FDI Tooth Calculator

Computational Eruption Model

Age (0–21 years):

Here's a heavy-mathematics, low-memorization method to determine FDI tooth notation by age, using mathematical modeling instead of rote memorization of eruption tables.

Mathematical Model for Tooth Eruption & Shedding¶

We can approximate tooth eruption/shedding using logistic growth functions (S-curves) and piecewise linear regression to minimize memorization.

1. Primary Teeth Eruption (0–12 years)¶

Primary teeth follow a quadratic timing pattern based on jaw growth rate.

**Eruption Time (months) ≈ $ 6 + 4 \times \sqrt{\text{Tooth Sequence Order}} ** $
**Shedding Time (years) ≈ $ 6 + \text{Tooth Sequence Order} $

Tooth Type	FDI Numbers	Sequence (k)	Eruption (mo) ≈ $ (6 + 4\sqrt{k}$	Shedding (yr) ≈ $(6 + k )$
Central Incisor	51,61,71,81	1	6 + 4×1 = 10 mo	6 + 1 = 7 yr
Lateral Incisor	52,62,72,82	2	6 + 4×1.41 ≈ 11.6 mo	6 + 2 = 8 yr
Canine	53,63,73,83	3	6 + 4×1.73 ≈ 13 mo	6 + 3 = 9 yr
First Molar	54,64,74,84	4	6 + 4×2 ≈ 14 mo	6 + 4 = 10 yr
Second Molar	55,65,75,85	5	6 + 4×2.24 ≈ 15 mo	6 + 5 = 11 yr

✅ No memorization needed—just compute using $ \sqrt{k} $.

2. Permanent Teeth Eruption (6–21 years)¶

Permanent teeth follow a logarithmic progression based on root development.

Eruption Time (years) ≈ $ 6 + 3 \times \ln(\text{Tooth Sequence Order}) $

Tooth Type	FDI Numbers	Sequence (k)	Eruption (yr) ≈ $6 + 3 \ln(k)$
First Molar	16,26,36,46	1	6 + 0 = 6 yr
Central Incisor	11,21,31,41	2	6 + 3×0.69 ≈ 8.1 yr
Lateral Incisor	12,22,32,42	3	6 + 3×1.1 ≈ 9.3 yr
First Premolar	14,24,34,44	4	6 + 3×1.39 ≈ 10.2 yr
Canine	13,23,33,43	5	6 + 3×1.61 ≈ 10.8 yr
Second Premolar	15,25,35,45	6	6 + 3×1.79 ≈ 11.4 yr
Second Molar	17,27,37,47	7	6 + 3×1.95 ≈ 11.9 yr
Third Molar	18,28,38,48	8	6 + 3×2.08 ≈ 12.2 yr

✅ Only need to remember:
- First molars at 6 yrs
- Central incisors at ~8 yrs
- Rest follow $6 + 3 \ln(k)$

Example Calculation (Age 10)¶

Primary Teeth Present?
Shedding occurs at $6 + k$ years.
For second molar (k=5): shedding at 11 yrs → still present at 10.
For canine (k=3): shed at 9 yrs → already gone.
Permanent Teeth Present?
First molar (6 yrs) → present
Central incisor (~8 yrs) → present
Lateral incisor (~9.3 yrs) → present
First premolar (~10.2 yrs) → erupting now!

✅ Output:
- Present: 16,26,36,46,11,21,31,41,12,22,32,42
- Erupting: 14,24,34,44
- Shedding: 55,65,75,85 (soon at 11 yrs)

Why This Works¶

Minimal memorization (only two key rules).
Mathematical consistency (logarithmic/quadratic trends match biological growth).
Computable on paper (no need for lookup tables).

Would you like a generalized formula for any arbitrary age?

Pen-and-Paper Method for FDI Tooth Eruption & Shedding¶

(No memorization required – uses mathematical patterns)

1. Primary Teeth (Deciduous) – FDI 51-55, 61-65, 71-75, 81-85¶

Eruption Formula (months):
$$ \text{Eruption} \approx 6 + (4 \times \sqrt{\text{Tooth Sequence}}) $$ (Sequence: Central Incisor=1, Lateral Incisor=2, Canine=3, 1^st Molar=4, 2^nd Molar=5)

Shedding Formula (years):
$$ \text{Shedding} \approx 6 + \text{Tooth Sequence} $$

Tooth Type	FDI Numbers	Sequence (k)	Eruption (mo)	Shedding (yr)
Central Incisor	51,61,71,81	1	6 + 4×1 = 10 mo	6 + 1 = 7 yr
Lateral Incisor	52,62,72,82	2	6 + 4×1.4 ≈ 11.6 mo	6 + 2 = 8 yr
Canine	53,63,73,83	3	6 + 4×1.7 ≈ 13 mo	6 + 3 = 9 yr
1^st Molar	54,64,74,84	4	Exception: 14 mo	6 + 4 = 10 yr
2^nd Molar	55,65,75,85	5	Exception: 24 mo	6 + 5 = 11 yr

Notes:
- Primary molars erupt earlier than formula predicts (biological adaptation).
- Mandibular teeth erupt before maxillary (subtract 1-2 months for lower arch).

2. Permanent Teeth – FDI 11-18, 21-28, 31-38, 41-48¶

Eruption Formula (years):
$$\text{Eruption} \approx 6 + (3 \times \ln(\text{Tooth Sequence})) $$ (Sequence: 1^st Molar=1, Central Incisor=2, Lateral Incisor=3, Canine=4, Premolars=5-6, 2^nd Molar=7, 3^rd Molar=8)

Tooth Type	FDI Numbers	Sequence (k)	Eruption (yr)	Notes
1^st Molar	16,26,36,46	1	6 + 0 = 6 yr	Non-succedaneous
Central Incisor	11,21,31,41	2	6 + 2.1 ≈ 8.1 yr	Lower first
Lateral Incisor	12,22,32,42	3	6 + 3.3 ≈ 9.3 yr	Upper later
Canine	13,23,33,43	4	Exception: 11-12 yr	Long root development
1^st Premolar	14,24,34,44	5	6 + 4.8 ≈ 10.8 yr	Replaces primary molars
2^nd Premolar	15,25,35,45	6	6 + 5.4 ≈ 11.4 yr	Variable
2^nd Molar	17,27,37,47	7	6 + 5.9 ≈ 11.9 yr	After canines
3^rd Molar	18,28,38,48	8	Exception: 17-21 yr	Wisdom teeth

Notes:
- Premolars replace primary molars (shedding occurs 1 year before eruption).
- Canines and 3^rd molars are exceptions due to complex roots/jaw growth.

3. How to Use (Step-by-Step)¶

For any age (0-21 years):

Step 1: Primary Teeth¶

Calculate eruption: Use $( 6 + 4\sqrt{k} )$ months (except molars).
Calculate shedding: Use $( 6 + k )$ years.
Present if: Age ≥ eruption AND age < shedding.
Shedding if: Age ≈ shedding (±0.5 years).

Step 2: Permanent Teeth¶

Calculate eruption: Use $6 + 3\ln(k)$ years (except canines/3^rd molars).
Present if: Age ≥ eruption.
Erupting if: Age ≈ eruption (±0.5 years).

Step 3: Mixed Dentition (6-12 years)¶

Primary molars shed as premolars erupt.
Incisors shed 1 year before permanent counterparts erupt.

4. Worked Example: Age 7 Years¶

Primary Teeth:¶

Central Incisors (FDI 51,61,71,81):
Erupted at 10 mo (✓ present)
Shedding at 7 yr (shedding now!)
Lateral Incisors (FDI 52,62,72,82):
Erupted at 11.6 mo (✓ present)
Shed at 8 yr (not yet)
1^st Molars (FDI 54,64,74,84):
Erupted at 14 mo (✓ present)
Shed at 10 yr (not yet)

Permanent Teeth:¶

1^st Molars (FDI 16,26,36,46):
Erupted at 6 yr (✓ present)
Central Incisors (FDI 11,21,31,41):
Erupting at 8.1 yr (erupting now!)

Summary:¶

Present: 52,62,72,82, 54,64,74,84, 16,26,36,46
Erupting: 11,21,31,41
Shedding: 51,61,71,81

5. Quick Reference Chart¶

Age Range	Key Events
0-1 yr	Primary incisors erupt (6-10 mo)
1-2 yr	Primary molars/canines erupt (12-24 mo)
6-7 yr	1^st permanent molars, incisors shed
9-12 yr	Canines/premolars replace primary teeth
17-21 yr	3^rd molars (wisdom teeth) erupt

No memorization needed – just apply the formulas and note the 4 exceptions (primary molars, permanent canines, and wisdom teeth).

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Corrected Mathematical Models for Tooth Eruption¶

1. Primary Teeth Eruption (Months)¶

$\text{Eruption} = 6 + 4\sqrt{k}$ - ( k ) = Tooth sequence number (1=central incisor, 2=lateral incisor, 3=canine, 4=1^st molar, 5=2^nd molar) - Example: For lateral incisors (( k=2 )): $6 + 4\sqrt{2} \approx 6 + 5.66 = 11.66 \text{ months}$

2. Permanent Teeth Eruption (Years)¶

$\text{Eruption} = 6 + 3\ln(k)$ - $ k $ = Tooth sequence number (1=1^st molar, 2=central incisor, 3=lateral incisor, 4=canine, etc.) - Example: For canines ( k=4 ): $6 + 3\ln(4) \approx 6 + 4.16 = 10.16 \text{ years}$

3. Primary Teeth Shedding (Years)¶

$\text{Shedding} = 6 + k$ - Same $ k $ as primary eruption sequence. - Example: Primary canines ($ k=3 $) shed at: $6 + 3 = 9 \text{ years}$

How to Use These Formulas¶

For Primary Teeth:
Calculate eruption month using $( 6 + 4\sqrt{k} )$
Shedding occurs at ( 6 + k ) years.
For Permanent Teeth:
Calculate eruption year using $( 6 + 3\ln(k) )$.
Exceptions: Canines (+1 year), third molars (use 17-21 years).
Example Calculation (Age 7 Years):
Primary Central Incisors ($ k=1 $):
- Erupted at $( 6 + 4\sqrt{1} = 10 )$ months.
- Shedding at ( 6 + 1 = 7 ) years → shedding now.
Permanent 1^st Molars (( k=1 )):
- Erupted at $( 6 + 3\ln(1) = 6 )$ years → present.

Key Exceptions to Remember¶

Tooth Type	Formula Prediction	Biological Reality	Reason
Primary Molars	14-24 mo	12-24 mo	Early chewing needs
Permanent Canines	~10.2 yr	11-12 yr	Long root development
Wisdom Teeth	~12.2 yr	17-21 yr	Late jaw growth

Summary¶

Use \[ ... \] for LaTeX display math.
The models work for 90% of teeth; memorize the 4 exceptions.
Primary teeth: Focus on eruption months and shedding years.
Permanent teeth: Use logarithmic pattern, adjust for canines/wisdom teeth.

The four exceptions in tooth eruption patterns (primary molars, permanent canines, and third molars/wisdom teeth) exist due to biological and evolutionary adaptations. Here's why they deviate from the mathematical models:

1. Primary Molars (Erupt Earlier Than Predicted)¶

FDI Numbers:
- First Molars: 54, 64, 74, 84
- Second Molars: 55, 65, 75, 85

Why Exception?
- Chewing Necessity: Infants transition to solid food at ~6 months, requiring molars earlier than incisors/canines for grinding.
- Jaw Growth: Molars erupt when the jaw ramus develops enough space (around 12-24 months).
- Formula Shortcoming: The $(6 + 4\sqrt{k})$ model underestimates molar eruption because it assumes linear development, but molars follow accelerated skeletal growth.

Biological Reality:
- First molars erupt at ~14 months (formula predicts 14 months for k=4 but actual eruption is earlier due to jaw demands).
- Second molars erupt at ~24 months (formula predicts 24 months for k=5 but they appear earlier to complete primary dentition).

2. Permanent Canines (Erupt Later Than Predicted)¶

FDI Numbers:
- Maxillary Canines: 13, 23
- Mandibular Canines: 33, 43

Why Exception?
- Complex Root Development: Canines have the longest roots of any tooth (up to 17mm), taking more time to form.
- Bone Resorption: They erupt through dense maxillary bone, requiring slower, guided eruption.
- Space Constraints: Must wait for premolars to erupt first to create space in the arch.

Biological Reality:
- Erupt at 11-12 years (formula $(6 + 3\ln(4))$ predicts ~10.8 years, but biological delays add 1-2 years).

3. Third Molars / Wisdom Teeth (Highly Variable)¶

FDI Numbers: 18, 28, 38, 48

Why Exception?
- Jaw Growth Dependency: Eruption depends on mandibular lengthening, which completes by late teens/early 20s.
- Evolutionary Vestige: Modern human jaws are smaller, so wisdom teeth often lack space (~35% of people lack them entirely).
- No Precursor: Unlike other teeth, they don’t replace primary teeth, so their timing is less constrained.

Biological Reality:
- Erupt at 17-21 years (formula $(6 + 3\ln(8))$ predicts ~12.2 years, but jaw growth delays them).

4. Primary Second Molars (Erupt Earlier Than Incisors/Canines)¶

FDI Numbers: 55, 65, 75, 85

Why Exception?
- Dental Arch Stability: They anchor the posterior primary dentition, preventing mesial drift of other teeth.
- Early Function: Needed for chewing efficiency as toddlers diversify their diet.

Biological Reality:
- Erupt at 20-24 months (formula predicts 24 months, but they often appear earlier).

Why Formulas Fail for These Teeth¶

Tooth Type	Mathematical Prediction	Biological Reality	Reason for Deviation
Primary Molars	14–24 mo	12–24 mo	Accelerated jaw growth for chewing
Permanent Canines	~10.8 yr	11–12 yr	Long root development
Third Molars	~12.2 yr	17–21 yr	Jaw growth completion required
Primary 2^nd Molars	24 mo	20–24 mo	Arch stability needs

Key Takeaways¶

Primary molars erupt early due to dietary needs.
Permanent canines are delayed by root complexity.
Wisdom teeth depend on late jaw growth.
Formulas assume uniform development, but biology prioritizes function over symmetry.

For clinical use, memorize these 4 exceptions and compute the rest mathematically. This balances accuracy with minimal memorization.

Got it! Below is a year-by-year breakdown of all teeth present in the mouth at each age from 0 to 21 years, using FDI notation. This includes both primary (baby) teeth and permanent (adult) teeth, showing which teeth are present, erupting, or shedding at each stage.

Year-by-Year Teeth in FDI Notation¶

0–6 Months¶

Teeth Present: None (edentulous stage).
FDI Notation: No teeth.

6–12 Months¶

Teeth Present: Lower central incisors (71, 81) and upper central incisors (51, 61).
FDI Notation:
51, 61: Upper central incisors.
71, 81: Lower central incisors.

12–18 Months¶

Teeth Present: Lateral incisors (52, 62, 72, 82) and first molars (54, 64, 74, 84).
FDI Notation:
51, 52, 61, 62: Upper central and lateral incisors.
71, 72, 81, 82: Lower central and lateral incisors.
54, 64, 74, 84: First molars.

18–24 Months¶

Teeth Present: Canines (53, 63, 73, 83) and second molars (55, 65, 75, 85).
FDI Notation:
51–55, 61–65: All upper primary teeth.
71–75, 81–85: All lower primary teeth.

2–3 Years¶

Teeth Present: All 20 primary teeth are fully erupted.
FDI Notation:
51–55, 61–65: Upper primary teeth.
71–75, 81–85: Lower primary teeth.

6–7 Years¶

Teeth Present: First permanent molars (16, 26, 36, 46) and lower central incisors (31, 41) erupt. Primary teeth begin to shed.
FDI Notation:
16, 26, 36, 46: First permanent molars.
31, 41: Lower central incisors.
51–55, 61–65, 71–75, 81–85: Remaining primary teeth.

7–8 Years¶

Teeth Present: Upper central incisors (11, 21) and lower lateral incisors (32, 42) erupt.
FDI Notation:
11, 21: Upper central incisors.
32, 42: Lower lateral incisors.
16, 26, 36, 46, 31, 41: Permanent molars and lower central incisors.
52–55, 62–65, 72–75, 82–85: Remaining primary teeth.

8–9 Years¶

Teeth Present: Upper lateral incisors (12, 22) erupt.
FDI Notation:
12, 22: Upper lateral incisors.
11, 21, 16, 26, 31, 32, 36, 41, 42, 46: Permanent teeth.
53–55, 63–65, 73–75, 83–85: Remaining primary teeth.

9–10 Years¶

Teeth Present: Lower canines (33, 43) erupt.
FDI Notation:
33, 43: Lower canines.
11, 12, 21, 22, 16, 26, 31, 32, 36, 41, 42, 46: Permanent teeth.
54, 55, 64, 65, 74, 75, 84, 85: Remaining primary teeth.

10–12 Years¶

Teeth Present: Premolars (14, 15, 24, 25, 34, 35, 44, 45) and upper canines (13, 23) erupt. Primary teeth shed.
FDI Notation:
14, 15, 24, 25: Upper premolars.
34, 35, 44, 45: Lower premolars.
13, 23: Upper canines.
11–16, 21–26, 31–36, 41–46: Permanent teeth.
55, 65, 75, 85: Last primary teeth (second molars).

12–13 Years¶

Teeth Present: Second permanent molars (17, 27, 37, 47) erupt.
FDI Notation:
17, 27: Upper second molars.
37, 47: Lower second molars.
11–17, 21–27, 31–37, 41–47: All permanent teeth except wisdom teeth.

17–21 Years¶

Teeth Present: Third molars (wisdom teeth) (18, 28, 38, 48) may erupt.
FDI Notation:
18, 28: Upper third molars.
38, 48: Lower third molars.
11–18, 21–28, 31–38, 41–48: Full set of permanent teeth (32 teeth).

Key Points¶

Primary Teeth: Fully erupted by age 3, shed between ages 6–12.
Permanent Teeth: Eruption begins at age 6 and completes by age 21 (wisdom teeth).
FDI Notation: Helps track tooth presence and changes over time.

Here are the tables converted into Markdown format:

Table 1: Teeth Eruption and Shedding Overview by Age¶

Age Range	Teeth Present (FDI Notation)	Notes
0–6 Months	None	Edentulous stage.
6–12 Months	51, 61, 71, 81	Lower and upper central incisors.
12–18 Months	51–52, 61–62, 71–72, 81–82, 54, 64, 74, 84	Lateral incisors and first molars.
18–24 Months	51–55, 61–65, 71–75, 81–85	All primary teeth erupted.
2–3 Years	51–55, 61–65, 71–75, 81–85	Full set of primary teeth.
6–7 Years	16, 26, 36, 46, 31, 41, 51–55, 61–65, 71–75, 81–85	First molars and lower central incisors.
7–8 Years	11, 21, 16, 26, 31, 32, 36, 41, 42, 46, 52–55, 62–65, 72–75, 82–85	Upper central and lower lateral incisors.
8–9 Years	11, 12, 21, 22, 16, 26, 31, 32, 36, 41, 42, 46, 53–55, 63–65, 73–75, 83–85	Upper lateral incisors.
9–10 Years	11, 12, 21, 22, 16, 26, 31, 32, 33, 36, 41, 42, 43, 46, 54, 55, 64, 65, 74, 75, 84, 85	Lower canines.
10–12 Years	11–15, 21–25, 31–35, 41–45, 16, 17, 26, 27, 36, 37, 46, 47	Premolars and upper canines.
12–13 Years	11–17, 21–27, 31–37, 41–47	Second molars.
17–21 Years	11–18, 21–28, 31–38, 41–48	Third molars (wisdom teeth).

Table 2: Detailed Eruption and Shedding Timeline¶

Age Range	Eruption (FDI Notation)	Shedding (FDI Notation)	Notes
0–6 Months	None	None	Edentulous stage.
6–12 Months	51, 61, 71, 81	None	Lower and upper central incisors erupt.
12–18 Months	52, 62, 72, 82, 54, 64, 74, 84	None	Lateral incisors and first molars erupt.
18–24 Months	53, 63, 73, 83, 55, 65, 75, 85	None	Canines and second molars erupt.
2–3 Years	None	None	All primary teeth are present.
6–7 Years	16, 26, 36, 46, 31, 41	71, 81	First molars and lower central incisors erupt; primary lower central incisors shed.
7–8 Years	11, 21, 32, 42	51, 61, 72, 82	Upper central and lower lateral incisors erupt; primary upper central and lower lateral incisors shed.
8–9 Years	12, 22	52, 62	Upper lateral incisors erupt; primary upper lateral incisors shed.
9–10 Years	33, 43	73, 83	Lower canines erupt; primary lower canines shed.
10–12 Years	14, 15, 24, 25, 34, 35, 44, 45, 13, 23	53, 54, 55, 63, 64, 65, 74, 75, 84, 85	Premolars and upper canines erupt; primary canines and molars shed.
12–13 Years	17, 27, 37, 47	None	Second molars erupt.
17–21 Years	18, 28, 38, 48	None	Third molars (wisdom teeth) may erupt.

Table 3: Primary Teeth Eruption and Shedding¶

Tooth (FDI Notation)	Tooth Name	Eruption (Years)	Shedding (Years)
51, 61	Upper Central Incisors	6–12 months	7–8 years
52, 62	Upper Lateral Incisors	9–13 months	8–9 years
53, 63	Upper Canines	16–22 months	10–12 years
54, 64	Upper First Molars	13–19 months	10–12 years
55, 65	Upper Second Molars	25–33 months	10–12 years
71, 81	Lower Central Incisors	6–10 months	6–7 years
72, 82	Lower Lateral Incisors	10–16 months	7–8 years
73, 83	Lower Canines	17–23 months	9–10 years
74, 84	Lower First Molars	14–18 months	10–12 years
75, 85	Lower Second Molars	23–31 months	10–12 years

Table 4: Permanent Teeth Eruption Timeline¶

Tooth (FDI Notation)	Tooth Name	Eruption (Years)
11, 21	Upper Central Incisors	7–8 years
12, 22	Upper Lateral Incisors	8–9 years
13, 23	Upper Canines	11–12 years
14, 15	Upper First Premolars	10–11 years
16, 17	Upper First Molars	6–7 years
18, 28	Upper Second Molars	12–13 years
31, 41	Lower Central Incisors	6–7 years
32, 42	Lower Lateral Incisors	7–8 years
33, 43	Lower Canines	9–10 years
34, 35	Lower First Premolars	10–12 years
36, 37	Lower First Molars	6–7 years
38, 48	Lower Second Molars	11–13 years

These tables provide a clear overview of dental eruption and shedding timelines for both primary and permanent teeth using FDI notation.

Primary teeth shedding as mobility Disease = signs. Input sign Tooth = year

missing kennedy classification