Perimeter, Area, Volume & Equation of a Line - Cheat sheet
Your last-minute revision companion for mensuration formulas and line equations — the most tested geometry topics in UPCAT
Perimeter Formulas
Formulas
P = 2(l + w)
Meaning
P = perimeter, l = length, w = width
Watch out
Don't square the values — perimeter is just the sum of sides
When to use
For rectangles when both length and width are given
P = 4s
Meaning
P = perimeter, s = side length
Watch out
Remember it's 4 times ONE side, not the area formula
When to use
For squares when one side is given
P = a + b + c
Meaning
P = perimeter, a, b, c = the three sides
Watch out
Make sure all three sides are in the same units
When to use
For any triangle when all three sides are known
C = 2πr = πd
Meaning
C = circumference, r = radius, d = diameter
Watch out
If given diameter, don't forget to divide by 2 for radius formulas
When to use
For circles — use 2πr when radius is given, πd when diameter is given
P = ns
Meaning
P = perimeter, n = number of sides, s = side length
Watch out
Only works for REGULAR polygons where all sides are equal
When to use
For regular polygons (all sides equal)
Common values
Value
3.14159...
Symbol
π
Quantity
Pi (π)
Value
22/7
Symbol
π
Quantity
Pi approximation
Important facts
- Perimeter units are always linear (cm, m, ft) — never squared
- For composite shapes, add the outer edges only
- π ≈ 3.14159 or use 22/7 for quick calculations
Key definitions
Perimeter
Example
Walking around the edge of a basketball court
The total distance around the outside of a 2D shape
Circumference
Example
The distance around a wheel's rim
The perimeter of a circle
Diagrams to know
- Rectangle with labeled length and width
- Circle with labeled radius and diameter
Area Formulas
Formulas
A = lw
Meaning
A = area, l = length, w = width
Watch out
Make sure length and width are perpendicular to each other
When to use
For rectangles and when you have perpendicular sides
A = s²
Meaning
A = area, s = side length
Watch out
Don't confuse with perimeter — this one is SQUARED
When to use
For squares when one side is given
A = ½bh
Meaning
A = area, b = base, h = height (perpendicular to base)
Watch out
Height must be perpendicular to the base, not just any side
When to use
For triangles when base and height are known
A = πr²
Meaning
A = area, r = radius
Watch out
Don't forget to SQUARE the radius — most common mistake
When to use
For circles when radius is given
A = bh
Meaning
A = area, b = base, h = height
Watch out
Use height, not the slanted side length
When to use
For parallelograms when base and height are known
A = ½(b₁ + b₂)h
Meaning
A = area, b₁ and b₂ = parallel sides, h = height
Watch out
Add the two parallel bases first, then multiply by height and divide by 2
When to use
For trapezoids with two parallel sides
A = ½d₁d₂
Meaning
A = area, d₁ and d₂ = diagonals
Watch out
This only works for rhombus — don't use for other quadrilaterals
When to use
For rhombus when both diagonals are given
Important facts
- Area units are always squared (cm², m², ft²)
- For composite shapes, break into familiar pieces and add/subtract areas
- Heron's formula for triangles: A = √[s(s-a)(s-b)(s-c)] where s = (a+b+c)/2
Key definitions
Area
Example
The floor space of a room
The amount of space inside a 2D shape
Base
Example
The bottom edge of a triangle
The bottom side of a shape, perpendicular to height
Height
Example
The vertical distance in a triangle from base to top vertex
The perpendicular distance from base to opposite side
Diagrams to know
- Triangle with base and height marked
- Circle with radius marked
- Trapezoid with parallel sides and height labeled
Reactions or equations
A = √[s(s-a)(s-b)(s-c)]
Note
s is the semi-perimeter: s = (a+b+c)/2
Conditions
When triangle has three known sides but no clear height
Volume Formulas
Formulas
V = s³
Meaning
V = volume, s = side length
Watch out
Cube the side length — it's to the third power
When to use
For cubes when one side is given
V = lwh
Meaning
V = volume, l = length, w = width, h = height
Watch out
Make sure all three dimensions are in the same units
When to use
For rectangular prisms (boxes) when all dimensions are known
V = πr²h
Meaning
V = volume, r = radius, h = height
Watch out
Don't forget to square the radius first
When to use
For cylinders when radius and height are known
V = ⅓πr²h
Meaning
V = volume, r = radius, h = height
Watch out
Don't forget the ⅓ factor — very common mistake
When to use
For cones when radius and height are known
V = ⅓lwh
Meaning
V = volume, l = length, w = width, h = height
Watch out
Don't forget the ⅓ factor for pyramids
When to use
For pyramids with rectangular base
V = (4/3)πr³
Meaning
V = volume, r = radius
Watch out
Cube the radius and remember the 4/3 coefficient
When to use
For spheres when radius is given
Common values
Value
1000 cm³
Symbol
L
Quantity
1 liter
Important facts
- Volume units are always cubed (cm³, m³, liters)
- Cones and pyramids always have ⅓ in their volume formulas
- 1 liter = 1000 cm³ = 0.001 m³
Key definitions
Volume
Example
The water that fits in a bottle
The amount of 3D space occupied by a solid
Cylinder
Example
A can of soda
A solid with circular base and top connected by curved surface
Cone
Example
An ice cream cone
A solid with circular base tapering to a point
Diagrams to know
- Cylinder with radius and height labeled
- Cone with radius and height labeled
- Sphere with radius marked
Equation of a Line
Formulas
y = mx + b
Meaning
m = slope, b = y-intercept, (x,y) = any point on line
Watch out
b is where line crosses y-axis, not x-axis
When to use
When you know slope and y-intercept, or need to graph quickly
y - y₁ = m(x - x₁)
Meaning
m = slope, (x₁,y₁) = known point, (x,y) = any point on line
Watch out
Use the coordinates of the KNOWN point for x₁ and y₁
When to use
When you know one point and the slope
Ax + By = C
Meaning
A, B, C = constants, (x,y) = any point on line
Watch out
To find x-intercept, set y = 0; for y-intercept, set x = 0
When to use
Standard form — useful for finding intercepts
m = (y₂ - y₁)/(x₂ - x₁)
Meaning
m = slope, (x₁,y₁) and (x₂,y₂) = two points on line
Watch out
Keep y-coordinates in numerator, x-coordinates in denominator
When to use
When you have two points and need to find slope
Important facts
- Positive slope: line goes up from left to right
- Negative slope: line goes down from left to right
- Zero slope: horizontal line
- Undefined slope: vertical line
- Parallel lines: m₁ = m₂
- Perpendicular lines: m₁ × m₂ = -1
Key definitions
Slope
Example
A slope of 2 means up 2, right 1
The steepness of a line, rise over run
Y-intercept
Example
Point (0, 3) means y-intercept is 3
Where the line crosses the y-axis (x = 0)
Parallel lines
Example
Lines with slopes 2 and 2 are parallel
Lines with equal slopes that never intersect
Perpendicular lines
Example
Slopes 2 and -½ are perpendicular
Lines whose slopes multiply to -1
Diagrams to know
- Coordinate plane with line showing slope and y-intercept
- Two parallel lines with same slope
- Two perpendicular lines forming 90° angle
Reactions or equations
m₁ × m₂ = -1
Note
One slope is negative reciprocal of the other
Conditions
For perpendicular lines
Distance and Midpoint
Formulas
d = √[(x₂-x₁)² + (y₂-y₁)²]
Meaning
d = distance, (x₁,y₁) and (x₂,y₂) = two points
Watch out
Don't forget to take the square root at the end
When to use
To find distance between any two points
M = ((x₁+x₂)/2, (y₁+y₂)/2)
Meaning
M = midpoint, (x₁,y₁) and (x₂,y₂) = endpoints
Watch out
Add coordinates first, then divide by 2 for each coordinate
When to use
To find the point exactly halfway between two points
Important facts
- Distance formula comes from Pythagorean theorem
- Distance is always positive
- Midpoint coordinates are averages of endpoint coordinates
Key definitions
Distance
Example
Distance from (0,0) to (3,4) is 5 units
The straight-line length between two points
Midpoint
Example
Midpoint of (1,2) and (5,8) is (3,5)
The point exactly halfway between two other points
Diagrams to know
- Two points connected by line segment with midpoint marked
- Right triangle showing distance formula derivation
Must remember
- Area units are SQUARED, volume units are CUBED
- Cones and pyramids have ⅓ in volume formula
- For circles: C = 2πr, A = πr² (don't mix them up)
- Perpendicular slopes multiply to -1
- Slope = rise/run = (y₂-y₁)/(x₂-x₁)
- Triangle area: A = ½bh (height must be perpendicular to base)
- If given diameter, divide by 2 to get radius
- Parallel lines have equal slopes
- Distance formula: d = √[(x₂-x₁)² + (y₂-y₁)²]
- For composite figures: break into simple shapes, then add/subtract
Last minute tips
- Draw and label diagrams — most mistakes come from using wrong measurements
- Check units — convert everything to same unit before calculating
- For word problems, identify the shape first, then choose the right formula
- When finding line equations, always identify what information you have (slope? points? intercepts?)
- For circle problems, check if you're given radius or diameter — don't assume!
Comparison tables
Perimeter vs Area vs Volume
| Concept | Dimension | Units | Example Formula |
|---|---|---|---|
| Perimeter | 1D (length) | cm, m, ft | P = 2l + 2w |
| Area | 2D (surface) | cm², m², ft² | A = lw |
| Volume | 3D (space) | cm³, m³, ft³ | V = lwh |
Line Forms Quick Reference
| Form | Equation | Best Used When |
|---|---|---|
| Slope-intercept | y = mx + b | Graphing or slope/y-intercept given |
| Point-slope | y - y₁ = m(x - x₁) | One point and slope given |
| Standard | Ax + By = C | Finding intercepts |
Diagrams
Mensuration Formula Overview
Complete overview of all mensuration formulas organized by type
Problem-Solving Decision Tree
Decision tree to help choose the correct formula based on what you're trying to find
Line Equation Problem-Solving Steps
Step-by-step approach to solving line equation problems