Activity 29

Building a Sundial
(Part 3 - Marking the Dial Plate)
GRADE LEVEL: Grades 10-12

Problem: Assuming that the meridian line (or noontime mark) has been established and the gnomon has been angled, we must find a way to mark the hour lines on the dial plate. Create a mathematical model that uses the angle of the sun on the style (top of the gnomon that creates the shadow) to mark hour lines on the dial plate of the sundial.

Using angle ø as the base angle of the gnomon, and angle t as the angle of the arc the sun passes through in a given time frame, we should be able to calculate angle h by using the length of the resulting shadow.

One Possible Solution: Angle t is perhaps the first angle that needs to be defined. Because the earth rotates through a central angle of 360^o in 24 hours, we can assume that each hour is defined by a 15^o arc of the sun's apparent movement over the surface of the earth. This is true at any longitude. Angle t then measures 15^o for each hour away from the noon hour.

If the length of the style on the gnomon is known, the vertical side of the gnomon can be calculated as follows: Height = sin ø (S) ; where S is the length of the style

At 11:00, angle t = 15^o. So, if we want to mark the 11:00 hour on the dial plate, angle h can be calculated by measuring the length of the shadow from the noon mark and subsequently estimating the length of the adjacent side as equal to the length of the style, so that the tangent ratio can be used.

The side of the shadow opposite angle h = (tan t)(sin ø)S

Also, tan h = [(tan t)(sin ø )S]/S ; since the tangent ratio is opp/adj and S is being substituted for the adjacent side in this ratio

Therefore: tan h = (tan t)(sin ø)

Because we know the latitude of our specific location, we also know the measure of angle ø. Let us assume for the sake of easy calculations that we are at a 30^o latitude, and that our style length is 8 inches. Because we are marking the 11:00 hour line, angle t = 15^o. We need to find angle h for several different hours in order to mark the dial plate appropriately. The following is a test calculation:

tan h = tan 15^o(sin 30^o)
tan h = (.268)(.5)
tan h = (.268)(.5)
h = tan^-1(.134) = 7.63^o

Therefore the 11:00 hour line would lie at an angle of 7.63^o to the left of the meridian mark. Also, since the hour marks are symmetrical with respect to the noon mark, it is an angle of 7.63^o to the right of noon for 1:00.

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Problem: Assuming that the meridian line (or noontime mark) has been established and the gnomon has been angled, we must find a way to mark the hour lines on the dial plate. Create a mathematical model that uses the angle of the sun on the style (top of the gnomon that creates the shadow) to mark hour lines on the dial plate of the sundial.	Using angle ø as the base angle of the gnomon, and angle t as the angle of the arc the sun passes through in a given time frame, we should be able to calculate angle h by using the length of the resulting shadow.
One Possible Solution: Angle t is perhaps the first angle that needs to be defined. Because the earth rotates through a central angle of 360^o in 24 hours, we can assume that each hour is defined by a 15^o arc of the sun's apparent movement over the surface of the earth. This is true at any longitude. Angle t then measures 15^o for each hour away from the noon hour.