A sundial
- Round
- Theory
- Points
- 10
- Level
- Bronze
- Taught in
- Section 2.8
Adapted from IOAA 2024 (Rio de Janeiro), theory problem 1. We have restated it in our own words, drawn our own sundial from a calculation and reworded statement (e), as the solution explains. The solution was last checked on 17 October 2026.
Figure 2.10 shows a horizontal sundial from above. Its gnomon is a flat triangle standing upright on the heavy line, with its tip directly above the point F, and the shadow of the tip falls on the dial. The sundial is either between the Tropic of Cancer and the Arctic Circle or between the Tropic of Capricorn and the Antarctic Circle. Each hour has a solid line and a dashed line, and the Sun's hour angles on the two lines for the same hour differ by the difference in longitude between the sundial and the central meridian of its time zone. Through the year the shadow of the tip of the gnomon always falls between curves A and C. Is each of the following statements true or false?
(a) The sundial works properly only in the southern hemisphere.
(b) Curve A is the path of the shadow of the tip of the gnomon on the day of the winter solstice, in the sundial's own hemisphere.
(c) Line B is the path of the shadow of the tip on the days of the equinoxes.
(d) The solid lines give local mean solar time.
(e) The figure of eight around the dashed line for 12 shows where the shadow of the tip falls at 12:00 zone time through the year.
- (a) The statement is true. A shadow points away from the Sun. Outside the tropics in the northern hemisphere the Sun moves from east through south to west, so the shadow turns from west through north to east, which is clockwise seen from above. In the southern hemisphere the Sun passes north of the zenith, and the shadow turns from west through south to east, anticlockwise. The hours on this dial increase anticlockwise, so it is built for the southern hemisphere, and its gnomon points to the south celestial pole.
- (b) The statement is false. Curve A is nearest to F, so it holds the shortest shadows, which come when the Sun is highest, at the summer solstice. In the southern hemisphere that is the December solstice.
- (c) The statement is true. At an equinox the Sun's declination is zero and it moves along the celestial equator, so the lines from the tip of the gnomon to the Sun all lie in one plane through the tip, parallel to the equator. That plane cuts the dial in a straight line. On any other day the lines form a cone, which cuts the dial in a curve.
- (d) The statement is false. The solid lines are where the shadow falls when the Sun's hour angle at the sundial is a whole number of hours, so they give local apparent solar time, the time kept by the real Sun (Section 2.2). Local mean time differs from it by the equation of time, by up to 16.4 minutes.
- (e) The statement is true. At 12:00 zone time the mean sun is on the central meridian of the time zone (Section 2.3). The real Sun is then
AnswerStatements (a), (c) and (e) are true, and (b) and (d) are false, as in the official answers. The original statement (e) described the figure of eight as the shadow at true solar noon on the central meridian, but at that moment the shadow lies on the dashed line for 12 itself. The figure of eight shows it at mean noon on the central meridian, which is 12:00 zone time, so we have reworded the statement, and its answer stays true. We drew the sundial for Christchurch in New Zealand (43.53° S, 172.64° E), whose standard time is the mean time of 180° E, so each dashed line is 7.36° of hour angle, or 29 minutes, from the solid line for the same hour.
This problem also appears among the olympiad problems of Section 2.8, Problems for Chapter 2, which teaches the ideas it uses. Equation and figure numbers refer to the course.