DECLINING VERTICAL SUNDIAL CALCULATOR
Enter data into the blue panel and click the calculate button.
 Latitude Declination Gnomon side angle Hour angle Length BC Length AB Shadow angle FCG Length AD Shadow length CF Length BD Horizontal dist' FG Length CD Vertical dist' CG Angle ACD To run another calculation, just edit in the blue panel as required. Only press reset to clear all boxes. Angle ACB Angle BCD Altitude of sun Azimuth from south

## Direct South Facing Vertical Sundial

The Direct South facing sundial can easily be calculated for your latitude by following the Quick Start Guide.
As can be seen, all the shadow angle lines radiate from the top at point 'C'. The values input for the dial shown are:-

• Latitude: 53
• Gnomon side angle: 90
• Length BC: 2.5
• Declination: 12
• Hour angles of: 15, 30 and 45

The Hour Angle (or to be correct the Local Hour Angle) of 45 degrees for 9am and 3pm calculates to a Shadow Angle of 31.04 degrees, and is the same for both the East and West faces of the dial as it is South facing.

The position of point 'F' corresponds to the point where the tip of the Gnomons shadow at point 'A' would touch the dial face at 2pm when the sun is at a Declination of plus 12 degrees.

To plot a line of Declination point 'F' would be plotted for each Shadow Angle and a line drawn through all the points. The process then repeated for other Declination lines as required. It is worth noting that for the Equinoxes the line is straight, and all others it is a curve. Line 'FG' is at right angles to vertical line 'CG' and angle 'FCG' is the Shadow Angle.

The Gnomon for the above dial is shown to the left and points 'B' and 'C' correspond with those on the dial face.

In this picture the gnomon of the Direct South facing dial is fitted and stands perpendicular to the dial face, so the Gnomon side angle is 90 degrees.
The shadow cast by the Gnomon is approaching the 11am shadow angle line.

## West Declining Vertical Sundial

The Declining Vertical sundial is not south facing, but is rotated either to the East or the West of south facing by some angle.

This angle has to be measured before any dial calculations can be started.
There are various ways of going about this, but would typically involve setting up a simple dial with a stick gnomon mounted on the surface that the sundial will be mounted on.
A reading of the suns azimuth angle is taken, and this is compared with the calculated azimuth of the sun at that time and date.

When the angle that the dial is to be rotated from due south has been obtained, it is converted to the Gnomon side angle.

In the example shown here, the dial is rotated 30 degrees round towards the west.
The Gnomon side angle is the angle between the dial face and the side face of the Gnomon, so as the Gnomon points due south along line 'B A' the side angle on the East face of the dial is 60 degrees, and on the West face 120 degrees.
Unlike the Direct South facing dial, the calculations for the East And West faces are carried out separately, each having different Gnomon side angles.

The values input for this Declining dial are:-

• Latitude: 53
• Length BC: 2.5
• Declination: 12
• Hour angles of: 15, 30 and 45

Two sets of calculations are obtained, one with the Gnomon side angle of 60 for the East face, and one with the Gnomon side angle of 120 for the West face of the dial.

Notice how in this action shot the shadow is following the 12 degree line of declination and has reached the 3pm time line.

Two Gnomons are shown here, the green one is same one used on the Direct South and Declining sundials shown above.
The purpose of the yellow one is to offer an alternative that stands squarly on the dial face, unlike the green one that is angled to point South.
The yellow Gnomon marked 'ACD' stands on dial face line 'CD' with a side angle of 90 degrees.
The Gnomon edge 'AC' that casts the shadow over the dial face is in exactly the same place relative to the dial face which ever type of Gnomon is chosen.

Here is another view of the Declining dial only this time its fitted with the 'ACD' gnomon.

The shadow is approaching the 3pm line and is following the 12 degree line of Declination that is plotted from point 'F' on each hour line.

As can be seen, an obvious difference between Direct south and Declining dials is the lack of symmetry between East and West faces on the Declining dial. The hour line angles are different East to West, and the lines of declination follow a different curve as they pass the vertical noon line.

## Preliminary Measurements

Before the vertical dial can be calculated, we need to know if the wall or surface intended for mounting the dial is direct south facing, and if not by what angle it is rotated from south facing.

A simple test dial can be made as shown that has a stick gnomon mounted perpendicular to the dial face with a vertical line projecting down from the center line of the gnomon.
If the shadow from the gnomon is in line with the vertical line, the sun must be directly opposite the dial.

Example 1:-
If the time is 12 noon (standard not summer time, gmt in the uk) and the gnomons shadow is vertical,
and the dial is situated on your local time meridian (zero degree longitude in the uk),
and the Equation of time is zero, then the dial must be direct south facing.

Example 2:-
If the time is 16 minutes after noon and the gnomon shadow is vertical,
and the dial is situated on the meridian of 4 degrees West of your time meridian,
and the eqation of time is zero.
A longitude of 4 degrees represents a time of 16 minutes, as one hour is 15 degrees (360/24), so the dial must be direct south facing.

Example 3:-
If the time is 1hr 40min in the afternoon and the gnomon shadow is vertical,
and the local time meridian is 16 minutes west of your time meridian,
and the equation of time is minus 6 minutes.
The dial is not situated on a time meridian, so it must be rotated with respect to due south.
The time taken for the sun to move from the Southern meridian to a position where it is facing the dial is 1hr 40min minus 16 minutes longitude plus -6 minutes for the equation of time, a total of 1hr 18min.
Whether you add or subtract Longitude and The Equation of time needs evaluating for each calculation.

We now need to convert the time to degrees, as there are 15 degrees to an hour the angle is 19.5 degrees.

There are two more item of information required, the latitude, say 55 degrees for this example, and the Declination of the sun on the day the reading was taken, in this case the date was July the 24th when the Declination is plus 20 degrees.

We now use the calculator to convert the hour angle to an angle of azimuth.
The angle of azimuth can be imagined as an angle around a vertical rod standing perpendicular in the ground, and for our purposes, zero degree is due south.

In the blue panel of the calculator enter as follows:-
• Latitude: 55
• Gnomon side angle: 1
• Length BC: 1
• Declination: 20
• Hour angle: 19.5

Now click the Calculate button and you will see the calculated value for Azimuth from the South is 30.646 degrees.
This angle would be zero degree if the dial was direct South facing.
In order to use the azimuth angle when calculating the sundial, we now convert the azimuth angle to Gnomon side angles.

The gnomon of the sundial (not necessarily the test dial) points south.
As the test dial shadow was vertical after noon, then the dial must be rotated towards the west.
The Gnomon side angle on the dials West face is 90 + 30.646 = 120.646
The Gnomon side angle on the dials East face is 90 - 30.646 = 59.354

A book that perhaps explains more clearly the setting up of declining sundials is by Albert E. Waugh and titled "Sundials Their Theory and Construction". The book contains various useful tables including the declination of the sun.

After all the foregoing work in calculating your sundial, it may be a good idea to produce a quick to make prototype of the dial to prove it out befor expending a lot of time on the final version, just to make sure it all worked out as expected...Good luck.

Back to the top
QUICK START GUIDE:-
To calculate the shadow angles for a direct south facing wall dial fill in the boxes in the blue panel as follows.
Latitude:-Your latitude in decimal degrees, eg for Derby enter 52.92
Gnomon side angle:- Enter 90
Length BC:- Enter 1
Declination:- Enter 0
Hour angle:- Hour angle in decimal degrees.
One hour =15 for 11am and 1pm shadow angles,
30 for 10am and 2pm shadow angles etc.
The Calculated shadow angle appears in the yellow panel after clicking the calculate button.

Please note that Javascript needs to be enabled to use the calculator.

DEFINITIONS:-

ANGLES...
Any Angle defined in Degrees, minutes and seconds need to be changed to decimal degrees, this can be achieved using the available Conversion calculator

LATITUDE...
In the UK your angle of latitude can be found from an Ordnance Survey map.
You can also use an online Latitude / Longitude locator.

GNOMON SIDE ANGLE...
For a dial mounted on a south facing wall the gnomon side angle is 90°. This is the angle the side of the gnomon makes with the dial face. The gnomon points south along line BA. So if the dial face is declined 30 degrees towards the west, the gnomon side angle on the west face of the dial is 120°, and on the east face 60°, (the two angles always add up to 180°).
Except when the angle is 90°, run calculations for both side angles. Note that when calculating for shadow angles with the dial west face, the resultant marking out of the west face will be for the morning sun as it comes up from the east.

LENGTH BC...
This is the vertical length of the gnomon where it joins the wall.
To the calculator the input value is in units, so it could represent Inches, Centimetres, Cubits or whatever you prefer, the calculated length values in the yellow panel will be in the same units.

DECLINATION...
The range is 23.44 for the sun's highest point in the summer, to -23.44 for the mid winter sun. The equinoxes have a declination of 0. On the dial, the lines of declination across the dial face are tracked by the shadow from the end of the gnomon, point `A', this shows as the tip of the shadow at point `F'.

HOUR ANGLE...
One hour is represented by 15 degrees, (360/24). Multiples of 15 up to 90 represent the range of hours on one side of the the dial. West face and east face being calculated separately except when the wall angle is 90. Any angles can be calculated in the range 0.1 to 90.

GNOMON...
The geometery of the gnomon is shown as lengths 'AB' 'AD' 'BD' and 'CD'. And angles 'ACB ' 'ACD' and 'BCD'.
Length 'AC' represents the shadow casting style of the gnomon. Length 'AB' projects from the dial pointing south. A gnomon can be made as triangles 'ACB' or 'ACD'.

The shadow angle is measured from vertical line 'CG' and radiates from point 'C'
The length of the shadow, as measured from point 'C' to point 'F' (shadow length CF) is affected by the declination of the sun for a given Hour angle. Distance 'FG' and 'CG' can be used as an alternative way of finding point 'F'.

LONGITUDE...
In the UK the time meridian is zero degrees Longitude that passes through Greenwich.(Greenwich Mean Time).
A sundial that is situated 3 degree West of Greenwich would be 12 minutes slow by the clock, that is the sun will be on the local Southern meridian 12 minutes after it was on the Greewich meridian.
Each hour of time represents 15 degrees of Longitude.
Longitude correction can be applied to the hour angles of a sundial, but care should be taken as in the above case, the correction would be subtracted from the dials East face and added to the hour angles of the West face. These corrections are applied to the Hour angle values that are input to the blue panel of the calculator.

THE EQUATION OF TIME...
The Equation of time is the variation in time shown by our clocks, and that shown on our sundials.

ALTITUDE and AZIMUTH...
Knowing the Altitude of the sun and in particular its Azimuth value from the southern meridian can be helpfull in finding the orientation of a wall that it is proposed to mount a sun dial on.
See the section on Preliminary Measurements for a detailed description.