Navigation calculations

Ten questions with at most 4 mistakes. They are solved with formulas and a calculator.

Practise this unit Print

Great circle

cos D = sin l1 sin l2 + cos l1 cos l2 cos ΔL. Turn D into minutes to get miles.

Initial course: tan R = sin ΔL / (cos l1 tan l2 − sin l1 cos ΔL), then convert to circular.

Position triangle

sin a = sin l sin d + cos l cos d cos P. Latitude and declination of opposite names: d negative.

Meridian: z = 90º − a. Facing south, l = d + z. Facing north, l = d − z.

Activities

• Sight reductionGreat circles, computed altitude, azimuth and meridian latitude, step by step.
• AlmanacHour angle, declination and the time of the Sun’s meridian passage.
• SextantBring the sun to the horizon, read the arc and correct the reading.

Questions in this unit

• Calculate the Great Circle distance from point A at l=42º23’S and L=147º42’E to point B at l=56º00’S and L=67º13’W.
• Calculate the initial Great Circle course from point A at l=43º23’S and L=147º42’E to point B at l=56º00’S and L=67º13’W.
• What is the shortest distance between the points on Earth at 25º N, 110º W and 45º N, 156º E? (round to the mile)
• Initial course to sail by great circle from 25º N, 110º W to 45º N, 156º E. (round to the degree)
• Initial course to sail by great circle from 36º 30' S, 092º 45' W to 21º 20' S, 163º 50' E (round to the degree).
• Great circle distance between 36º 30' S, 092º 45' W and 21º 20' S, 163º 50' E (round to the mile).
• Initial course to sail by great circle from 33º N, 025º E to 66º S, 085º E (round to the degree).
• Shortest distance between the points on Earth at 33º N, 025º E and 66º S, 085º E (round to the mile).
• From DR position 48º 19' S, 142º 28' E, two bodies are observed simultaneously, giving: Body 1: Z = 152º, Δa = -5.7'; Body 2: Z = 239º, Δa = +2.1'. Calculate the position.
• From DR position 26º 42' S, 75º 18' E, two bodies are observed simultaneously, giving: Body 1: Z = 313º, Δa = +5.1'; Body 2: Z = 024º, Δa = -3.3'. Calculate the position.
• Calculate graphically the observed position from two simultaneous position lines at DR position 24°50.3´ N; 150°58.0´E, given: Kochab: Z= N14W; Altitude difference= +3. Hamal: Z= N75E; Altitude difference= +2.
• 29 June 2021: at Greenwich civil time 130058, they are at DR position 40º 07.2'N - 014º 25.0'W. At that moment the Sun passes the yacht’s moving meridian, and they get a true Sun altitude of 72º 59.5' and declination +23º 11.7’. Having worked out the zenith distance, they find the observed latitude:
• On 24.11.2024, the Sun’s lower limb is observed with sextant altitude 34º 01.5’. Index error Ei = 1.5’ to the right. Height of eye 5 metres. Calculate the Sun’s true altitude.
• On 15.03.2025, the Sun’s lower limb is observed with sextant altitude 34º 06’. Index error Ei = 2` (-). Height of eye 5 metres. Calculate the Sun’s true altitude.
• A ship at Longitude = 142º 36’ W has zone time 20h56m18s on the 14th. Calculate its local civil time:
• At Greenwich civil time 17:43:40, in DR longitude 056º 35.0′E and with a Greenwich hour angle of the star Antares at that time of 292º 20.7′, calculate the meridian angle (P) between the hour angle of Antares and the observer’s longitude.
• At the same time a sextant altitude of the star Antares is taken as 75 09.5′ (height of eye 4 metres and index correction −2′), and its true altitude is calculated.
• With declination −26º 28.1′, DR latitude 15º 46.0′S and a meridian angle of 11º 04.3′ (E), the skipper of the yacht “Eagle” calculates the computed altitude of the star Antares using the position triangle formulas, and the altitude difference, given a true altitude of 75º 03.6′:
• On Friday 9 March 2018, heading for the Strait of Gibraltar, at local civil time 221010 and in DR position 36º 15′N 005º 05′W, the students of the sixth watch calculate the Greenwich civil time at that moment... :
• At Greenwich civil time 1330, still sailing in the same conditions, the DR position is 23º 27′S 041º 30′W and the Sun’s Greenwich hour angle is 020º 45.7′. Calculate the meridian angle (P).
• At Greenwich civil time 1350, still sailing towards Cape Town, the DR position is 23º 27.2′S 041º 27.2′W, with the Sun’s declination + 00º 39.3′ and meridian angle 15º 41.5′ E. Calculate the Sun’s true azimuth, in circular notation, using the position triangle formulas.
• On Saturday 10 March 2018, at Greenwich civil time (GCT) 103000, they are at DR position 34º 18′N – 006º 36′W and, having worked out the Sun’s declination for that moment = -4º 02.5′ and the meridian angle (P) to the Sun = 31º 40.2′ (E), they work out the Sun’s true azimuth at that moment (using the formulas for the sides and angles of the position triangle), obtaining:
• Having obtained the Sun’s true altitude, with the same Greenwich civil time = 1350, DR position 23º 27.2′S - 041º 27.2′W, meridian angle = 15º 41.5′E and Sun’s declination = +00º 39.3′, work out the Sun’s computed altitude, using the formulas that solve the sides and angles of the position triangle.
• They study the great circle route from -A- 32º14.2′N – 064º43.1′W (Hamilton) to -B- 36º50.3′N – 008º57.5′W (south of San Vicente, Portugal) and work out the great circle distance:
• GCT = 0600, south of Cabo de Gata, at point -A- 36º 20.0′N – 002º 17.7′W, the Captain works out the great circle distance to point -B- at Sapri (Amalfi coast, Salerno, Italy) at 39º 53.0′N – 015º 34.8′E:
• At longitude 010º 00' W the local civil time (HcL) is 22h 40m on the 14th. Calculate the local civil time at longitude 020º 00' E.
• When Greenwich civil time (HcG) is 22h 30m on 10 April 2020, what are the standard time (Ho) and date in the Seychelles?
• At longitude 132º 20' E on 3 December 2022, when local civil time (HcL) is 3h 24m 10s, calculate the Greenwich civil time (HcG) and zone time (Hz).
• On 22 November 2025, ship A is at 14º 14’ N, 120º 28' E and ship B is at 10º 21’ S, 098º 09' W, both keeping zone time (Hz). When ship A's clock reads 08:00, find Greenwich civil time (HcG) and ship B's zone time (Hz).
• On 26 April 2025 at local civil time (HcL) 23:01:00, in position l = 40° 36.0' N, L = 135º 15' E, calculate the Greenwich civil time (HcG) and zone time (Hz).
• On 23 November 2024 at local civil time (HcL) 22h 57m 30s, at longitude L = 120º 15' W, calculate the Greenwich civil time (HcG) and zone time (Hz).
• On the 6th at longitude 140º 18' E the local civil time (HcL) is 3h 42m 10s. At that same instant, calculate the HcL and zone time (Hz) at longitude 086º 24' W.
• At position A (l = 40º 30' N, L = 30º 30' E), the local civil time HcL(A) is 12h 40m on 20 April 2024. What is the local civil time at position B (l = 25º 15' S, L = 10º 00' W)?
• When Greenwich civil time (HcG) is 23h 00m on 23 April 2023, what are the zone time (Hz) and date at longitude L = 172º 30' W?
• At longitude 110º 45' W on 25 October 2023, when local civil time (HcL) is 23h 44m 50s, calculate the Greenwich civil time (HcG) and zone time (Hz).
• After departing Puerto Sherry at zone time (Hz) = 200000 from the departure waypoint (36º 32.5’ N, 006º 19.2' W), calculate the Greenwich civil time (HcG) at that instant.
• In the Indian Ocean near Mauritius, when Greenwich civil time (HcG) is 2230 on 28 March, calculate the local civil time (HcL) and date at DR position 24º 20.2’ S, 066º 20.0’ E:
• In the North Atlantic near New York at longitude 070º 40.2’ W, when our zone time (Hz) is 0720 on 29 March, calculate the Greenwich civil time and date:
• On 16 April 2021 at dawn, departing Nassau at ship's clock set to zone time (Hz) = 08:00 from point A (25º 42.1' N, 076º 42.4' W), calculate the Greenwich civil time (HcG) and date:
• On 25 June 2021 at ship's clock set to zone time (Hz) = 0600, at departure point A (49º 59' N, 006º 44’ W), calculate the Greenwich civil time at that instant:
• On 16 November 2021 at zone time (Hz) 0800, in position 27º 39.3' N, 015º 34.8' W, calculate the Greenwich civil time (HcG) at that instant:
• At California standard time = 04:00 with a +2h daylight-saving advance in force, at DR position 32º 40.2' N, 119º 26.5' W, calculate the Greenwich civil time:
• On 10 April at zone time Hz = 06h 13m 30s, ship A is at longitude L = 132º 42’ W. At what longitude is ship B if her local civil time is HcL = 09h 47m 06s on the same day?
• On 14 March 2026 at local civil time HcL = 14h 06m 52s, at DR position l = 35º 30’ N, L = 011º 30’ W, calculate the Greenwich civil time (HcG) at that instant.
• In Faro (37º 02' N, 007º 56' W) the zone time (Hora Legal) is 23h 23m 23s on 23 April 2020. What is the local civil time (HcL) in Carboneras (37º 00' N, 001º 54' W)?
• On 28 February 2021, when UTC is 13:12:08, find the local civil time at a point of longitude 060º 23’ W.
• Calculate the great-circle distance between Position A (19° 42.0' N, 112° 09.0' W) and Position B (20° 18.0' N, 154° 15.0' W).
• Calculate the great-circle distance from point A (l = 30° 37.0’ N, L = 143° 38.0’ E) to point B (l = 29° 28.0’ N, L = 127° 30.0’ W).
• Calculate the initial great-circle course from departure l = 36º 41.8’ N, L = 008º 56.9’ W to arrival l’ = 03º 59.8’ S, L’ = 033º 43.7’ W.
• Calculate the great-circle distance between Vigo (42º 14.1' N, 008º 43.6' W) and Rio de Janeiro (22º 54.4' S, 043º 11.8' W).
• Calculate the initial course to sail by the shortest route from a place at longitude 015º 00′ E and any latitude (except 90º) to another at longitude 075º 00′ W and latitude 00º 00′.
• Calculate the great-circle distance separating the points 33° 41′ S, 132° 27′ E and 16° 19′ N, 081° 52′ W.
• Great-circle sailing: calculate the distance between 65° 33' S, 093° 14' W and 12° 31' N, 169° 56' E.
• Great-circle distance between the points 47º 39′ S, 91º 52′ E and 18º 21′ N, 103º 46′ W:
• Calculate the shortest distance separating the points on the Earth's surface at 25º 43' S, 110º 17' E and 45º 58' S, 156º 39' W.
• In position 35° 00.3' N, 019° 02.0' E, given that the Sun's corrected Greenwich hour angle (hGc) is 300° 13.2' and its corrected declination is +23° 25.5', find the Sun's computed altitude.
• Calculate the Sun's altitude from position l = 22° 57.0' N, L = 160° 36.0' W on 30 March 2022 at 18:00 UT (given GHA = 88º 54.2' and declination = +3º 57.9').
• On 25 November 2023, shortly after crossing the Tropic of Cancer on Rv = 340°, at meridian transit the Sun's lower-limb sextant altitude is Ai = 43° 17.7', declination δ = −20° 40.1', height of eye 3 m, and index error Ei = −4'. Find the observed latitude.
• On 17 April 2021 at Greenwich civil time (HcG) = 17:30, in DR position 27º 17.6' N, 073º 26.4' W, given the Sun's Greenwich hour angle (hGO) = 82º 37.8', calculate the meridian angle (P) to the Sun:
• On 18 April 2021 at HcG = 14:50, at DR position 28º 03.6' N, 070º 57.1' W, given the Sun's declination = +11º 07.6' and meridian angle P = 28º 16.4' (E), calculate the Sun's circular true azimuth:
• On 20 April 2021 at Greenwich civil time (HcG) = 23:26, in DR position 31º 12.0' N, 066º 14.4' W, given the Greenwich hour angle of Aries (HGY) = 200º 45.4', calculate the local hour angle of Aries (HlY):
• On 21 April 2021 at HcG = 02:26, in DR position 31º 32.8' N, 065º 53.3' W, given the Greenwich hour angle of Aries (HGY) = 245º 52.8' and the sidereal hour angle (A.S.) of Arcturus = 145º 50.4', calculate the Greenwich hour angle of Arcturus (HG*):
• On 27 June 2021, sailing on compass course Ra = 214º at HcG = 1015 in DR position 44º 22.2' N, 008º 36.4' W, the Sun's compass azimuth is Za = 121º, declination = +23º 29.5', and meridian angle P = 35º 34.9' (E). Calculate the true course (Rv):
• At Greenwich civil time 10:30, becalmed without way at longitude 119º 50.0' W, given the Sun's Greenwich hour angle (hGO) = 337º 24.1', calculate the time interval until the Sun reaches that fixed meridian:
• A vessel sailing on compass course Ra = 165° at latitude l = 36° N takes a relative bearing on the Sun at true sunset of 95° to starboard, when the Sun's declination is 13° 45’ S. Calculate the total compass error (Ct):
• In position 43° 30′ S, 89° 24′ E, given that the Sun's declination is 18° 00′ S, calculate the Sun's azimuth at true sunrise.
• From position 51° 37.8' S, 173° 11.1' W, we observe a body with local hour angle = 104° 05.8' and declination = 19° 38.4' S. At what nautical (circular) azimuth will we see it?
• Calculate the Sun's azimuth at true sunset from position 61° 27.0' S, 122° 47' W, when the Sun's declination is 21° 12.6' N.
• On 20 April 2024, sailing on Rv = 130° after crossing the Tropic of Capricorn, at meridian transit the Sun's lower-limb sextant altitude is ai = 51° 50.4', declination +11° 52.5', index error Ei = +0.6', and height of eye 7 m. Find the observed latitude.
• Find the azimuth and computed altitude of a body when the observer's latitude is 25º 00.0’ N, the body's declination is 38º 47.3’ N, and the local hour angle is 351º 18.9’.
• Find the total compass error (Ct) given that the compass azimuth of Polaris is Za = 005º.
• On 18 December 2020 the Sun's sextant altitude is 35° 45.7', height of eye is 2.4 m above sea level, and the zero of the micrometer drum lies 0.2' below the zero of the vernier. Calculate the Sun's true altitude.
• On 23 June 2015 the Sun's sextant altitude is 43° 21.7', height of eye is 14.2 m above the water, and the zero of the micrometer drum lies 0.2' above the zero of the vernier. Calculate the Sun's true altitude.
• On 22 November 2025 the Sun's lower-limb sextant altitude is 36º 25.4’, height of eye is 3 m above sea level, and the sextant index error is 0.8’ (-). Calculate the Sun's true altitude.
• On 26 April 2025 the sextant altitude of the star Sirius is 20º 31.0', height of eye is 8.2 m above sea level, and the sextant index error is 0.1' (-). Calculate the star's true altitude.
• Determine the true altitude (av) of the Sun's lower limb on 23 April 2023, given sextant altitude = 40º 00’, Ci = -3', and height of eye = 1.8 m.
• Determine the true altitude of the star Vega given sextant altitude = 40º 20', Ci = -4', and height of eye = 5 m.
• At HcG = 10:00 in DR position 31º 33.2' N, 121º 38.2' W, the sextant altitude of the star Vega is 76º 50' 22'' with index correction = +4' and height of eye 3 m. Calculate Vega's true altitude:
• On 26 November 2021 we obtain a lower-limb Sun sextant altitude of 36º 25.4' with height of eye 10 m and index correction 1' (-). Find the true altitude.
• On 15 September 2020 the Sun's lower-limb sextant altitude is 38° 30.0', height of eye is 7.85 m, and the zero of the micrometer drum lies 0.2' above the zero of the vernier. Calculate the Sun's true altitude.
• At Greenwich civil time 123000, berthed in Puerto Sherry, the master of the yacht “Albatros” works out the great circle distance to Mindelo: Departure WPT = 36º 32.5'N – 006º 19.2'W. Arrival WPT = 21º 05.4'N – 016º 22.8'W.
• Calculate the great circle distance between point A in the Canaries = 29º 40.6’N 015º 20.3’W and point B near Puerto de la Guaira (Venezuela) = 10º 36.5’N 066º 56.9’W:
• 15 April 2021. The Master works out the great circle distance from point A in Nassau (Bahamas) = 25º 42.1´N – 076º 42.4´W to point B south of Cape St Vincent (Portugal) = 36º 53.8´N – 009º 00.0´W:
• 24 June 2021. The Master studies the great circle route from departure point A 49º59'N - 006º 01’W west of Plymouth to arrival point B 28º44'N – 018º 04'W, near Tazacorte on La Palma (Canaries), and works out the distance
• 15 November 2021, the Master studies the great circle route from departure point A 27º 39.3´N – 015º 34.8´W to arrival point B 18º 25.1´N – 064º 20.1´W and works out the distance:
• The great circle distance from departure point A 32º 45.8´N – 117º 38.2´W (San Diego) to arrival point B 21º 09.3´N – 157º 38.2´W (Honolulu):
• At l = 43º 09.6’ N and L = 009º 47.4’ W, calculate the initial course (Ri) and great circle distance to sail to point P at lp = 06º 48’ N and Lp = 058º 08’ W.
• On 15 March 2025 we leave l: 43º 27`N L:008º 23`W for a point at l:36º 54`N L: 075º 50´W. Calculate the great circle distance.
• Calculate the great circle distance between point A at l: 34º38`N L: 139º 51`E and point B at l: 37º 47`N L: 122º 38`W
• Calculate the great circle distance from point A (l=05º35’N and L=077º28’W) to point B (l=50º47’N and L=156º38’E).
• Calculate the initial great circle course from point A (l=05º35’N and L=077º28’W) to point B (l=50º47’N and L=156º38’E).
• Calculate the Initial Great Circle Course to sail most effectively from Tokyo (latitude: 35º 25.0’N; Longitude: 139º 46.0’E) to Lima (latitude: 10º 16.3’S and Longitude: 077º 53.0’W).
• Calculate the initial great circle course (Ri) to sail most effectively from departure point A (latitude: 32º 39´ N Longitude: 117º 17´ W) to arrival point B (latitude: 34º 36`N L:139º 58` E).
• On 16 March 2025 the schooner “Tramuntana” is near the landfall buoy of Acapulco at l: 16º 45`N L: 099º 55´W, setting course for a point in Fiji at l: 17º 40`N L: 178º 55`E. Calculate the initial great circle course in circular notation.
• Calculate the initial great circle course from point A near Bali = 10º 10.0’S – 126º 20.0’E to point B east of Durban (South Africa) = 30º 30.0’S 040º 10.0'E:
• In the North Atlantic, near the Canaries, on 28 March 2020, in the morning, at DR position 30º 10.7’N 015º 10.0’W, we first obtain for the Sun: Declination = +03º 15.8’; Angle at the pole (P) = 27º 40.2’ E. We calculate the computed altitude of the Sun using the formulas that solve the sides of the position triangle.
• 19 November 2021, at Greenwich civil time 1020, at DR position 26º 40.2´N – 021º 27.2´W, having obtained the Sun’s declination -19º 33.4 and angle at the pole 42º 41.0’ (E), they work out the Sun’s computed altitude with the formulas for the sides of the position triangle:
• In Portuguese waters 100 miles west of Figueira da Foz, in the early hours of 29 March 2020, at Greenwich civil time 0430, at DR position 40º 10.6’N 011º 03.2’W, we obtain for the star Arcturus: Declination = +19º 04.6' and angle at the pole (P) = 21º 30.4’W. We calculate the star’s true azimuth in circular notation.
• We are at DR latitude le = 25º 19’ N, the Angle at the Pole of the star Regulus being P = 26º 46.7’ E and its declination d= +11º 52.2’. Calculate its nautical azimuth (Zn).
• On 15 March 2025 we are at DR position le = 45º 15’ N and Le: 003 20`E, the Angle at the Pole of the star Regulus (no. 50) being P = 38º 47.8’ and its declination d= +11º 50.5’. Calculate its nautical azimuth (Zn).
• At DR position le = 43º 56.8’ N and Le = 045º 39’ W, the Sun is observed on the upper meridian with a true meridian altitude of 62º 25.0’, its declination being +16º 22.9’. Calculate the observed latitude.
• At DR position le = 40º 15` N and Le = 003º 20’ E, the sun is observed on the upper meridian with a true meridian altitude of 47º 48`, its declination being - 1º 55.7`. Calculate the observed latitude.
• On 14 March 2026, at DR position l=35º05’N and L=011º00’W, the Sun is observed on the upper meridian of the place with a true meridian altitude of 52º31.2’, the Sun’s declination being -2º 24.7’. Calculate the observed latitude.
• On 13 June 2026, at DR position l=40º30’N and L=010º30’W, the Sun is observed on the upper meridian of the place with a true meridian altitude of 72º 41.1’, the Sun’s declination being +23º13.5’. Calculate the observed latitude.
• A ship at L = 18º 30º W observed facing South a true meridian altitude of the Sun of 32º 15.2’, the Sun’s declination being 18º 14.9 `S. Find the observed latitude.
• The Sun is observed on the upper meridian of the place with a true altitude of 62°. Calculate the latitude, knowing that the culmination is seen facing North (Z = 000°) and that the Sun’s declination is 08° S.
• Calculate the observed position from two simultaneous position lines at DR position le=33°05.0´N; Le=061°27.7´W, given: Markab (True azimuth =085º, Altitude difference =2.0 (-)) and Dubhe (True azimuth =330.5º, Altitude difference = 0.5 (-)).
• Calculate the observed position from two simultaneous position lines at DR position le=27º 00.0´N; Le:016°31.9´W, given: Atria (True azimuth: 005º, Altitude difference: - 3.5) and Formalhaut (True azimuth: 120º, Altitude difference: +2.5).
• At DR position 19º 44' S, 132º 32' W, two bodies are observed simultaneously. For the first we calculate Z = 067º, Δa = –4.3’ and for the second Z = 349º, Δa = +3.2’. Calculate the observed position.
• A ship on compass course 150° at latitude 25° N takes, at sunset, a compass azimuth of the Sun of 263°, its declination being 15° 47’ S. Calculate the total correction (CT):
• At 14 38 23 UT on 18 January 2023 the Sun’s lower limb is observed with a sextant altitude of 31º 29.8'. Index correction = +8', height of eye = 6 m. Calculate the true altitude.