Tables of Cell Statistics Across Resolutions

Cell counts

We list the number of hexagons and pentagons at each H3 resolution. There are always exactly $12$ pentagons at every resolution.

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 given by WGS84/EPSG:4326.

Average area in km²

The area of an H3 cell varies based on its position relative to the icosahedron vertices. We show the average hexagon areas for each resolution. All pentagons within a resolution have the same area.

Res	Average Hexagon Area (km2)	Pentagon Area* (km2)	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 m²

Here are the same areas, but in m².

Res	Average Hexagon Area (m2)	Pentagon Area* (m2)
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. We compute the minimum and maximum values for the hexagon areas (excluding the pentagons) at each resolution, and show their ratio.

Res	Min Hexagon Area (km^2)	Max Hexagon 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 given by WGS84/EPSG: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

Hexagons have 7 hexagon children. Pentagons have 6 children: 5 hexagons and 1 pentagon.

Cell counts

By definition, 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.

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.

Finally, using the closed forms for $h(n)$ and $p(n)$, and the fact that (by definition) 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.

Cell areas

Cell areas are computed with a spherical model of the earth using the authalic radius given by WGS84/EPSG:4326.

The h3-py-notebooks repo has notebooks for producing the average cell area table and the min/max area table.