<P> A body of icy or rocky material in outer space may, if it can build and retain sufficient heat, develop a differentiated interior and alter its surface through volcanic or tectonic activity . The length of time through which a planetary body can maintain surface - altering activity depends on how well it retains heat, and this is governed by its surface area - to - volume ratio . For Vesta (r = 263 km), the ratio is so high that astronomers were surprised to find that it did differentiate and have brief volcanic activity . The moon, Mercury and Mars have radii in the low thousands of kilometers; all three retained heat well enough to be thoroughly differentiated although after a billion years or so they became too cool to show anything more than very localized and infrequent volcanic activity, of which none is evident at present . Venus and Earth (r> 6,000 km) have sufficiently low surface area - to - volume ratios (roughly half that of Mars and much lower than all other known rocky bodies) so that their heat loss is minimal . </P> <Table> <Tr> <Th> Shape </Th> <Th> </Th> <Th> Characteristic Length a (\ displaystyle a) </Th> <Th> Surface Area </Th> <Th> Volume </Th> <Th> SA / V ratio </Th> <Th> SA / V ratio for unit volume </Th> </Tr> <Tr> <Td> Tetrahedron </Td> <Td> </Td> <Td> side </Td> <Td> 3 a 2 (\ displaystyle (\ sqrt (3)) a ^ (2)) </Td> <Td> 2 a 3 12 (\ displaystyle (\ frac ((\ sqrt (2)) a ^ (3)) (12))) </Td> <Td> 6 6 a ≈ 14.697 a (\ displaystyle (\ frac (6 (\ sqrt (6))) (a)) \ approx (\ frac (14.697) (a))) </Td> <Td> 7.21 </Td> </Tr> <Tr> <Td> Cube </Td> <Td> </Td> <Td> side </Td> <Td> 6 a 2 (\ displaystyle 6a ^ (2)) </Td> <Td> a 3 (\ displaystyle a ^ (3)) </Td> <Td> 6 a (\ displaystyle (\ frac (6) (a))) </Td> <Td> 6 </Td> </Tr> <Tr> <Td> Octahedron </Td> <Td> </Td> <Td> side </Td> <Td> 2 3 a 2 (\ displaystyle 2 (\ sqrt (3)) a ^ (2)) </Td> <Td> 1 3 2 a 3 (\ displaystyle (\ frac (1) (3)) (\ sqrt (2)) a ^ (3)) </Td> <Td> 3 6 a ≈ 7.348 a (\ displaystyle (\ frac (3 (\ sqrt (6))) (a)) \ approx (\ frac (7.348) (a))) </Td> <Td> 5.72 </Td> </Tr> <Tr> <Td> Dodecahedron </Td> <Td> </Td> <Td> side </Td> <Td> 3 25 + 10 5 a 2 (\ displaystyle 3 (\ sqrt (25 + 10 (\ sqrt (5)))) a ^ (2)) </Td> <Td> 1 4 (15 + 7 5) a 3 (\ displaystyle (\ frac (1) (4)) (15 + 7 (\ sqrt (5))) a ^ (3)) </Td> <Td> 12 25 + 10 5 (15 + 7 5) a ≈ 2.694 a (\ displaystyle (\ frac (12 (\ sqrt (25 + 10 (\ sqrt (5))))) ((15 + 7 (\ sqrt (5))) a)) \ approx (\ frac (2.694) (a))) </Td> <Td> 5.31 </Td> </Tr> <Tr> <Td> Capsule </Td> <Td> </Td> <Td> radius (R) </Td> <Td> 4 π a 2 + 2 π a ∗ 2 a = 8 π a 2 (\ displaystyle 4 \ pi a ^ (2) + 2 \ pi a * 2a = 8 \ pi a ^ (2)) </Td> <Td> 4 π a 3 3 + π a 2 ∗ 2 a = 10 π a 3 3 (\ displaystyle (\ frac (4 \ pi a ^ (3)) (3)) + \ pi a ^ (2) * 2a = (\ frac (10 \ pi a ^ (3)) (3))) </Td> <Td> 12 5 a (\ displaystyle (\ frac (12) (5a))) </Td> <Td> 5.251 </Td> </Tr> <Tr> <Td> Icosahedron </Td> <Td> </Td> <Td> side </Td> <Td> 5 3 a 2 (\ displaystyle 5 (\ sqrt (3)) a ^ (2)) </Td> <Td> 5 12 (3 + 5) a 3 (\ displaystyle (\ frac (5) (12)) (3 + (\ sqrt (5))) a ^ (3)) </Td> <Td> 12 3 (3 + 5) a ≈ 3.970 a (\ displaystyle (\ frac (12 (\ sqrt (3))) ((3 + (\ sqrt (5))) a)) \ approx (\ frac (3.970) (a))) </Td> <Td> 5.148 </Td> </Tr> <Tr> <Td> Sphere </Td> <Td> </Td> <Td> radius </Td> <Td> 4 π a 2 (\ displaystyle 4 \ pi a ^ (2)) </Td> <Td> 4 π a 3 3 (\ displaystyle (\ frac (4 \ pi a ^ (3)) (3))) </Td> <Td> 3 a (\ displaystyle (\ frac (3) (a))) </Td> <Td> 4.83515 </Td> </Tr> </Table> <Tr> <Th> Shape </Th> <Th> </Th> <Th> Characteristic Length a (\ displaystyle a) </Th> <Th> Surface Area </Th> <Th> Volume </Th> <Th> SA / V ratio </Th> <Th> SA / V ratio for unit volume </Th> </Tr> <Tr> <Td> Tetrahedron </Td> <Td> </Td> <Td> side </Td> <Td> 3 a 2 (\ displaystyle (\ sqrt (3)) a ^ (2)) </Td> <Td> 2 a 3 12 (\ displaystyle (\ frac ((\ sqrt (2)) a ^ (3)) (12))) </Td> <Td> 6 6 a ≈ 14.697 a (\ displaystyle (\ frac (6 (\ sqrt (6))) (a)) \ approx (\ frac (14.697) (a))) </Td> <Td> 7.21 </Td> </Tr>

What is the effect of a large surface area to volume ratio