Geometry

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×WidthA = \text{Length} \times \text{Width}A=Length×Width

Volume of a Rectangular Prism:

V=Length×Width×HeightV = \text{Length} \times \text{Width} \times \text{Height}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:

a2+b2=c2a^2 + b^2 = c^2a2+b2=c2

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:

A=12×Base×HeightA = \frac{1}{2} \times \text{Base} \times \text{Height}A=21​×Base×Height

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×HeightA = \text{Base} \times \text{Height}A=Base×Height
  • Rectangle Area: A=Length×WidthA = \text{Length} \times \text{Width}A=Length×Width
  • Rectangle Perimeter: P=2(Length+Width)P = 2(\text{Length} + \text{Width})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: C=πDorC=2πrC = \pi D \quad \text{or} \quad C = 2\pi rC=πDorC=2πr
  • Area of a Circle: A=πr2A = \pi r^2A=πr2

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: SA=6s2SA = 6s^2SA=6s2
  • Rectangular Prism: SA=2lw+2lh+2whSA = 2lw + 2lh + 2whSA=2lw+2lh+2wh
  • Cylinder: SA=2πrh+2πr2SA = 2\pi rh + 2\pi r^2SA=2πrh+2πr2
  • Sphere: SA=4πr2SA = 4\pi r^2SA=4πr2

Volume Formulas:

  • Cylinder: V=πr2hV = \pi r^2 hV=πr2h
  • Sphere: V=43πr3V = \frac{4}{3} \pi r^3V=34​πr3
  • Cone: V=13πr2hV = \frac{1}{3} \pi r^2 hV=31​πr2h

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−y1x2−x1m = \frac{y_2 – y_1}{x_2 – x_1}m=x2​−x1​y2​−y1​​

  • Measures the steepness of a line.

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

 

Ready to test your math knowledge?