---
title: 'Measuring the Earth with Terence Tao'
source: 'https://youtube.com/watch?v=uT4-WVa-YfU'
video_id: 'uT4-WVa-YfU'
date: 2026-08-10
duration_sec: 82
---

# Measuring the Earth with Terence Tao

> Source: [Measuring the Earth with Terence Tao](https://youtube.com/watch?v=uT4-WVa-YfU)

## Summary

A short video featuring mathematician Terence Tao explaining the method of measuring the Earth's radius using a mountain top and trigonometry. The approach, attributed to Matian Aloni, yields a result within 1% of the actual value.

### Key Points

- **Historical Measurement** [00:01] — Matian Aloni measured the Earth's radius with an accuracy of 1% by climbing a mountain and measuring the angle at which the horizon dips.
- **The Principle of Horizon Dip** [00:16] — From a mountain top, the horizon appears at an angle, not horizontal. This forms a right triangle with the Earth's radius and the mountain height.
- **Trigonometric Calculation** [00:29] — Using trigonometry (the cosine function), the Earth's radius can be calculated from the mountain height and the declination angle.
- **Determining Mountain Height** [00:56] — The method raises the question of how the mountain height is determined; the interviewer asks about it, and the answer involves walking around and taking more angles.

## Transcript

matian aloni who measured the the radius of the Earth to within 1% what he did was that he climbed a mountain with an ASE he measured The Horizon The Horizon dipped at an angle here's the Earth and here's a mountain not a scale okay so so
if you're standing on top of a mountain the The Horizon is actually not horizontal it's at an angle right yeah here's the raise of the earth and you have a right angle triangle the hypotenuse is uh the radius plus the
is Theta then by trigonometry this angle is also Theta and so uh there is an cosine [Music] Theta just from elementary trigonometry you can relate the radius of the Earth
angle and so you can compute the radius angle and so you can compute the radius I think it's something like H Co so the r of the Earth from the height of your mountain and the angle of
declination how did you know the height of the Mountain uh that's a good question uh I guess you could you could walk around a bit and take angles and enough trigonometry you can do it this comes from an interview with teren to
about all of the clever ways that Humanity has measured distances in the the most distant galaxies if you want the full story click the link at the the full story click the link at the bottom of this short