Question: A sundial's gnomon casts a shadow of length $5\,\text{cm}$ when the sun's elevation angle is $60^\circ$. What is the height of the gnomon?
["Title: How to Calculate the Height of a Sundial Gnomon Using Sun’s Elevation Angle – A Step-by-Step Guide", "---", "Keywords: sundial gnomon height, sun elevation angle, shadow length, trigonometry, gnomon length calculation, solar geometry", "---", "### Introduction", "Ever noticed how a sundial’s vertical stick (gnomon) casts a shadow cast by the sun? Understanding the relationship between the gnomon’s height, sun elevation angle, and shadow length can unlock fascinating insights into both ancient timekeeping and basic trigonometry. A common question arises: If a sundial’s gnomon casts a shadow 5 cm long when the sun’s elevation angle is 60°, what is the height of the gnomon?", "In this article, we’ll explore how to solve this practical problem using fundamental trigonometric principles. By the end, you’ll understand how to calculate the gnomon’s height step-by-step, whether you’re an astronomy enthusiast, a student, or someone curious about how sundials work.", "---", "### Understanding the Physics: Sun Elevation and Shadow", "The gnomon in a sundial acts as a vertical reference. When sunlight hits it at a specific elevation angle, it casts a shadow on the dial surface. The length of that shadow depends directly on the gnomon’s height and the sun’s angle above the horizon.", "This relationship is governed by basic trigonometry. Specifically, the tangent of the sun’s elevation angle relates the height of the gnomon (( h )) to the length of the shadow (( s )):", "[\n\ an(\ heta) = \frac{\ ext{height of gnomon}}{\ ext{length of shadow}} = \frac{h}{s}\n]", "where:\n- ( \ heta = 60^\circ )\n- Shadow length ( s = 5,\ ext{cm} )\n- ( h ) = height of the gnomon (to be calculated)", "---", "### Applying Trigonometry", "Start with the tangent function:", "[\n\ an(60^\circ) = \frac{h}{5}\n]", "We know from trigonometric values:", "[\n\ an(60^\circ) = \sqrt{3} \approx 1.732\n]", "So,", "[\n\sqrt{3} = \frac{h}{5}\n]", "Multiply both sides by 5:", "[\nh = 5 \cdot \sqrt{3}\n]", "---", "### Calculating the Gnomon Height", "[\nh = 5 \cdot 1.732 = 8.66,\ ext{cm}\n]", "Thus, the height of the gnomon is approximately 8.66 cm, or more precisely:", "[\n\boxed{h = 5\sqrt{3},\ ext{cm}}\n]", "---", "### Why This Matters", "This simple calculation reveals how sundials are ingeniously designed—using geometry and solar positioning to mark time accurately. The gnomon’s height is no arbitrary number; it’s carefully calculated to ensure reliable shadow length across varying sunlight angles. For horologists, astronomers, and educators, understanding this principle connects historical instruments to modern physics and math.", "---", "### Summary", "- The gnomon’s height ( h ) relates to shadow length ( s ) and sun elevation angle ( \ heta ) via ( h = s \cdot \ an(\ heta) ).\n- For ( s = 5,\ ext{cm} ) and ( \ heta = 60^\circ ), ( h = 5\sqrt{3},\ ext{cm} ).\n- This exemplifies the elegant fusion of astronomy, geometry, and practical engineering in sundials.", "---", "### Further Reading", "- Trigonometry fundamentals for timekeeping devices\n- How to build a functional sundial at home\n- The role of shadow angles in solar energy and architecture", "---", "Understanding the shadow is understanding the sun — and in sundials, the gnomon’s height is the key to unlocking time itself. 🌞⌚", "---", "Meta Description:\nLearn how to calculate the height of a sundial’s gnomon using the sun’s elevation angle. With a 5 cm shadow and 60° solar elevation, the gnomon’s height is exactly ( 5\sqrt{3} ) cm—perfect for precision timekeeping.", "---", "Internal Link Suggestions:\n- How shadow angles affect time measurement\n- From sundials to modern solar panels: ancient wisdom today\n- Geometry in ancient engineering: other sundial designs"]
