Perimeter, Area, Volume & Equation of a Line - Revision notes
This chapter covers essential mensuration formulas (perimeter, area, volume) and linear equations - two topics that frequently appear together in UPCAT and other major Philippine entrance exams. Understanding these concepts is crucial as geometry problems often require setting up linear equations on coordinate planes.
Perimeter - Distance Around Figures
Formulas
P = 2(l + w)
Example
Rectangle with length 8cm and width 5cm: P = 2(8 + 5) = 2(13) = 26cm
Variables
l = length, w = width
Application
Rectangle perimeter
P = 4s
Example
Square with side 7m: P = 4(7) = 28m
Variables
s = side length
Application
Square perimeter
P = a + b + c
Example
Triangle with sides 3cm, 4cm, 5cm: P = 3 + 4 + 5 = 12cm
Variables
a, b, c = side lengths
Application
Triangle perimeter
C = 2πr or C = πd
Example
Circle with radius 6cm: C = 2π(6) = 12π ≈ 37.68cm
Variables
r = radius, d = diameter
Application
Circle circumference
Exam tips
- Always check if measurements are in the same units before calculating
- Draw and label diagrams to identify all boundaries clearly
- For circles, remember π ≈ 3.14 for approximations
Key points
- Perimeter measures the total distance around the boundary of a shape
- Always expressed in linear units (cm, m, ft) - never squared
- For composite figures, add up all the outer boundaries
- Circle perimeter is called circumference
Definitions
Perimeter
The total distance around the outside of a two-dimensional shape
Importance
Essential for calculating fencing, borders, or any boundary measurements
Common mistakes
- Confusing perimeter with area - perimeter is LINEAR measurement
- Forgetting to convert diameter to radius when using circle formulas
- Not identifying all sides in composite figures
Area - Surface Coverage
Formulas
A = lw
Example
Rectangle 10m × 6m: A = 10 × 6 = 60m²
Variables
l = length, w = width
Application
Rectangle area
A = s²
Example
Square with side 9cm: A = 9² = 81cm²
Variables
s = side length
Application
Square area
A = ½bh
Example
Triangle with base 8cm and height 6cm: A = ½(8)(6) = 24cm²
Variables
b = base, h = height (perpendicular to base)
Application
Triangle area
A = πr²
Example
Circle with radius 5m: A = π(5²) = 25π ≈ 78.54m²
Variables
r = radius
Application
Circle area
A = bh
Example
Parallelogram with base 12cm and height 8cm: A = 12 × 8 = 96cm²
Variables
b = base, h = height
Application
Parallelogram area
A = ½(b₁ + b₂)h
Example
Trapezoid with parallel sides 10cm and 6cm, height 4cm: A = ½(10 + 6)(4) = 32cm²
Variables
b₁, b₂ = parallel sides, h = height between them
Application
Trapezoid area
Exam tips
- Always identify the perpendicular height, not just any side
- For circles, if given diameter, divide by 2 to get radius first
- Label your diagram clearly to avoid confusion
- Check if answer should be exact (with π) or approximate
Key points
- Area measures the amount of surface a shape covers
- Always expressed in square units (cm², m², ft²)
- For composite figures, break into familiar shapes and add/subtract areas
- Height must be perpendicular to the base in area calculations
Definitions
Area
The amount of space inside the boundary of a flat two-dimensional shape
Importance
Used for calculating paint coverage, flooring, land measurements
Heron's Formula
A = √[s(s-a)(s-b)(s-c)] where s = (a+b+c)/2
Importance
Calculate triangle area when only three sides are known
Common mistakes
- Using slant height instead of perpendicular height
- Forgetting to square the radius in circle area formula
- Mixing up base and height in triangle calculations
- Not converting diameter to radius
Volume - Space Occupied by Solids
Formulas
V = s³
Example
Cube with side 4cm: V = 4³ = 64cm³
Variables
s = side length
Application
Cube volume
V = lwh
Example
Box 8m × 5m × 3m: V = 8 × 5 × 3 = 120m³
Variables
l = length, w = width, h = height
Application
Rectangular prism volume
V = πr²h
Example
Cylinder with radius 3cm and height 10cm: V = π(3²)(10) = 90π ≈ 282.74cm³
Variables
r = radius, h = height
Application
Cylinder volume
V = ⅓πr²h
Example
Cone with radius 6m and height 9m: V = ⅓π(6²)(9) = 108π ≈ 339.29m³
Variables
r = radius, h = height
Application
Cone volume
V = ⅓lwh
Example
Pyramid with base 12cm × 8cm and height 15cm: V = ⅓(12)(8)(15) = 480cm³
Variables
l = length, w = width, h = height
Application
Rectangular pyramid volume
V = (4/3)πr³
Example
Sphere with radius 3m: V = (4/3)π(3³) = 36π ≈ 113.10m³
Variables
r = radius
Application
Sphere volume
Exam tips
- Remember: cones and pyramids always have ⅓ in their volume formula
- Double-check if the problem gives radius or diameter
- Volume answers are always in cubic units
- For spheres, cube the radius: r³, not r²
Key points
- Volume measures the amount of three-dimensional space a solid occupies
- Always expressed in cubic units (cm³, m³, L)
- Cones and pyramids have volume = ⅓ × base area × height
- Remember the ⅓ factor - most common source of errors
Definitions
Volume
The amount of three-dimensional space occupied by a solid object
Importance
Essential for calculating capacity, storage, and material requirements
Common mistakes
- Forgetting the ⅓ factor for cones and pyramids
- Confusing radius and diameter in sphere/cylinder formulas
- Using wrong height (slant height vs. perpendicular height)
- Mixing up area and volume formulas
Equation of a Line
Formulas
y = mx + b
Example
Line y = 2x + 3 has slope 2 and y-intercept 3
Variables
m = slope, b = y-intercept
Application
Slope-intercept form (easiest to graph)
y - y₁ = m(x - x₁)
Example
Line through (2, 5) with slope 3: y - 5 = 3(x - 2) → y = 3x - 1
Variables
m = slope, (x₁, y₁) = known point
Application
Point-slope form
Ax + By = C
Example
3x + 2y = 12 can be rewritten as y = -1.5x + 6
Variables
A, B, C = constants (A and B not both zero)
Application
Standard form
m = (y₂ - y₁)/(x₂ - x₁)
Example
Points (1, 3) and (4, 9): m = (9-3)/(4-1) = 6/3 = 2
Variables
(x₁, y₁) and (x₂, y₂) = two points on the line
Application
Slope between two points
Exam tips
- Always reduce slope fractions to lowest terms
- For perpendicular lines: if slope is a/b, perpendicular slope is -b/a
- Graph points carefully to check your line equation
- Remember: horizontal lines have slope 0, vertical lines have undefined slope
Key points
- A line can be expressed in three main forms: slope-intercept, point-slope, and standard form
- Slope represents the rate of change (rise over run)
- Parallel lines have equal slopes; perpendicular lines have negative reciprocal slopes
- Understanding slope is crucial for identifying line relationships
Definitions
Slope
The measure of steepness of a line, calculated as rise over run
Importance
Determines if lines are parallel, perpendicular, or neither
Y-intercept
The y-coordinate where the line crosses the y-axis (when x = 0)
Importance
Starting point for graphing in slope-intercept form
Parallel Lines
Lines with the same slope (m₁ = m₂) that never intersect
Importance
Key concept in coordinate geometry and real-world applications
Perpendicular Lines
Lines whose slopes multiply to -1 (m₁ × m₂ = -1)
Importance
Form 90° angles, common in construction and design problems
Common mistakes
- Confusing x and y coordinates when calculating slope
- Forgetting negative signs when working with negative reciprocals
- Not simplifying slope fractions
- Mixing up parallel and perpendicular slope relationships
Distance and Midpoint Formulas
Formulas
d = √[(x₂-x₁)² + (y₂-y₁)²]
Example
Distance between (1, 2) and (4, 6): d = √[(4-1)² + (6-2)²] = √[9 + 16] = √25 = 5
Variables
(x₁, y₁) and (x₂, y₂) = two points
Application
Distance between two points
M = ((x₁+x₂)/2, (y₁+y₂)/2)
Example
Midpoint of (2, 4) and (8, 10): M = ((2+8)/2, (4+10)/2) = (5, 7)
Variables
(x₁, y₁) and (x₂, y₂) = endpoints
Application
Midpoint of line segment
Exam tips
- Always plot points to visualize the problem
- Double-check coordinate substitution carefully
- Simplify square roots when possible
- Midpoint coordinates are averages - they should be between the original points
Key points
- Distance formula comes from the Pythagorean theorem
- Midpoint is the average of x-coordinates and y-coordinates
- These formulas connect algebra with geometry
- Essential for solving coordinate geometry problems
Definitions
Distance Formula
Formula to find the straight-line distance between two points in a coordinate plane
Importance
Based on Pythagorean theorem, essential for geometric calculations
Midpoint
The point that is exactly halfway between two given points
Importance
Used in finding centers, bisecting segments, and solving geometric problems
Common mistakes
- Forgetting to take the square root in distance formula
- Mixing up coordinates when substituting into formulas
- Not simplifying square roots completely
- Confusing midpoint with distance calculations
Connections
- Perimeter and area calculations often appear in coordinate geometry problems
- Linear equations help solve optimization problems involving area and volume
- Distance formula connects algebraic calculations with geometric measurements
- Slope concepts apply to real-world rate problems and physics applications
- These topics frequently combine in UPCAT word problems involving rectangular lots, circular gardens, and construction scenarios
Quick review questions
What is the area of a triangle with base 12 cm and height 8 cm?
Answer
48 cm²
Using A = ½bh: A = ½(12)(8) = ½(96) = 48 cm²
Find the slope of the line passing through points (2, 3) and (6, 11).
Answer
2
Using m = (y₂-y₁)/(x₂-x₁): m = (11-3)/(6-2) = 8/4 = 2
What is the volume of a cone with radius 4 cm and height 9 cm?
Answer
48π cm³ or approximately 150.8 cm³
Using V = ⅓πr²h: V = ⅓π(4²)(9) = ⅓π(16)(9) = 48π cm³
If a line has slope 3/4, what is the slope of a line perpendicular to it?
Answer
-4/3
For perpendicular lines, slopes are negative reciprocals. If m₁ = 3/4, then m₂ = -4/3
What is the circumference of a circle with diameter 14 cm?
Answer
14π cm or approximately 43.96 cm
Using C = πd: C = π(14) = 14π cm. Note: we use diameter directly, not radius
Diagrams
Complete Chapter Overview - Formulas and Concepts
Comprehensive overview of all major formulas and concepts covered in this chapter, organized by topic area
Problem-Solving Decision Tree - Which Formula to Use
Decision tree to help students choose the correct formula based on what information is given and what needs to be found
Step-by-Step Solution Process for Line Equations
Systematic approach for solving line equation problems with built-in verification step
Common UPCAT Question Distribution
Typical distribution of problem types in UPCAT mathematics sections covering this chapter
Exam strategy
Focus on memorizing the core formulas, but more importantly, understand when to apply each one. Practice identifying which formula to use based on given information. For composite figures, break them into familiar shapes. In coordinate geometry, always plot points when possible to visualize the problem. Remember the ⅓ factor for cones and pyramids, and always check units in your final answer. Most UPCAT problems will test your ability to combine these concepts, so practice multi-step problems that involve both mensuration and coordinate geometry.