6.4 Coordinate Geometry, Perimeter, Area & Volume Formulas

Armed Forces Classification Test AFCT

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1 Chapter 1: Guide Scope, AFQT & Service-Specific Policy

1.1 Editorial Scope and Service-Specific AFCT Policy 1.2 AFQT Meaning and Study Priorities 1.3 Service Composites and Targeted Skill Planning 1.4 Format-Agnostic Test Preparation and Pacing

2 Chapter 2: General Science (GS)

2.1 Life Science: Cell Biology, Human Anatomy & Genetics 2.2 Physical Science: Matter, Energy, Forces & Motion 2.3 Earth & Space Science: Geology, Meteorology & Astronomy 2.4 Chemistry Fundamentals: Atomic Structure, Elements & Chemical Reactions

3 Chapter 3: Arithmetic Reasoning (AR)

3.1 Word Problems: Rates, Ratios, Proportions & Percentages 3.2 Financial & Distance Math: Interest, Profit/Loss, Speed & Time 3.3 Work & Mixture Problems: Joint Rates & Percentage Concentrations 3.4 Multi-Step Problem Solving & Algebraic Word Problems

4 Chapter 4: Word Knowledge (WK)

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5 Chapter 5: Paragraph Comprehension (PC)

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6 Chapter 6: Mathematics Knowledge (MK)

6.1 Pre-Algebra & Number Operations: Exponents, Radicals & Order of Operations 6.2 High School Algebra: Linear Equations, Inequalities & Polynomials 6.3 Geometry Fundamentals: Angles, Triangles, Quadrilaterals & Circles 6.4 Coordinate Geometry, Perimeter, Area & Volume Formulas

7 Chapter 7: Electronics Information (EI)

7.1 Electrical Principles: Voltage, Current, Resistance & Ohm's Law (V=IR, P=VI) 7.2 Electric Circuits: Series, Parallel & Combination Circuit Analysis 7.3 Electrical Components: Resistors, Capacitors, Inductors & Transformers 7.4 Electronic Semiconductors, Transistors, Diodes & Logic Gates

8 Chapter 8: Auto & Shop Information (AS)

8.1 Automotive Systems: Internal Combustion Engines, Fuel & Ignition Systems 8.2 Automotive Mechanical Systems: Transmission, Suspension, Brakes & Cooling 8.3 Hand & Power Tools: Identification, Uses & Fasteners 8.4 Shop Practice, Woodworking, Metalworking & Safety Rules

9 Chapter 9: Mechanical Comprehension (MC)

9.1 Simple Machines: Levers, Pulleys, Inclined Planes & Mechanical Advantage 9.2 Force, Motion, Work & Power: Newton's Laws & Energy Transformations 9.3 Fluid Dynamics & Pressure: Pascal's Principle, Hydraulics & Pneumatics 9.4 Mechanical Components: Gears, Belts, Springs & Friction

10 Chapter 10: Spatial Assembly Practice (AO Skill)

10.1 Spatial Connections and Assembly Patterns

Free exam prep AFCT Armed Forces Classification Test Practice Questions 100 questions Study Guide 37 topics Cheat Sheet 58 facts Flashcards 50 cards

6.4 Coordinate Geometry, Perimeter, Area & Volume Formulas

Key Takeaways Explain This Topic AI

Calculate distance d = sqrt((x2 - x1)^2 + (y2 - y1)^2) and midpoint M = ((x1 + x2)/2, (y1 + y2)/2) on the Cartesian coordinate plane.

Determine line slopes m = (y2 - y1)/(x2 - x1), remembering parallel lines have equal slopes and perpendicular lines have negative reciprocal slopes.

Master 2D perimeter and area formulas for rectangles, triangles, parallelograms, trapezoids, and circles.

Compute 3D volume and surface area for rectangular prisms, cubes, cylinders, cones, and spheres.

Break complex composite figures into basic geometric shapes to solve combined area and volume problems efficiently. Last updated: July 2026

6.4 Coordinate Geometry, Perimeter, Area & Volume Formulas Coordinate geometry merges algebraic equations with geometric space, while measurement formulas allow you to calculate perimeter, area, surface area, and volume of two- and three-dimensional figures. These topics build broad Mathematics Knowledge skills. The Cartesian Coordinate Plane The Cartesian coordinate plane is formed by two perpendicular number lines: the horizontal x-axis and the vertical y-axis , intersecting at the origin ( 0 , 0 ) (0,0) ( 0 , 0 ) . The axes divide the plane into four quadrants: Quadrant I: ( + x , + y ) (+x, +y) ( + x , + y ) Quadrant II: ( − x , + y ) (-x, +y) ( − x , + y ) Quadrant III: ( − x , − y ) (-x, -y) ( − x , − y ) Quadrant IV: ( + x , − y ) (+x, -y) ( + x , − y ) Essential Coordinate Formulas For any two points A ( x 1 , y 1 ) A(x_1, y_1) A ( x 1 , y 1 ) and B ( x 2 , y 2 ) B(x_2, y_2) B ( x 2 , y 2 ) : Distance Formula: Derived directly from the Pythagorean Theorem. d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 Midpoint Formula: Calculates the average of the x-coordinates and y-coordinates. M = ( x 1 + x 2 2 , y 1 + y 2 2 ) M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) M = ( 2 x 1 + x 2 , 2 y 1 + y 2 ) Slope Formula ( m m m ): Measures the steepness and direction of a line ( rise / run \,\text{rise} / \text{run}\, rise / run ). m = y 2 − y 1 x 2 − x 1 ( x 1 ≠ x 2 ) m = \frac{y_2 - y_1}{x_2 - x_1} \quad (x_1 \neq x_2) m = x 2 − x 1 y 2 − y 1 ( x 1  = x 2 ) Slope Classifications & Line Relationships Positive Slope ( m > 0 m > 0 m > 0 ): Line rises from left to right. Negative Slope ( m < 0 m < 0 m < 0 ): Line falls from left to right. Zero Slope ( m = 0 m = 0 m = 0 ): Horizontal line ( y = c y = c y = c ). Undefined Slope: Vertical line ( x = c x = c x = c ). Parallel Lines ( L 1 ∥ L 2 L_1 \parallel L_2 L 1 ∥ L 2 ): Slopes are identical ( m 1 = m 2 m_1 = m_2 m 1 = m 2 ). Perpendicular Lines ( L 1 ⊥ L 2 L_1 \perp L_2 L 1 ⊥ L 2 ): Slopes are negative reciprocals ( m 1 ⋅ m 2 = − 1 ⟹ m 2 = − 1 m 1 m_1 \cdot m_2 = -1 \implies m_2 = -\frac{1}{m_1} m 1 ⋅ m 2 = − 1 ⟹ m 2 = − m 1 1 ). Example: If m 1 = 3 4 m_1 = \frac{3}{4} m 1 = 4 3 , then m 2 = − 4 3 m_2 = -\frac{4}{3} m 2 = − 3 4 . Linear Equations of Lines

Form of Line Equation Standard Formula Parameters Slope-Intercept Form y = m x + b y = mx + b y = m x + b m = slope , b = y-intercept ( 0 , b ) m = \text{slope}, b = \text{y-intercept } (0,b) m = slope , b = y-intercept ( 0 , b ) Point-Slope Form y − y 1 = m ( x − x 1 ) y - y_1 = m(x - x_1) y − y 1 = m ( x − x 1 ) m = slope , ( x 1 , y 1 ) = known point m = \text{slope}, (x_1, y_1) = \text{known point} m = slope , ( x 1 , y 1 ) = known point Standard Form A x + B y = C Ax + By = C A x + B y = C A , B , C A, B, C A , B , C are integers, A ≥ 0 A \ge 0 A ≥ 0 Worked Example: Perpendicular Line Equation Find the equation of the line passing through point ( 4 , − 2 ) (4, -2) ( 4 , − 2 ) that is perpendicular to the line y = 2 x + 5 y = 2x + 5 y = 2 x + 5 . Identify given slope: m 1 = 2 m_1 = 2 m 1 = 2 . Determine perpendicular slope: m 2 = − 1 2 m_2 = -\frac{1}{2} m 2 = − 2 1 . Use Point-Slope form with point ( 4 , − 2 ) (4, -2) ( 4 , − 2 ) : y − ( − 2 ) = − 1 2 ( x − 4 ) ⟹ y + 2 = − 1 2 x + 2 y - (-2) = -\frac{1}{2}(x - 4) \implies y + 2 = -\frac{1}{2}x + 2 y − ( − 2 ) = − 2 1 ( x − 4 ) ⟹ y + 2 = − 2 1 x + 2 Subtract 2 2 2 from both sides: y = − 1 2 x y = -\frac{1}{2}x y = − 2 1 x 2D Perimeter & Area Formulas Perimeter ( P P P ) measures total distance around a 2D boundary (linear units). Area ( A A A ) measures total surface enclosed (square units).

Shape Perimeter Formula Area Formula Rectangle P = 2 l + 2 w P = 2l + 2w P = 2 l + 2 w A = l ⋅ w A = l \cdot w A = l ⋅ w Square P = 4 s P = 4s P = 4 s A = s 2 A = s^2 A = s 2 Triangle P = a + b + c P = a + b + c P = a + b + c A = 1 2 b h A = \frac{1}{2} b h A = 2 1 bh Parallelogram P = 2 a + 2 b P = 2a + 2b P = 2 a + 2 b A = b ⋅ h A = b \cdot h A = b ⋅ h Trapezoid P = a + b 1 + c + b 2 P = a + b_1 + c + b_2 P = a + b 1 + c + b 2 A = 1 2 ( b 1 + b 2 ) h A = \frac{1}{2}(b_1 + b_2) h A = 2 1 ( b 1 + b 2 ) h Rhombus / Kite P = 4 s P = 4s P = 4 s A = 1 2 d 1 d 2 A = \frac{1}{2} d_1 d_2 A = 2 1 d 1 d 2 Circle C = 2 π r C = 2\pi r C = 2 π r A = π r 2 A = \pi r^2 A = π r 2 Heron's Formula for Triangle Area: Given three side lengths a , b , c a, b, c a , b , c , calculate semi-perimeter s = a + b + c 2 s = \frac{a+b+c}{2} s = 2 a + b + c . Then Area A = s ( s − a ) ( s − b ) ( s − c ) A = \sqrt{s(s-a)(s-b)(s-c)} A = s ( s − a ) ( s − b ) ( s − c ) . 3D Geometry: Volume & Surface Area Formulas Volume ( V V V ) measures space enclosed within a 3D object (cubic units). Surface Area ( S A SA S A ) measures total outer surface area (square units). 3D Geometric Formulas Rectangular Prism (Box): Volume: V = l ⋅ w ⋅ h V = l \cdot w \cdot h V = l ⋅ w ⋅ h Surface Area: S A = 2 ( l w + l h + w h ) SA = 2(lw + lh + wh) S A = 2 ( lw + l h + w h ) Cube: Volume: V = s 3 V = s^3 V = s 3 Surface Area: S A = 6 s 2 SA = 6s^2 S A = 6 s 2 Right Circular Cylinder: Volume: V = π r 2 h V = \pi r^2 h V = π r 2 h Lateral Area: L A = 2 π r h LA = 2\pi r h L A = 2 π r h Total Surface Area: S A = 2 π r h + 2 π r 2 SA = 2\pi r h + 2\pi r^2 S A = 2 π r h + 2 π r 2 Right Circular Cone: Volume: V = 1 3 π r 2 h V = \frac{1}{3}\pi r^2 h V = 3 1 π r 2 h Slant Height: l = r 2 + h 2 l = \sqrt{r^2 + h^2} l = r 2 + h 2 Total Surface Area: S A = π r l + π r 2 SA = \pi r l + \pi r^2 S A = π r l + π r 2 Sphere: Volume: V = 4 3 π r 3 V = \frac{4}{3}\pi r^3 V = 3 4 π r 3 Surface Area: S A = 4 π r 2 SA = 4\pi r^2 S A = 4 π r 2 Square Pyramid: Volume: V = 1 3 s 2 h V = \frac{1}{3} s^2 h V = 3 1 s 2 h Composite Figures & Strategy Many high-level practice questions present composite 2D or 3D figures combining multiple basic shapes (e.g., a rectangle surmounted by a semicircle, or a cylinder topped by a cone). Strategy for Composite Figures Deconstruct: Break the figure into standard geometric components. Calculate: Find individual areas or volumes using standard formulas. Combine: Add component measurements (or subtract cutouts) to find total area or volume.

Test Your Knowledge

What is the equation of the line passing through point (4, -2) that is perpendicular to the line y = 2x + 5?

y = 2x - 10

y = -(1/2) * x + 4

y = (1/2) * x - 4

y = -(1/2) * x Check Answer Need a hint? AI

Test Your Knowledge

What is the distance between points A(-3, 2) and B(5, -4) on the Cartesian coordinate plane?

14 Check Answer Need a hint? AI

Test Your Knowledge

A right circular cylinder has a height of 10 cm and a base radius of 3 cm. What is its total surface area?

60 * pi sq cm

90 * pi sq cm

78 * pi sq cm

84 * pi sq cm Check Answer Need a hint? AI

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A trapezoid has parallel bases of length 14 ft and 20 ft, and a height of 8 ft. What is its area?

136 sq ft

272 sq ft

160 sq ft

112 sq ft Check Answer Need a hint? AI

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