuse (longest side).
Special Right Triangles:
- 30°-60°-90° Triangle: Side ratio = 1 : √3 : 2
- 45°-45°-90° Triangle: Side ratio = 1 : 1 : √2
Acute vs. Obtuse Triangles:
- Acute Triangle: All angles < 90°
- Obtuse Triangle: One angle > 90°
Area of a Triangle:
Area= 1/2 × base × height
Where:
base is the length of the bottom side of the triangle,
height is the perpendicular distance from the base to the opposite vertex.
4. Quadrilaterals
Quadrilaterals are four-sided polygons with specific properties.
Angles in a Quadrilateral:
- Sum of interior angles = 360°.
Types of Quadrilaterals:
| Quadrilateral Type | Properties |
| Parallelogram | Opposite sides are parallel and congruent. |
| Rectangle | Special parallelogram with all 90° angles. |
| Square | A rectangle with all sides equal. |
| Rhombus | Four equal sides, opposite angles are equal. |
| Trapezoid | Only one pair of parallel sides. |
| Kite | Two pairs of adjacent equal sides. |
Formulas for Quadrilaterals:
- Parallelogram Area: A=Base×Height
- Rectangle Area: A=Length×Width
- Rectangle Perimeter: P=2(Length+Width)
5. Circles
Circles have unique properties involving radius, diameter, and π.
Key Circle Terms:
| Term | Definition |
| Radius (r) | Distance from the center to the edge. |
| Diameter (D) | Distance across the circle through the center. D=2rD = 2rD=2r |
| Pi (π) | Approximation: 3.14; Ratio of circumference to diameter. |
Circle Formulas:
- Circumference: 2 × π × radius or π × diameter
- Area of a Circle: π × radius²
6. Properties of Parallel Lines & Transversals
When a transversal crosses parallel lines, it creates special angle relationships.
Angle Pairs:
| Type | Property |
| Corresponding Angles | Equal |
| Alternate Interior Angles | Equal |
| Alternate Exterior Angles | Equal |
| Consecutive Interior Angles | Sum to 180° |
7. Surface Area & Volume of 3D Shapes
Surface area covers the outside of a solid, while volume measures internal space.
Surface Area Formulas:
- Cube: 6 × side²
- Rectangular Prism: 2 × (length × width + length × height + width × height)
- Cylinder: 2 × π × radius × (radius + height)
- Sphere: 4 × π × radius²
Volume Formulas:
- Cylinder: π × radius² × height
- Sphere: (4/3) × π × radius³
- Cone: (1/3) × π × radius² × height
8. Combined Figures
Complex shapes can be broken into simpler geometric shapes for calculations.
Strategy:
- Break down into rectangles, triangles, and circles.
- Find individual areas/volumes and add or subtract as needed.
9. Coordinate Geometry
Coordinate geometry involves plotting points and finding slopes.
Slope Formula:
(m) = (y₂ – y₁) ÷ (x₂ – x₁)
- Measures the steepness of a line.
Example:
Find the slope between the points (2, 3) and (6, 11).
Using the formula:
m = (11 – 3) ÷ (6 – 2)
m = 8 ÷ 4
m = 2
So, the slope is 2.
Equation of a Line (Slope-Intercept Form):
y=mx+by = mx + by=mx+b
- m = slope, b = y-intercept.
Colinear Points:
- Points that lie on the same straight line.
Quadrants of the Cartesian Plane:
| Quadrant | Sign of (x, y) |
| I | (+, +) |
| II | (-, +) |
| III | (-, -) |
| IV | (+, -) |
Final Review
- Master area and volume formulas for 2D & 3D shapes.
- Understand angle relationships, triangle types, and quadrilateral properties.
- Memorize circle formulas (circumference, area).
- Apply coordinate geometry concepts for slopes and equations.
Next Section: Statistics and Probabilities
Previous Section: Algebra