I’ll admit it. I hate math. But I also love learning about Astronomy. So does the interest beat the frustration? Apparently, yes. This is a quick tutorial on how to find the approximate diameter of the planets of the Solar System, except Earth of course. Why use this method? There are two reasons. First, doing this makes you feel smart. Secondly, why not practice some math?
I first learned of this method via Khan Academy, which is where I do my mathematics, and I applied it with the given information from NASA to Mars. So I figured then why not try this (simplified) method for all of the other planets?
The first one I tried was Venus. To find the necessary information, I used Wikipedia’s article on Venus for the angular diameter in arc seconds and Universe Today’s article on the distance of Venus from Earth for distances. You’ll also need your computer’s calculator or an online one capable of storing loads of digits. Desk calculators won’t work here.
You should set up your equation like this (preferably with a vertical fraction bar).
d = δ/206,265 ∙ D
d = diameter of Venus
δ = angular diameter in arc seconds
D = distance from the Earth, in km
206265 is the constant used to convert radians to arc seconds
First we will use the figures from Venus’ closest approach to Earth, although you could do the furthest approach first if you wanted.
d = 66/206,265 ∙ 38,000,000
d = 66/206,265 ∙ 38,000,000 = 12,159.1157006731
12,159.1157006731 ≈ 12,159.1157
That diameter figure will be used later to calculate an average. Now onto the farthest point from Earth.
d = 9.7/206,265 ∙ 261,000,000
d = 9.7/206,265 ∙ 261,000,000 = 12,274.01643516835
12,274.01643516835 ≈ 12,274.0164
You can also round the figures beforehand and spare step three, just remember to use that ≈ sign beforehand instead of the regular = sign.
Now all you need to do is average the two diameters out.
(12,159.1157+ 12,274.0164 )/2
(12,159.1157+ 12,274.0164 )/2 = 12,216.56605 km
Averaged Diameter of Venus: 12,216.56605 km
Actual Diameter of Venus: Approx. 12,100 km
You can also do this for the other planets and Pluto. Here are my diameters for each.
Angular Diameter in Arc seconds: 13 (nearest), 4.5 (farthest).
Distance from the Earth: 77 million km (nearest), 222 million km (farthest)
Averaged Diameter: 4,848.13225 km
Actual Diameter: 4,875 km
Note that Mars is a bit tricky since the closest distance from Mars to Earth varies by millions of kilometers each orbit. This set of calculations uses Mars’ closest point to Earth in 2003, which was at 55.7 million km (NASA Mars Fact Sheet).
Angular Diameter in Arc seconds: 25.1 (nearest), 3.5 (farthest)
Distance from the Earth: 55.7 million km (nearest), 401 million km (farthest)
Averaged Diameter: 6,791.19095 km
Actual Diameter: 6,800 km
Angular Diameter in Arc seconds: 50.1 (closest), 29.8 (farthest)
Distance from the Earth: 628,743,036 km (closest), 928,081,020 km (farthest)
Averaged Diameter: 143,400.09335 km
Actual Diameter: 143,000 km
Angular Diameter in Arc seconds (excluding rings): 20.1 (closest), 14.5 (farthest)
Distance from the Earth: 1,200,000,000 km (closest), 1,670,000,000 km (farthest)
Averaged Diameter: 117,167.2363 km
Actual Diameter: 120,540 km
Angular Diameter in Arc Seconds: 4.1 (nearest), 3.3 (farthest)
Distance from the Earth: 2,570,000,000 km (nearest), 3,150,000,000 (farthest)
Averaged Diameter: 50,740.5522 km
Actual Diameter: 51,120 km
Angular Diameter in Arc Seconds: 2.4 (nearest), 2.2 (farthest)
Distance from the Earth: 4,301,000,000 km (nearest), 4,553,000,000 km (farthest)
Averaged Diameter: 49,303.081 km
Actual Diameter:49,500 km
Angular Diameter in Arc Seconds: 0.11 (nearest), 0.06 (farthest)
Distance from the Earth: 4,200,000,000 km (nearest), 7,500,000,000 km (farthest)
Averaged Diameter: 2,210.7483 km
Actual Diameter: 2,300 km (before New Horizons)
Actual Diameter Figures are from the DK Visual Space Encyclopedia (one of my brother’s space books).