Latitude and Longitude
Open a map app and drop a pin on downtown Chicago. The screen reads $41.88^\circ$ N, $87.63^\circ$ W. Those two numbers pin down any spot on Earth, just like $(x, y)$ pins down a point on a coordinate plane. The difference is that the "sheet of paper" is a sphere, so the coordinates are angles. By the end of this page you will be able to decode them, measure distances with them, and even work out what time it is on the other side of the world.
Meridians and parallels
Picture the Earth as a sphere that spins around an axis through the North and South Poles.
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Meridians are half-circles that run from pole to pole. The reference meridian is the prime meridian, which passes through Greenwich, England. It is meridian $0^\circ$.
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Parallels are circles perpendicular to the axis. The biggest one is the Equator (parallel $0^\circ$). It splits the Earth into the Northern and Southern Hemispheres.
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The prime meridian, together with the $180^\circ$ meridian on the opposite side, splits the Earth into the Eastern and Western Hemispheres.
Drag the sliders and see how a point moves:
Imagine you are a beginner cartographer. On your first map, you must locate the most important reference lines of the Earth. Look at the drawing and match each letter…
Imagine you are a beginner cartographer. On your first map, you must locate the most important reference lines of the Earth. Look at the drawing and match each letter with its corresponding name:
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Equator.
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Greenwich Meridian.
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North Pole.
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Any parallel.
State in which hemisphere (North or South for latitude, East or West for longitude) the following geographical points are located: Point A: $34^\circ$ N, $120^\circ$ E…
State in which hemisphere (North or South for latitude, East or West for longitude) the following geographical points are located:
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Point A: $34^\circ$ N, $120^\circ$ E
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Point B: $12^\circ$ S, $45^\circ$ W
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Point C: $89^\circ$ N, $5^\circ$ W
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Point D: $40^\circ$ S, $170^\circ$ E
Latitude and longitude
Here is how to read a flat map (a planisphere). Find the horizontal line for the latitude and the vertical line for the longitude, then see where they cross.
Point $P$ is $15^\circ$ south of the Equator (halfway between the $10^\circ$ S and $20^\circ$ S lines) and halfway between the $50^\circ$ W and $40^\circ$ W lines. So $P$ is at $15^\circ$ S, $45^\circ$ W.
An explorer has mistakenly written down the following coordinates in a diary. Explain why these coordinates are geographically impossible: a) Latitude: $105^\circ$ S,…
An explorer has mistakenly written down the following coordinates in a diary. Explain why these coordinates are geographically impossible:
- a) Latitude: $105^\circ$ S, Longitude: $20^\circ$ E.
- b) Latitude: $45^\circ$ N, Longitude: $210^\circ$ W.
Two ships, Alfa and Beta, are sailing in the Atlantic Ocean. Their coordinates are: Ship Alfa: $20^\circ$ N, $30^\circ$ W Ship Beta: $20^\circ$ N, $60^\circ$ W Answer…
Two ships, Alfa and Beta, are sailing in the Atlantic Ocean. Their coordinates are:
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Ship Alfa: $20^\circ$ N, $30^\circ$ W
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Ship Beta: $20^\circ$ N, $60^\circ$ W
Answer with reasoning:
- a) Are they on the same parallel or on the same meridian?
- b) Which of the two ships is closer to the Greenwich Meridian?
Look at the following flat representation (planisphere) of an ocean area. Identify the coordinates (Latitude and Longitude) of point $P$ marked with a star.
Look at the following flat representation (planisphere) of an ocean area. Identify the coordinates (Latitude and Longitude) of point $P$ marked with a star.
Antipodes and degrees-minutes
The antipode of a place is the point diametrically opposite it on the sphere. To find it:
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Latitude: keep the number, switch N ↔ S.
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Longitude: subtract from $180^\circ$ and switch E ↔ W.
Angles are often given in degrees and minutes, with $1^\circ = 60'$. To subtract from $180^\circ$ with minutes, write $180^\circ$ as $179^\circ 60'$.
If you travel by airplane from Chicago ($40^\circ$ N, $3^\circ$ W) to a city that is exactly at its antipodes (the opposite point on the Earth's sphere): a) What would…
If you travel by airplane from Chicago ($40^\circ$ N, $3^\circ$ W) to a city that is exactly at its antipodes (the opposite point on the Earth's sphere):
- a) What would the latitude of that point be? (Same value but opposite direction).
- b) If the longitude of a point and its antipode must add up to $180^\circ$ and be in opposite hemispheres, what would the longitude of that point be?
The antipodes of a place are said to be the diametrically opposite point on the Earth's surface. If the coordinates of your city are $40^\circ$ $25'$ N latitude and…
The antipodes of a place are said to be the diametrically opposite point on the Earth's surface. If the coordinates of your city are $40^\circ$ $25'$ N latitude and $3^\circ$ $41'$ W longitude:
- a) What would the latitude of your antipodes be?
- b) What would the longitude of your antipodes be?
(Hint: For the latitude, change the hemisphere; for the longitude, find the meridian that completes $180^\circ$ and change the E/W hemisphere).
Distances along a meridian
Two cities on the same meridian are joined by an arc of a great circle. A full trip around a meridian circle is $2\pi R$. In these problems $R = 6{,}371$ mi, so the full circle is about $40{,}030$ mi. If the two cities are $\alpha$ degrees apart in latitude, the distance is the fraction $\alpha/360$ of that circle.
Two cities, Sunville and Moonville, are located on the same meridian. If Sunville has a latitude of $45^\circ$ N and Moonville has a latitude of $20^\circ$ S, how many…
Two cities, Sunville and Moonville, are located on the same meridian. If Sunville has a latitude of $45^\circ$ N and Moonville has a latitude of $20^\circ$ S, how many degrees of meridian arc separate them?
A humpback whale migrates following exactly the same meridian from a feeding ground located at $65^\circ$ N latitude to a breeding ground at $15^\circ$ S. If the mean…
A humpback whale migrates following exactly the same meridian from a feeding ground located at $65^\circ$ N latitude to a breeding ground at $15^\circ$ S. If the mean radius of the Earth is considered to be $6\,371$ miles, calculate the total distance in miles that the whale travels on its outward journey.
Calculate the shortest distance between two points, P and Q, located on the Earth's surface if their coordinates are: P: $(40^\circ \text{ N}, 10^\circ \text{ O})$ Q:…
Calculate the shortest distance between two points, P and Q, located on the Earth's surface if their coordinates are:
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P: $(40^\circ \text{ N}, 10^\circ \text{ O})$
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Q: $(40^\circ \text{ S}, 10^\circ \text{ O})$
Consider the Earth to be a perfect sphere of radius $6371 \text{ mi}$. Express the result in miles. Does the traveler move along a meridian or a parallel? Justify your answer.
Working backward: from distance to angle
Sometimes you know the distance and need the angle. Solve the same proportion for $\alpha$:
$$\alpha = \frac{360\, d}{2\pi R} = \frac{d}{111.195}\ \text{degrees}$$(with $R = 6{,}371$ mi). On the Equator, longitude works the same way, because the Equator is a great circle.
So $1$ nautical mile $= \dfrac{111.195}{60} \approx 1.853$ mi with this radius. Also, each degree of latitude is exactly $60$ nautical miles, whatever the radius is.
If a problem gives its own conversion for knots or nautical miles, use the one it gives.
In navigation, the "nautical mile" is used, which is defined as the length of a meridian arc of $1'$ (one minute of a degree). a) Calculate how many miles a nautical…
In navigation, the "nautical mile" is used, which is defined as the length of a meridian arc of $1'$ (one minute of a degree).
- a) Calculate how many miles a nautical mile is equivalent to (Radius of the Earth = $6\,371$ miles).
- b) If a sailboat has sailed along a meridian from a latitude of $30^\circ$ N to $35^\circ$ N, how many nautical miles has it traveled?
Two oceanographic research ships are on the Equator. Ship A is at a longitude of $12^\circ$ E and ship B is at a distance of $2\,500$ miles to the West of ship A.…
Two oceanographic research ships are on the Equator. Ship A is at a longitude of $12^\circ$ E and ship B is at a distance of $2\,500$ miles to the West of ship A. Determine the exact longitude of ship B, knowing that the radius of the Earth is $6\,371$ miles.
Distances along a parallel
A parallel at latitude $\alpha$ is a circle smaller than the Equator. Slice the sphere through the axis and the picture is a right triangle: the radius $r$ of the parallel is the horizontal leg, and the Earth's radius $R$ is the hypotenuse.
Handy values (use degree mode on your calculator): $\cos 30^\circ \approx 0.866$, $\cos 45^\circ \approx 0.707$, $\cos 60^\circ = 0.5$.
For a school project, you need to calculate the length of the $40^\circ$ N parallel of latitude. Knowing that the Earth's radius is $R \approx 6\,371$ miles, calculate:…
For a school project, you need to calculate the length of the $40^\circ$ N parallel of latitude. Knowing that the Earth's radius is $R \approx 6\,371$ miles, calculate:
- a) The radius of the circle formed by this parallel (use the expression $r = R \cdot \cos(\text{latitud})$).
- b) The total length (perimeter) of this parallel.
Two cities, P and Q, are located on the same meridian. City P is at $42^\circ$ N and city Q is at $15^\circ$ N. Another city, R, is located on the same parallel as Q,…
Two cities, P and Q, are located on the same meridian. City P is at $42^\circ$ N and city Q is at $15^\circ$ N. Another city, R, is located on the same parallel as Q, but at a distance of $20^\circ$ of longitude to the East.
Calculate, using $R = 6\,371$ miles:
- a) The distance between P and Q along the meridian.
- b) The distance between Q and R along the parallel.
Which of the two distances is greater?
Two cities, A and B, are located on the same parallel at a latitude of $45^\circ \text{ N}$. City A has a longitude of $20^\circ \text{ E}$ and city B a longitude of…
Two cities, A and B, are located on the same parallel at a latitude of $45^\circ \text{ N}$. City A has a longitude of $20^\circ \text{ E}$ and city B a longitude of $70^\circ \text{ E}$.
Knowing that the radius of the Earth is approximately $6371 \text{ mi}$:
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Calculate the radius of the $45^\circ \text{ N}$ parallel (Remember that the radius of a parallel at a latitude $\alpha$ is calculated as $r = R \cdot \cos \alpha$. For $45^\circ$, you can use $0,707$ as an approximation of the cosine).
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Determine the distance in miles that separates both cities following the arc of the parallel.
Time zones
The Earth turns $360^\circ$ in 24 hours, which is $360^\circ \div 24 = 15^\circ$ per hour. So there are 24 theoretical time zones, each $15^\circ$ of longitude wide. Zone $0$ is centered on the prime meridian. The Sun rises first in the east, so moving east it is later, and moving west it is earlier.
To find the time zone number, divide the longitude by $15$ and round to the nearest whole number. East is $+$ and west is $-$. UTC (Coordinated Universal Time) is the time at Greenwich.
$$\text{local time} = \text{UTC} + \text{zone number}$$The safest method for any problem is to convert to UTC, do the flight or the wait, then convert to the destination's zone.
We know that the Earth takes 24 hours to complete a $360^\circ$ rotation on its axis. a) How many degrees of longitude correspond to a one-hour difference? b) If it is…
We know that the Earth takes 24 hours to complete a $360^\circ$ rotation on its axis.
- a) How many degrees of longitude correspond to a one-hour difference?
- b) If it is 12:00 noon in a city located at $0^\circ$ longitude (Greenwich), what time will it be in a city located at $30^\circ$ E?
A group of friends in Madrid (longitude $3^\circ$ $41'$ W) wants to watch a live space rocket launch from Cape Canaveral (longitude $80^\circ$ $36'$ W). If the launch is…
A group of friends in Madrid (longitude $3^\circ$ $41'$ W) wants to watch a live space rocket launch from Cape Canaveral (longitude $80^\circ$ $36'$ W). If the launch is scheduled for 2:30 p.m. local time in Florida, and knowing that each $15^\circ$ time zone is equivalent to a theoretical one-hour difference, at what time should they tune into the live stream in Spain? Use the theoretical time zone diagram to carry out the calculation.
A reconnaissance plane takes off from a base located at coordinates $(30^\circ \text{ N}, 15^\circ \text{ O})$ at 8:00 a.m. local time. It flies directly East following…
A reconnaissance plane takes off from a base located at coordinates $(30^\circ \text{ N}, 15^\circ \text{ O})$ at 8:00 a.m. local time. It flies directly East following the same parallel until it reaches a city located at $60^\circ$ longitude East. The flight has a total duration of 9 hours.
At what local time of the destination city will the plane land? Take into account the change of time zones during the journey (consider theoretical time zones of $15^\circ$ each).
The date line and crossing 180°
The $180^\circ$ meridian (the antimeridian of Greenwich) is where the International Date Line roughly runs. Cross it and the calendar date changes. A date change happens exactly when a route crosses the $180^\circ$ meridian. Crossing the prime meridian does not change the date.
If you use zone numbers (east $+$, west $-$) and convert through UTC, the date line takes care of itself. The date is whatever the arithmetic gives.
The shortest longitude difference
To find how many degrees of longitude separate two points:
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Same hemisphere (both E or both W): subtract.
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Opposite hemispheres: add. If the sum is more than $180^\circ$, the short way goes across the $180^\circ$ meridian, so use $360^\circ -$ the sum.
An airplane takes off from Tokyo ($35^\circ$ N, $139^\circ$ E) at 8:00 a.m. on Monday bound for San Francisco ($37^\circ$ N, $122^\circ$ W). If the flight lasts 10…
An airplane takes off from Tokyo ($35^\circ$ N, $139^\circ$ E) at 8:00 a.m. on Monday bound for San Francisco ($37^\circ$ N, $122^\circ$ W). If the flight lasts 10 hours, what time and day will it be in San Francisco when the airplane lands? Take into account the date change when crossing the Greenwich antimeridian ($180^\circ$).
A submarine travels along the seabed from a point X $(0^\circ, 175^\circ \text{ E})$ to a point Y $(0^\circ, 170^\circ \text{ O})$. 1. How many degrees of longitude has…
A submarine travels along the seabed from a point X $(0^\circ, 175^\circ \text{ E})$ to a point Y $(0^\circ, 170^\circ \text{ O})$.
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How many degrees of longitude has it traveled if it followed the shortest route along the Equator?
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If the submarine travels at a constant speed of $20 \text{ nudos}$ ($1 \text{ nudo} \approx 1,85 \text{ mph}$), how long will it take to complete the journey? (Radius of the Earth: $6371 \text{ mi}$).
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Was there a change of date during the journey? Explain why.
Polyhedra: Euler's formula and surface area
Some longer problems on this page switch from the globe to 3-D solids. Here are the tools for the polyhedron ones.
When you truncate a solid (slice off its corners), a vertex where $k$ edges meet is replaced by $k$ new vertices and one new $k$-sided face. That gives a quick way to count:
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$F = $ (old faces) $+$ (number of vertices cut off).
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$V = $ (sum over the cut vertices of the edges meeting there).
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$E = \dfrac{3V}{2}$ when exactly 3 edges meet at every vertex.
For surface area, add up the areas of all faces. These formulas are enough:
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Square of side $s$: $s^2$.
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Equilateral triangle of side $s$: $\dfrac{\sqrt{3}}{4}s^2$.
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Regular hexagon of side $s$: $\dfrac{3\sqrt{3}}{2}s^2$.
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Regular polygon: $\tfrac{1}{2}\cdot(\text{perimeter})\cdot(\text{apothem})$.
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Prism: lateral area $=$ (perimeter of base) $\times$ (height).
A regular hexagon splits into 6 equilateral triangles, so its circumradius equals its side, and its apothem (center to the middle of a side) is $a = \dfrac{\sqrt{3}}{2}s$.
A truncated octahedron is constructed from a regular octahedron with an edge of $12 \text{ cm}$. To do this, each of the six vertices is cut off so that small regular…
A truncated octahedron is constructed from a regular octahedron with an edge of $12 \text{ cm}$. To do this, each of the six vertices is cut off so that small regular square-based pyramids are removed, turning each original triangular face into a regular hexagon.
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Determine the number of faces, vertices and edges of the resulting polyhedron and check if it satisfies Euler's formula.
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If the edge of the new polyhedron (that of the resulting hexagon) is $4 \text{ cm}$, calculate the total surface area of the truncated octahedron.
A water tank has the shape of a right regular hexagonal prism. It is known that the radius of the circumscribed circle of the hexagonal base measures $4 \text{ m}$ and…
A water tank has the shape of a right regular hexagonal prism. It is known that the radius of the circumscribed circle of the hexagonal base measures $4 \text{ m}$ and that the height of the prism is three times the apothem of the base.
Calculate the cost of waterproofing the side walls and the bottom (bottom base) if the price of the material is $12,50 \text{ \$/m}^2$. Round the result to two decimal places.
Volumes of curved solids
The last two challenge problems mix cones, spheres and cylinders. Here is the toolkit.
The frustum formula comes from a big cone minus the small cone that was cut off the top. Similar triangles tell you how tall the missing cone was.
Two more tips:
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Inscribed solids. If a sphere of radius $r$ touches the walls and both bases of a cylinder, then the cylinder has radius $r$ and height $2r$. The "free space" left over is $V_{\text{cylinder}} - V_{\text{sphere}}$.
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Capacity. $1 \text{ cm}^3 = 1 \text{ mL}$. To check whether a container holds a given number of milliliters, compare its volume in cm$^3$ with that number.
A perfume company designs a bottle in the shape of an inverted frustum of a cone topped by a hemisphere that fits perfectly on the top base. The design data is as…
A perfume company designs a bottle in the shape of an inverted frustum of a cone topped by a hemisphere that fits perfectly on the top base. The design data is as follows:
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Height of the frustum of a cone: $8 \text{ cm}$.
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Radius of the bottom base (where it rests): $3 \text{ cm}$.
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Radius of the top base (where the hemisphere fits): $6 \text{ cm}$.
Calculate the total volume of the bottle and determine if it is able to hold a standard $330 \text{ ml}$ bottle of essence. (Take $\pi \approx 3,14$).
There is a cylinder of radius $R$ and height $2R$. A sphere of radius $R$ is placed inside it, fitting tangentially with the walls and bases of the cylinder. 1. Prove…
There is a cylinder of radius $R$ and height $2R$. A sphere of radius $R$ is placed inside it, fitting tangentially with the walls and bases of the cylinder.
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Prove analytically the relationship between the volume of the sphere and the volume of the cylinder.
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If the volume of the cylinder is $54 \text{ cm}^3$, what volume of air is left free inside the cylinder once the sphere is inserted?
Back to the start: $41.88^\circ$ N means $41.88^\circ$ north of the Equator, and $87.63^\circ$ W means $87.63^\circ$ west of the prime meridian. That is Chicago, on the shore of Lake Michigan.