1. The Distance to the Horizon
Y Pellter i’r Gorwel
The Distance to the Horizon
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3. Er mwyn cyfrifo’r pellter P (mewn km) i’r gorwel,
Ffeindiwch (neu mesurwch) yr uchder U rhwng eich llygad a lefel y môr, mewn metrau.
Lluoswch efo 13.
Cyfrifwch yr ail isradd.

4. In order to calculate the distance D (in km) to the horizon,
Find (or measure) the height U between your eye level and the level of the sea, in metres.
Multiply by 13.
Calculate the square root.

5. P U
Mae radiws y Ddaear tua 6,500 km

6. D U
The radius of the earth is ≈ 6,500 km

7. P
Mae’r radiws a’r tangiad yn cyfarfod ar ongl sgwâr, felly gallwn ddefnyddio Theorem Pythagoras ar y triongl ongl sgwâr.
𝑅+𝑈 2 = 𝑅 2 + 𝑃 2
𝑅 2 +2𝑅𝑈+ 𝑈 2 = 𝑅 2 + 𝑃 2
2𝑅𝑈+ 𝑈 2 = 𝑃 2
Mewn gwirionedd, byddai’r uchder uwchben lefel y môr llawer llai na radiws y Ddaear, felly 2𝑅𝑈≈ 𝑃 2
𝑃≈ 2𝑅𝑈
O amcangyfrif radiws y Ddaear fel 6,500 km, mae 𝑃≈ 13,000𝑈
ond mae ein mesuriad ar gyfer 𝑈 mewn metrau.
Felly rhaid ei rannu efo 1,000 i’w newid i fod mewn km, ac felly
𝑃≈ 13𝑈 U R

8. D
The radius and tangent meet at a right angle, so we can use Pythagoras’ Theorem on the right–angled triangle.
𝑅+𝑈 2 = 𝑅 2 + 𝐷 2
𝑅 2 +2𝑅𝑈+ 𝑈 2 = 𝑅 2 + 𝐷 2
2𝑅𝑈+ 𝑈 2 = 𝐷 2
In reality, the height above sea level will be much less than the radius of the Earth, so that 2𝑅𝑈≈ 𝐷 2
𝐷≈ 2𝑅𝑈
Estimating the radius of the Earth to be 6,500 km, we obtain 𝐷≈ 13,000𝑈
but our measurement for 𝑈 is in metres.
Therefore, we need to divide by 1,000 to change into km, so that
𝑃≈ 13𝑈 U R