The solar altitude angle, or simply altitude, is the Sun’s angular position above the horizon from the perspective of an observer on the earth. It is an important factor to consider in fields such as solar energy, building energy, and agriculture.
It is easy to know the solar altitude angle for where you are at the present moment. Just go out, look for the Sun, and estimate its angle above the horizon. But how about for any given set of location, date, and time? For example, what is the solar altitude angle for Boston, MA at noon on the winter solstice?
We can figure this out using a little bit geometry. Take a look at the diagram below. We know that the latitude of Boston, MA, is 42° and the Earth’s tilt is 23°. On the winter solstice, the Sun reaches its southernmost point and the Earth’s North Pole tilts away from the Sun. Thus, the angular distance between the observer and the orbital plane is the sum of the latitude and the Earth tilt (42° + 23° = 65°). Assuming the orbital plane is parallel to the sunray, according to the Parallel Lines Theorem, the solar zenith angle (Z) is equal to the angular distance (65°) we just calculated. Then, according to the Complementary Angles Theorem, the solar altitude angle is 90° minus the solar zenith angle (90° - 65° = 25°).
We can verify our solution using Energy3D, a simulation-based engineering tool for designing green buildings and power stations. Energy3D can simulate the Sun’s apparent movement in the sky and its angular position for a given date and time for many major cities in the world. Download, install, and open Energy3D. Go to Tutorials → Solar Science Basics → Solar Angles to open a prebuilt model for exploring the Sun’s path. Set the location to Boston, MA, date on December 22 (winter solstice for the northern hemisphere), and time at 12:00pm. Then read the solar altitude angle (h) in the display area. The reading is 25°, the same as we previously calculated.
Now, try for yourself to find out the solar altitude angle for Boston, MA, at noon on the summer solstice (June 22) and verify your result using the Energy3D software.
Related Learning Standards:
CCSS.MATH.CONTENT.7.G.B.5
Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.
CCSS.MATH.CONTENT.HSG.CO.C.9
Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.
NGSS. ESS1.A The Universe and Its Stars
Patterns of the apparent motion of the sun, the moon, and stars in the sky can be observed, described, predicted, and explained with models. (MS-ESS1–1)