Geometry — Perimeter, Area, Circumference & Volume - Cheat Sheet
Your last-minute revision companion for mastering geometry calculations on Philippine entrance exams
Perimeter Formulas
Formulas
P = 4s
Meaning
P = perimeter, s = side length
Watch out
Don't confuse with area formula s²
When to use
Finding perimeter of any square
P = 2L + 2W or P = 2(L + W)
Meaning
P = perimeter, L = length, W = width
Watch out
Make sure to multiply BOTH length and width by 2
When to use
Finding perimeter of any rectangle
P = a + b + c
Meaning
P = perimeter, a, b, c = side lengths
Watch out
Must add ALL three sides, not just two
When to use
Finding perimeter of any triangle
P = 2(b + h)
Meaning
P = perimeter, b = base, h = height (slant side)
Watch out
h here is the slant height, not perpendicular height
When to use
Finding perimeter of parallelogram
P = a + b + c + d
Meaning
P = perimeter, a, b, c, d = all four sides
Watch out
Add all four individual sides, not just bases
When to use
Finding perimeter of trapezoid
Important facts
- Perimeter is always measured in linear units (cm, m, ft)
- For regular polygons: P = n × s (n = number of sides, s = side length)
- Equilateral triangle: all three sides are equal
- Perimeter problems often give total and ask for individual side
Key definitions
Perimeter
Example
Fence around a rectangular lot
Total distance around the boundary of a closed shape
Regular Polygon
Example
Square, equilateral triangle, regular hexagon
Polygon with all sides equal and all angles equal
Diagrams to know
- Square with labeled sides
- Rectangle with length and width marked
- Triangle with three sides labeled
Area Formulas
Formulas
A = s²
Meaning
A = area, s = side length
Watch out
Remember to square the side, not multiply by 4
When to use
Finding area of any square
A = L × W
Meaning
A = area, L = length, W = width
Watch out
Length times width, not length plus width
When to use
Finding area of any rectangle
A = (1/2) × b × h
Meaning
A = area, b = base, h = perpendicular height
Watch out
Must use perpendicular height, not slant side
When to use
Finding area of any triangle
A = b × h
Meaning
A = area, b = base, h = perpendicular height
Watch out
Use perpendicular height, not the slant side length
When to use
Finding area of parallelogram
A = (1/2) × (a + b) × h
Meaning
A = area, a and b = parallel sides, h = height
Watch out
Add the two parallel bases first, then multiply by height and divide by 2
When to use
Finding area of trapezoid
Important facts
- Area is always measured in square units (cm², m², ft²)
- Triangle area formula works for ALL triangles (right, acute, obtuse)
- For parallelograms, height must be perpendicular to base
- Trapezoid has exactly one pair of parallel sides
Key definitions
Area
Example
Floor space of a room
Amount of space inside a closed shape
Base
Example
Bottom side of triangle
Bottom side of a shape, or any side chosen as reference
Height
Example
Vertical distance in triangle
Perpendicular distance from base to opposite side
Diagrams to know
- Rectangle with dimensions labeled
- Triangle showing base and height
- Trapezoid with parallel sides and height marked
Circle Formulas
Formulas
C = 2πr
Meaning
C = circumference, π = 3.14, r = radius
Watch out
Don't forget to multiply by 2 when using radius
When to use
Finding distance around a circle
C = πd
Meaning
C = circumference, π = 3.14, d = diameter
Watch out
When given diameter, don't multiply by 2
When to use
Finding circumference when diameter is given
A = πr²
Meaning
A = area, π = 3.14, r = radius
Watch out
Must square the radius, not multiply by 2
When to use
Finding area inside a circle
d = 2r
Meaning
d = diameter, r = radius
Watch out
Diameter is always twice the radius
When to use
Converting between diameter and radius
Common values
Value
3.14 or 22/7
Symbol
π
Quantity
Pi
Important facts
- π (pi) ≈ 3.14 or 22/7 in most problems
- Radius is half the diameter: r = d/2
- Diameter is twice the radius: d = 2r
- Circumference is the perimeter of a circle
Key definitions
Circumference
Example
Length of wire needed to make a circular ring
Distance around the outside of a circle
Radius
Example
Half the width of a circular table
Distance from center to any point on the circle
Diameter
Example
Width of a circular pizza
Distance across circle through the center
Diagrams to know
- Circle with radius drawn from center
- Circle with diameter drawn across
- Circle showing both radius and diameter
Volume Formulas
Formulas
V = s³
Meaning
V = volume, s = side length
Watch out
Cube the side length (s × s × s), not square it
When to use
Finding volume of a cube
V = L × W × H
Meaning
V = volume, L = length, W = width, H = height
Watch out
Must multiply all three dimensions together
When to use
Finding volume of rectangular prism
V = πr²h
Meaning
V = volume, π = 3.14, r = radius, h = height
Watch out
Square the radius first, then multiply by π and height
When to use
Finding volume of cylinder
V = (4/3)πr³
Meaning
V = volume, π = 3.14, r = radius
Watch out
Must cube the radius and multiply by 4/3
When to use
Finding volume of sphere
V = (1/3) × B × h
Meaning
V = volume, B = base area, h = height
Watch out
Don't forget to divide by 3
When to use
Finding volume of pyramid or cone
Common values
Value
3.14
Symbol
π
Quantity
Pi
Important facts
- Volume is always measured in cubic units (cm³, m³, ft³)
- Prism volume = Base area × Height
- Pyramid/cone volume = (1/3) × Base area × Height
- Sphere is perfectly round in all directions
Key definitions
Volume
Example
Water capacity of a tank
Amount of space occupied by a three-dimensional object
Cube
Example
Dice, Rubik's cube
3D shape with all edges equal and all faces square
Cylinder
Example
Tin can, water tank
3D shape with circular base and top, straight sides
Diagrams to know
- Cube with side length labeled
- Cylinder with radius and height marked
- Rectangular prism with L, W, H labeled
Must remember
- Perimeter = add all sides; Area = multiply dimensions
- Circle: C = 2πr, A = πr²; π ≈ 3.14
- Triangle area = (1/2) × base × height (ALWAYS divide by 2)
- Volume always uses cubic units (cm³, m³)
- Cube volume = s³ (side cubed)
- Cylinder volume = πr²h (circle area × height)
- Radius = diameter ÷ 2; Diameter = radius × 2
- Rectangle area = length × width (NOT length + width)
- Trapezoid area = (1/2)(b₁ + b₂)h
- Always check units in your final answer
Last minute tips
- Draw a quick sketch of the shape and label given measurements
- If given perimeter, work backwards to find individual sides
- For word problems, identify the shape first, then choose the right formula
- Double-check whether you need perimeter (around) or area (inside)
- When calculating with π, leave answer as πr² or use 3.14 as specified
Comparison tables
Perimeter vs Area vs Volume
| Property | Perimeter | Area | Volume |
|---|---|---|---|
| Measures | Distance around | Space inside | Space occupied |
| Units | Linear (cm, m) | Square (cm², m²) | Cubic (cm³, m³) |
| Dimension | 1D boundary | 2D surface | 3D object |
Square vs Rectangle vs Triangle
| Shape | Perimeter | Area | Special Property |
|---|---|---|---|
| Square | P = 4s | A = s² | All sides equal |
| Rectangle | P = 2L + 2W | A = L × W | Opposite sides equal |
| Triangle | P = a + b + c | A = (1/2)bh | Three-sided polygon |
Diagrams
Geometry Formulas Overview
Complete overview of all geometry formulas organized by measurement type
Problem-Solving Process
Step-by-step flowchart for determining which formula to use in geometry problems
Common Exam Question Types
Distribution of geometry question types commonly found on Philippine entrance exams