How Ancient Indian Astronomers Calculated the Size of the Earth and the Sun Without Telescopes

In 499 CE, the mathematician and astronomer Aryabhata wrote in the Aryabhatiya that the Earth's circumference was 4,967 yojanas. Scholars who have converted the yojana to modern units place that figure between 39,736 km and 40,075 km. The actual circumference of the Earth is 40,075 km. The margin of error is under 1%. Aryabhata was 23 years old when he completed the text.
He did not guess. He calculated, using the same geometric logic that Eratosthenes had applied in Alexandria roughly a thousand years earlier, though there is no established evidence that Aryabhata had access to Eratosthenes' work. Both men arrived at the same method independently: measure the angle of the Sun's shadow at two known locations at the same moment, and use the difference to infer the curvature of the Earth below.The method depends on one assumption, that the Sun is far enough away that its rays arrive at the Earth essentially parallel. If you plant a vertical stick (a gnomon) in the ground at noon on the summer solstice, it casts a shadow whose angle tells you your latitude. If someone plants an identical stick at a different latitude and measures their shadow angle at the same moment, the difference between the two angles equals the arc of the Earth's surface between those two points. Divide 360 degrees by that angular difference, multiply by the known ground distance between the two locations, and you have the full circumference.
Indian astronomers called the gnomon the shanku. The Aryabhatiya and later texts like Brahmagupta's Brahmasphutasiddhanta (628 CE) describe shadow measurements with precision that implies systematic, repeated observation across multiple sites. This was not armchair mathematics, it required coordinated measurement, reliable distance estimates between cities, and the ability to identify solar noon accurately.Calculating the Sun's actual size requires two pieces of information: how far away it is, and how large it appears in the sky. Ancient Indian astronomers had a workable estimate of the angular diameter of the Sun, about half a degree, which is correct. The harder problem was distance.
Aryabhata placed the Sun at roughly 459,585 km from the Earth. The actual mean distance is about 149.6 million km, so that figure is significantly short. But the method he used, based on the geometry of lunar eclipses and the relative sizes of the Earth's shadow and the Moon, was structurally sound. The error came from the available data on parallax, not from a flaw in the geometric reasoning.

Brahmagupta later revised solar distance estimates. By the medieval period, the Surya Siddhanta, a Sanskrit astronomical text whose composition scholars date to somewhere between the 4th and 9th centuries CE, gave a solar diameter of 6,522 Earth diameters. The modern figure is about 109 Earth diameters, so the Surya Siddhanta overshot significantly. What that text got right was the Moon's diameter: it placed the Moon at 0.28 Earth diameters. The actual figure is 0.27.No telescopes. No satellites. The instruments ancient Indian astronomers used were the gnomon, the water clock (clepsydra), the armillary sphere (gola yantra), and careful naked-eye observation of celestial events, particularly lunar and solar eclipses, which provided the clearest geometric windows into the relative sizes and distances of the Earth, Moon, and Sun.
Eclipse geometry is especially powerful. During a lunar eclipse, the Earth's shadow falls on the Moon. By timing how long the Moon takes to cross that shadow, and knowing the Moon's angular speed across the sky, you can calculate the width of Earth's shadow at the Moon's distance. Combine that with the known angular size of the Moon and you can triangulate the Earth's actual diameter. Aryabhata's Earth diameter figure from this method, approximately 12,805 km, compares to the actual polar diameter of 12,714 km.

The institutional structure behind this work matters too. Aryabhata worked in Kusumapura, likely near present-day Patna, in what was then the Gupta Empire. Astronomical calculation was not a solitary pursuit, it was embedded in a tradition of royal patronage, temple calendar-making, and the practical demands of agriculture and navigation. Accuracy had consequences. A festival calculated on a wrong date, a monsoon predicted badly, these were not abstract failures.