25 guides

Guides

The math and the data behind the tools, with every step shown on real places and every quoted figure linked to its source.

01 — 6 guides

Distance and direction

  1. How the distance between two places is calculatedStraight-line distance uses the haversine formula on a sphere. Worked example: Boston to Providence is 40.6 miles, every step shown.
  2. How to find the midpoint between two coordinatesAveraging latitudes and longitudes misses the true midpoint by 201 miles on a Sacramento to Boston trip. The spherical formula, worked.
  3. How to calculate the bearing between two pointsBearing from latitude and longitude, with the formula and a worked example: Denver to Boston starts at 73° and ends at 96°.
  4. Great circle vs rhumb lineA great circle is the shortest path; a rhumb line keeps one compass heading. Sacramento to Boston, the rhumb line adds 38 miles.
  5. How far away is the horizon?From a standing eye height of 1.7 m the geometric horizon is 4.7 km (2.9 miles) away. Formula and a table by height.
  6. Which state capitals are closest together?Washington, DC and Annapolis, MD are the closest capitals at 30 miles. The nearest capital to all 51, in one table.

02 — 6 guides

Coordinates and maps

  1. How many miles are in a degree?A degree of latitude is about 69.1 miles. A degree of longitude is 69.17 miles at the equator and 34.67 at 60°.
  2. Latitude and longitude explainedLatitude runs north and south, longitude east and west. Juneau is the northernmost US capital at 58.30°N. All 51 capitals listed.
  3. How to convert DMS to decimal degreesDecimal degrees = degrees + minutes/60 + seconds/3600, negative for south and west. Worked example on Nashville and a 10 capital table.
  4. How many decimal places do GPS coordinates need?Five decimal places pin a point to about 1.1 m, six to 11 cm. Table of ground size per decimal at the equator and at 40°.
  5. UTM zones explainedUTM splits the world into 60 zones, each 6° of longitude wide. US counties fall in 18 zones. The zone for every state capital, with easting and northing.
  6. How much does the Mercator projection distort size?Mercator stretches lengths by 1/cos(latitude) and areas by its square. On Web Mercator, Alaska looks 8.1 times the size of Texas; on the ground it is 2.2.

03 — 7 guides

Counties and codes

  1. FIPS codes explainedA county FIPS code is 5 digits: 2 for the state, 3 for the county. Cook County, IL is 17031. Full state code table and 3,144 county codes.
  2. How many counties are in the United States?The 50 states and DC have 3,144 counties and county-equivalents in 2024 Census data: 2,999 counties plus parishes, boroughs and independent cities.
  3. What is an independent city?An independent city belongs to no county and counts as a county in Census data. There are 41: 38 in VA, 1 in MD, 1 in MO, 1 in NV.
  4. What is the most common county name?Washington is the most common county name, used in 31 states. The top 25 names with every state that uses them.
  5. What does a typical US county look like?The median US county has 26,122 residents on 616 sq mi of land. The average is 108,178 residents. Percentiles for size, people and income.
  6. Parishes, boroughs and planning regionsLouisiana has 64 parishes, Alaska 30 boroughs and census areas, Connecticut 9 planning regions. How each fits the county system.
  7. How county land area and internal points are measuredCensus land area leaves out water area. The 10 largest counties hold 13% of US land. How ALAND and the internal point are defined.

04 — 4 guides

Population and income data

  1. How population density is calculatedDensity = residents ÷ land area. The US averages 96.3 people per sq mi, yet the median county has 45.0. Worked example and county bands.
  2. Where is the center of population of each state?A population center computed from 3,144 counties lands 6 miles from the Census 2020 US center. The same method for all 50 states.
  3. How Census population estimates workBetween censuses, the Census Bureau estimates population each July 1 from births, deaths and migration. This site uses Vintage 2024: 340,110,988 residents.
  4. How county median household income is estimatedSAIPE models income for 3,143 counties. The median county has $62,851; Loudoun County tops the list at $173,655.

05 — 2 guides

Sunrise, sunset and daylight

  1. How sunrise and sunset times are calculatedSunrise is when the sun's center is 0.833° below the horizon. The calculation step by step, and June vs December daylight at all 51 US capitals.
  2. How many hours of daylight at each latitude?At 40°N days run from 9 h 20 min in December to 15 h 01 min in June. Daylight at every 10° of latitude on the solstices and equinoxes.