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---
id: restable
title: Tables of Cell Statistics Across Resolutions
sidebar_label: Tables of cell stats
slug: /core-library/restable
---
## Cell counts
We list the number of hexagons and pentagons at each H3 resolution.
[There are always exactly $12$ pentagons at every resolution](../core-library/overview.md).
| Res | Total number of cells | Number of hexagons | Number of pentagons |
|------:|------------------------:|---------------------:|----------------------:|
| 0 | 122 | 110 | 12 |
| 1 | 842 | 830 | 12 |
| 2 | 5,882 | 5,870 | 12 |
| 3 | 41,162 | 41,150 | 12 |
| 4 | 288,122 | 288,110 | 12 |
| 5 | 2,016,842 | 2,016,830 | 12 |
| 6 | 14,117,882 | 14,117,870 | 12 |
| 7 | 98,825,162 | 98,825,150 | 12 |
| 8 | 691,776,122 | 691,776,110 | 12 |
| 9 | 4,842,432,842 | 4,842,432,830 | 12 |
| 10 | 33,897,029,882 | 33,897,029,870 | 12 |
| 11 | 237,279,209,162 | 237,279,209,150 | 12 |
| 12 | 1,660,954,464,122 | 1,660,954,464,110 | 12 |
| 13 | 11,626,681,248,842 | 11,626,681,248,830 | 12 |
| 14 | 81,386,768,741,882 | 81,386,768,741,870 | 12 |
| 15 | 569,707,381,193,162 | 569,707,381,193,150 | 12 |
## Cell areas
:::caution
Cell areas are computed with a **spherical** model of the earth using the
[authalic radius](https://en.wikipedia.org/wiki/Earth_radius#Authalic_radius)
given by
[WGS84](https://en.wikipedia.org/wiki/WGS84)/[EPSG:4326](https://epsg.io/4326).
:::
### Average area in km<sup>2</sup>
The area of an H3 cell varies based on its position relative to the
[icosahedron vertices](../core-library/overview.md).
We show the **average** hexagon areas for each resolution.
All pentagons within a resolution have the same area.
| Res | Average <ins>Hexagon</ins> Area (km<sup>2</sup>) | Pentagon Area* (km<sup>2</sup>) | Ratio (P/H) |
|------:|---------------------------------------------------:|----------------------------------:|--------------:|
| 0 | 4,357,449.416078381 | 2,562,182.162955496 | 0.5880 |
| 1 | 609,788.441794133 | 328,434.586246469 | 0.5386 |
| 2 | 86,801.780398997 | 44,930.898497879 | 0.5176 |
| 3 | 12,393.434655088 | 6,315.472267516 | 0.5096 |
| 4 | 1,770.347654491 | 896.582383141 | 0.5064 |
| 5 | 252.903858182 | 127.785583023 | 0.5053 |
| 6 | 36.129062164 | 18.238749548 | 0.5048 |
| 7 | 5.161293360 | 2.604669397 | 0.5047 |
| 8 | 0.737327598 | 0.372048038 | 0.5046 |
| 9 | 0.105332513 | 0.053147195 | 0.5046 |
| 10 | 0.015047502 | 0.007592318 | 0.5046 |
| 11 | 0.002149643 | 0.001084609 | 0.5046 |
| 12 | 0.000307092 | 0.000154944 | 0.5046 |
| 13 | 0.000043870 | 0.000022135 | 0.5046 |
| 14 | 0.000006267 | 0.000003162 | 0.5046 |
| 15 | 0.000000895 | 0.000000452 | 0.5046 |
*: Within a given resolution, all pentagons have the same area.
### Average area in m<sup>2</sup>
Here are the same areas, but in m<sup>2</sup>.
| Res | Average <ins>Hexagon</ins> Area (m<sup>2</sup>) | Pentagon Area* (m<sup>2</sup>) |
|------:|--------------------------------------------------:|---------------------------------:|
| 0 | 4,357,449,416,078.392 | 2,562,182,162,955.496 |
| 1 | 609,788,441,794.134 | 328,434,586,246.469 |
| 2 | 86,801,780,398.997 | 44,930,898,497.879 |
| 3 | 12,393,434,655.088 | 6,315,472,267.516 |
| 4 | 1,770,347,654.491 | 896,582,383.141 |
| 5 | 252,903,858.182 | 127,785,583.023 |
| 6 | 36,129,062.164 | 18,238,749.548 |
| 7 | 5,161,293.360 | 2,604,669.397 |
| 8 | 737,327.598 | 372,048.038 |
| 9 | 105,332.513 | 53,147.195 |
| 10 | 15,047.502 | 7,592.318 |
| 11 | 2,149.643 | 1,084.609 |
| 12 | 307.092 | 154.944 |
| 13 | 43.870 | 22.135 |
| 14 | 6.267 | 3.162 |
| 15 | 0.895 | 0.452 |
*: Within a given resolution, all pentagons have the same area.
### Hexagon min and max areas
The area of an H3 cell varies based on its position relative to the
[icosahedron vertices](../core-library/overview.md).
We compute the minimum and maximum values for the **hexagon** areas (excluding
the pentagons) at each resolution, and show their ratio.
| Res | Min <ins>Hexagon</ins> Area (km^2) | Max <ins>Hexagon</ins> Area (km^2) | Ratio (max/min) |
|------:|-------------------------------------:|-------------------------------------:|------------------:|
| 0 | 4,106,166.334463915 | 4,977,807.027442012 | 1.212276 |
| 1 | 447,684.201817940 | 729,486.875275344 | 1.629468 |
| 2 | 56,786.622889474 | 104,599.807218925 | 1.841980 |
| 3 | 7,725.505769639 | 14,950.773301379 | 1.935248 |
| 4 | 1,084.005635363 | 2,135.986983965 | 1.970457 |
| 5 | 153.766244448 | 305.144308779 | 1.984469 |
| 6 | 21.910021013 | 43.592111685 | 1.989597 |
| 7 | 3.126836030 | 6.227445905 | 1.991613 |
| 8 | 0.446526174 | 0.889635157 | 1.992347 |
| 9 | 0.063780227 | 0.127090737 | 1.992635 |
| 10 | 0.009110981 | 0.018155820 | 1.992740 |
| 11 | 0.001301542 | 0.002593689 | 1.992782 |
| 12 | 0.000185933 | 0.000370527 | 1.992797 |
| 13 | 0.000026562 | 0.000052932 | 1.992802 |
| 14 | 0.000003795 | 0.000007562 | 1.992805 |
| 15 | 0.000000542 | 0.000001080 | 1.992805 |
## Edge lengths
:::caution
Edge lengths are computed with a **spherical** model of the earth using the
[authalic radius](https://en.wikipedia.org/wiki/Earth_radius#Authalic_radius)
given by
[WGS84](https://en.wikipedia.org/wiki/WGS84)/[EPSG:4326](https://epsg.io/4326).
Average edge lengths were calculated exactly for resolutions 0 through 6 and
extrapolated for finer resolutions.
:::
| Res | Average edge length (Km) |
|----:|-------------------------:|
| 0 | 1281.256011 |
| 1 | 483.0568391 |
| 2 | 182.5129565 |
| 3 | 68.97922179 |
| 4 | 26.07175968 |
| 5 | 9.854090990 |
| 6 | 3.724532667 |
| 7 | 1.406475763 |
| 8 | 0.531414010 |
| 9 | 0.200786148 |
| 10 | 0.075863783 |
| 11 | 0.028663897 |
| 12 | 0.010830188 |
| 13 | 0.004092010 |
| 14 | 0.001546100 |
| 15 | 0.000584169 |
## Appendix: Methodology
<div align="center">
<img src="/images/pentagon_hexagon_children.png" style={{width:'800px'}} /><br />
<i>Hexagons have 7 hexagon children. Pentagons have 6 children: 5 hexagons and 1 pentagon.</i>
</div>
### Cell counts
[By definition](../core-library/overview.md), resolution `0` has $110$
**hexagons** and $12$ **pentagons**, for a total of $122$ **cells**.
In fact, *every* H3 resolution has exactly $12$ **pentagons**, which are always
centered at the icosahedron vertices; the number of **hexagons** increases
with each resolution.
:::tip Formula
Accounting for both **hexagons** and **pentagons**,
the total number of **cells** at resolution $r$ is
$$
c(r) = 2 + 120 \cdot 7^r.
$$
:::
#### Derivation of the cell count formula
We can derive the formula above with the following steps.
First, let $h(n)$ be the number of
children $n \geq 0$ resolution levels below any single **hexagaon**.
Any **hexagon** has $7$ immediate children, so recursion gives us
that
$$
h(n) = 7^n.
$$
Next, let $p(n)$ be the number of children $n \geq 0$ resolution levels below
any single **pentagon**.
Any **pentagon** has $5$ hexagonal immediate children and $1$ pentagonal
immediate child.
Thus, $p(0) = 1$ and $p(1) = 6$.
For $n \geq 1$, we get the general recurrence relation
$$
\begin{aligned}
p(n) &= 5 \cdot h(n-1) + p(n-1) \\
&= 5 \cdot 7^{n-1} + p(n-1).
\end{aligned}
$$
For $n \geq 0$, after working through the recurrence, we get that
$$
\begin{aligned}
p(n) &= 1 + 5 \cdot \sum_{k=1}^n\ 7^{k-1} \\
&= 1 + 5 \cdot \frac{7^n - 1}{6},
\end{aligned}
$$
using the closed form for a
[geometric series](https://en.wikipedia.org/wiki/Geometric_series).
Finally, using the closed forms for $h(n)$ and $p(n)$,
and the fact that ([by definition](../core-library/overview.md))
resolution `0` has
$12$ **pentagons** and $110$ **hexagons**,
we get the closed form for the total number of **cells**
at resolution $r$ as
$$
\begin{aligned}
c(r) &= 12 \cdot p(r) + 110 \cdot h(r) \\
&= 2 + 120 \cdot 7^r.
\end{aligned}
$$
#### Jupyter notebook
A notebook to produce the cell count table above can be
[found here](https://github.com/uber/h3-py-notebooks/blob/master/notebooks/stats_tables/cell_counts.ipynb).
### Cell areas
Cell areas are computed with a **spherical** model of the earth using the
[authalic radius](https://en.wikipedia.org/wiki/Earth_radius#Authalic_radius)
given by
[WGS84](https://en.wikipedia.org/wiki/WGS84)/[EPSG:4326](https://epsg.io/4326).
The [`h3-py-notebooks` repo](https://github.com/uber/h3-py-notebooks)
has notebooks for producing the
[average cell area table](https://github.com/uber/h3-py-notebooks/blob/master/notebooks/stats_tables/avg_area_table.ipynb)
and the
[min/max area table](https://github.com/uber/h3-py-notebooks/blob/master/notebooks/stats_tables/extreme_hex_area.ipynb).
|
