Al-Biruni calculated the Earth's radius to be approximately 6,339.9 km (or 6,336 km in some sources), which is an incredibly close estimate for his time. He used a method involving trigonometry, the height of a mountain, and measuring the angle of the horizon's dip from the mountaintop. This calculation was highly accurate, with a margin of error of less than 1% compared to modern measurements.
Al-Biruni's method:
Concept: Al-Biruni understood that the Earth's curvature causes the horizon to appear lower from a mountaintop than from sea level.
Height measurement: He determined the height of a mountain by measuring the angle of elevation from two different points on flat ground.
Horizon dip: Using an astrolabe, he measured the angle of dip (or depression) of the horizon below the horizontal line from the mountaintop.
Calculation: He applied the law of sines to the large triangle formed by the Earth's center, the mountaintop, and the horizon to calculate the Earth's radius.
Result: His calculation for the radius was about 6,339.9 km, and his estimate for the circumference was within 0.3% of today's accepted value.