Prior Learning Formulae AI SL & AI HL: Complete Foundation Guide for IB Mathematics
Welcome to the definitive guide for Prior Learning Formulae in IB Mathematics Applications and Interpretation for both Standard Level and Higher Level students. Success in IB Math AI begins with a rock-solid foundation in fundamental mathematics. This comprehensive resource covers all prerequisite concepts including geometry and mensuration, algebra, coordinate geometry, trigonometry, and essential mathematical techniques that every AI SL and AI HL student must master. Whether you're just starting your IB journey or reviewing foundations before tackling advanced topics, this guide provides the complete mathematical toolkit you need.
Why Prior Learning is Critical for Both SL and HL
IB Mathematics AI at both Standard and Higher Level assumes competency with foundational mathematical concepts learned in previous years. These prior learning formulas aren't just prerequisite knowledge—they're tools you'll use daily throughout your IB Math AI course. From calculating areas in optimization problems to applying the distance formula in function analysis, prior learning concepts integrate seamlessly into every IB Math AI topic. Students who invest time mastering these foundations dramatically improve their performance, confidence, and exam results.
Geometry and Mensuration: 2D Shapes
Geometry and mensuration deal with measuring shapes—their perimeters, areas, volumes, and surface areas. These calculations appear throughout IB Math AI in modeling, optimization, and real-world applications. Explore our Triangle Area Calculator and Circle Area Calculator for practice.
Area Formulas for 2D Shapes
Area measures the amount of space inside a two-dimensional shape, always expressed in square units (cm², m², km²).
where \( l \) = length, \( w \) = width
where \( s \) = side length
where \( b \) = base, \( h \) = perpendicular height
where \( b \) = base, \( h \) = perpendicular height (not slant height!)
where \( a \) and \( b \) are the parallel sides, \( h \) = perpendicular height
where \( r \) = radius (\( \pi \approx 3.14159 \))
Find the area of a circle with radius 5 cm.
Solution:
\( A = \pi r^2 = \pi(5)^2 = 25\pi \approx 78.54 \text{ cm}^2 \)
Perimeter and Circumference Formulas
Perimeter measures the distance around the outside of a 2D shape, expressed in linear units (cm, m, km).
Sum of all three side lengths
where \( d = 2r \) is the diameter
| 2D Shape | Area Formula | Perimeter Formula |
|---|---|---|
| Rectangle | \( A = lw \) | \( P = 2(l+w) \) |
| Square | \( A = s^2 \) | \( P = 4s \) |
| Triangle | \( A = \frac{1}{2}bh \) | \( P = a+b+c \) |
| Parallelogram | \( A = bh \) | \( P = 2(a+b) \) |
| Trapezoid | \( A = \frac{1}{2}(a+b)h \) | Sum of all sides |
| Circle | \( A = \pi r^2 \) | \( C = 2\pi r \) |
Geometry and Mensuration: 3D Shapes
Three-dimensional shapes have volume (space inside) and surface area (total area of all faces). These concepts are essential for modeling real-world objects and optimization problems. Practice with our Sphere Volume Calculator.
Volume Formulas for 3D Shapes
Volume measures the three-dimensional space inside a solid object, expressed in cubic units (cm³, m³, L).
where \( l \) = length, \( w \) = width, \( h \) = height
where \( s \) = edge length
where \( A \) = cross-sectional area, \( h \) = height/length
Works for any prism (triangular, pentagonal, etc.)
where \( r \) = radius of circular base, \( h \) = height
where \( r \) = radius
where \( r \) = radius of base, \( h \) = perpendicular height
where \( B \) = area of base, \( h \) = perpendicular height
Notice that cones and pyramids have volumes exactly \( \frac{1}{3} \) of their corresponding cylinder or prism with the same base and height. This is because these "pointed" shapes occupy one-third the space of their "flat-topped" counterparts.
A cylindrical water tank has radius 2 m and height 5 m. Find its volume.
Solution:
\( V = \pi r^2 h = \pi(2)^2(5) = 20\pi \approx 62.83 \text{ m}^3 \)
Surface Area Formulas for 3D Shapes
Surface area is the total area of all faces of a 3D object, measured in square units.
Sum of areas of all six rectangular faces
Six identical square faces
Two circular bases (\( 2\pi r^2 \)) plus curved surface (\( 2\pi rh \))
Area of the curved surface only (excluding circular bases)
| 3D Shape | Volume Formula | Surface Area Formula |
|---|---|---|
| Cuboid | \( V = lwh \) | \( SA = 2(lw+lh+wh) \) |
| Cube | \( V = s^3 \) | \( SA = 6s^2 \) |
| Cylinder | \( V = \pi r^2h \) | \( SA = 2\pi r^2 + 2\pi rh \) |
| Sphere | \( V = \frac{4}{3}\pi r^3 \) | \( SA = 4\pi r^2 \) |
| Cone | \( V = \frac{1}{3}\pi r^2h \) | \( SA = \pi r^2 + \pi rl \)* |
| Pyramid | \( V = \frac{1}{3}Bh \) | \( SA = B + \text{lateral area} \) |
* \( l \) = slant height for cone surface area
Coordinate Geometry
Coordinate geometry (analytic geometry) uses algebraic methods to study geometric shapes on the coordinate plane. These formulas are fundamental for functions, graphing, and calculus in IB Math AI. Use our Distance Calculator for practice.
Distance Between Two Points
Distance between points \( (x_1, y_1) \) and \( (x_2, y_2) \)
Derived from Pythagorean theorem
Find the distance between \( A(1, 2) \) and \( B(7, 10) \)
Solution:
\( d = \sqrt{(7-1)^2 + (10-2)^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \) units
Midpoint of a Line Segment
Midpoint between \( (x_1, y_1) \) and \( (x_2, y_2) \)
Average the x-coordinates and average the y-coordinates
Find the midpoint of the line segment joining \( A(3, 4) \) and \( B(9, 12) \)
Solution:
\( M = \left(\frac{3+9}{2}, \frac{4+12}{2}\right) = (6, 8) \)
Slope of a Line
Slope between points \( (x_1, y_1) \) and \( (x_2, y_2) \)
- Positive slope: Line rises from left to right
- Negative slope: Line falls from left to right
- Zero slope: Horizontal line (\( y = c \))
- Undefined slope: Vertical line (\( x = c \))
Equations of Lines
where \( m \) = slope, \( c \) = y-intercept
Most common form for IB exams
Useful when you know slope and one point \( (x_1, y_1) \)
General linear equation form
Basic Trigonometry
Trigonometry studies relationships between angles and sides in triangles. These ratios extend to circular functions and wave modeling throughout IB Math AI.
Right Triangle Trigonometry
For right triangles: \( a, b \) are legs, \( c \) is hypotenuse
A right triangle has opposite side 3 cm and hypotenuse 5 cm. Find \( \sin\theta \).
Solution:
\( \sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{3}{5} = 0.6 \)
Basic Algebra
Solving Linear Equations
- Expand brackets if necessary
- Collect like terms on each side
- Move variable terms to one side, constants to the other
- Divide both sides by the coefficient of the variable
Expanding and Factoring
Number Operations and Properties
Fractions, Decimals, and Percentages
Fluency with number conversions is essential throughout IB Math AI. Use our Percentage Calculator and Fraction Calculator for practice.
Divide numerator by denominator
Interactive Geometry Calculator
Area Calculator for Common Shapes
Calculate areas of rectangles, triangles, and circles
Study Strategies for Prior Learning Success
Building Strong Foundations
- Daily Practice: Spend 15-20 minutes daily reviewing prior learning concepts. Consistent practice builds automaticity
- Create Formula Cards: Make flashcards for each formula. Include the formula, what each variable represents, and one example
- Draw Diagrams: Visualize geometry problems by sketching labeled diagrams before calculating
- Understand Units: Always include units in your answers. Area uses square units, volume uses cubic units, distance uses linear units
- Check Reasonableness: Estimate answers mentally before calculating to catch errors
- Connect Concepts: Recognize relationships (e.g., circle circumference and area both involve \( \pi r \))
Common Mistakes to Avoid
| Common Error | Correct Approach | Example |
|---|---|---|
| Confusing area and perimeter | Area = square units, Perimeter = linear units | Rectangle 5×3: Area = 15 cm², Perimeter = 16 cm |
| Forgetting to square in distance formula | Square BOTH differences before adding | \( d = \sqrt{(\Delta x)^2 + (\Delta y)^2} \) |
| Using diameter instead of radius | Check if given diameter or radius (\( r = \frac{d}{2} \)) | Circle with \( d = 10 \): use \( r = 5 \) |
| Wrong formula for trapezoid | Average of parallel sides × height | \( A = \frac{1}{2}(a+b)h \), not \( (a+b)h \) |
| Mixing up volume formulas | Remember \( \frac{1}{3} \) for cones and pyramids | Cone: \( V = \frac{1}{3}\pi r^2h \), not \( \pi r^2h \) |
Prior Learning Connections to IB Math AI Topics
Understanding how prior learning connects to IB Math AI content helps you see the relevance and importance of these foundational formulas.
Where Prior Learning Appears in AI SL & HL
- Geometry in Optimization: Finding maximum areas and minimum perimeters in calculus applications
- Volume in Rates of Change: Calculating how quickly a container fills or empties
- Coordinate Geometry in Functions: Understanding domain, range, and transformations
- Distance Formula in Modeling: Calculating distances for real-world scenarios
- Trigonometry in Periodic Functions: Modeling waves, seasons, and cyclical phenomena
- Area Calculations in Statistics: Understanding probability distributions and histograms
Technology and Calculator Skills
- Memory Storage: Store values of \( \pi \) and intermediate calculations
- Equation Solver: Verify algebraic solutions
- Table Mode: Generate value tables for patterns
- Fraction Display: Show exact fractional answers
- Degree/Radian Mode: Ensure correct mode for trigonometry
Self-Assessment Checklist
Can you confidently do these without assistance?
- ✓ Calculate area and perimeter of rectangles, triangles, circles
- ✓ Find volume of cuboids, cylinders, and spheres
- ✓ Apply distance and midpoint formulas to coordinate points
- ✓ Calculate slope and write equations of lines
- ✓ Use trigonometric ratios in right triangles
- ✓ Apply Pythagorean theorem
- ✓ Expand brackets and factor simple expressions
- ✓ Solve linear equations
- ✓ Convert between fractions, decimals, and percentages
- ✓ Calculate percentage increase and decrease
Additional RevisionTown Resources
Strengthen your prior learning foundation with these comprehensive RevisionTown tools and guides:
- Functions Formulae AI SL & AI HL - Build on coordinate geometry foundations
- Number and Algebra Formulae AI SL & AI HL - Extend algebraic skills
- Triangle Area Calculator - Practice geometry calculations
- Circle Area Calculator - Master circle formulas
- Sphere Volume Calculator - 3D geometry practice
- Distance Calculator - Coordinate geometry tool
- Percentage Calculator - Number operations
- Fraction Calculator - Work with fractions
- Quadratic Equation Calculator - Algebra practice
- IB Mathematics AA vs AI Guide - Understand your course
- IB Diploma Points Calculator - Track your progress
Time Management for Prior Learning Review
Week 1-2: Geometry and Mensuration (2D shapes)
- Area formulas for all common shapes
- Perimeter and circumference
- Practice mixed problems
Week 3-4: 3D Geometry
- Volume formulas
- Surface area calculations
- Real-world application problems
Week 5-6: Coordinate Geometry
- Distance and midpoint
- Slope and line equations
- Parallel and perpendicular lines
Week 7-8: Trigonometry and Algebra
- SOH-CAH-TOA
- Pythagorean theorem
- Basic equation solving
Ongoing: 15 minutes daily review across all topics
Conclusion
Mastering prior learning formulas creates the essential foundation for success in IB Math AI at both Standard and Higher Level. These concepts aren't merely review—they're fundamental tools you'll apply throughout your IB journey and beyond. From calculating optimal dimensions in real-world modeling to analyzing function behavior in calculus, prior learning concepts integrate seamlessly into every advanced topic.
Students who invest time building automaticity with these foundational formulas develop confidence, speed, and accuracy that translates directly into better exam performance. The geometry, coordinate geometry, trigonometry, and algebra skills you master now become second nature, freeing your mental energy to focus on the complex problem-solving and critical thinking that characterize IB Math AI.
Remember that mathematical proficiency develops through consistent practice. Use the interactive calculators, work through varied examples, and regularly self-assess your understanding. Connect prior learning to the advanced topics you're studying currently, recognizing how these foundations enable sophisticated mathematical analysis.
Continue building your IB Mathematics expertise through RevisionTown's comprehensive collection of IB Mathematics resources, practice with interactive calculators, and explore how prior learning connects to every aspect of your IB Math AI course. Success begins with a solid foundation—master these prior learning formulas and watch your mathematical confidence soar.
