How is size of black hole is related temperure?

To answer the question we have to introduce the following concepts:

Black Hole Radiation

Black Hole Entropy

Relation of Black Hole mass with Temperature

Black Hole Radiation:

Classically, black holes are black. Quantum mechanically, black holes radiate. Recent attempts to understand black holes on a quantum level have indicated that they radiate thermally (they have a finite temperature, though one incredibly low if the black hole is of reasonable size) that is proportional to the gradient of the gravity field.

This radiation known as Hawking radiation, after the British physicist Stephen Hawking who first proposed it. Hawking radiation has a blackbody (Planck) spectrum with a temperature T given by

kT = hbar g / (2 pi c) = hbar c / (4 pi rs)

where k is Boltzmann's constant, hbar = h / (2 pi) is Planck's constant divided by 2 pi, and g = G M / rs2 is the surface gravity at the horizon, the Schwarzschild radius rs, of the black hole of mass M. Numerically, the Hawking temperature is T = 4 × 10-20 g Kelvin if the gravitational acceleration g is measured in Earth gravities (gees).

Hawking Radiation is due to the capture of virtual particles decaying from the vacuum at the horizon. These are created in pairs and one of them is caught in the black hole and the other is radiated externally. This has been interpreted by Hawking as a tunneling effect and as a form of Unruh radiation.

Black Hole entropy and Hawking’s Area law:

Black hole entropy is the entropy carried by a black hole. If black holes carried no entropy, it would be possible to violate the second law of thermodynamics by throwing mass into the black hole. The only way to satisfy the second law is to admit that the black holes have entropy whose increase more than compensates for the decrease of the entropy carried by the object that was swallowed.

Starting from some theorems proved by Stephen Hawking, Jacob Bekenstein conjectured that the black hole entropy was proportional to the area of its event horizon. Later, Stephen Hawking showed that black holes emit thermal Hawking radiation corresponding to a certain temperature (Hawking temperature). Using some arguments rooted in thermodynamics, Hawking was also able to calculate the entropy that the black hole must carry. The result confirmed Bekenstein's conjecture:

S_BH = k A / {4 (l_p)^2}

l_p is Planck’s length, a very small length scale.

The second law of black hole mechanics states:

The horizon area is, (assuming the weak energy condition), a non-decreasing function of time.

That is

dA >= 0

i.e. the area of a black hole never decreases.

The second law is the statement of Hawking's area theorem. Analogously, the second law of thermodynamics states that the entropy of a closed system is a non-decreasing function of time, suggesting a link between entropy and the area of a black hole horizon. However, this version violates the second law of thermodynamics by matter losing (its) entropy as it falls in, giving a decrease in entropy. Generalized second law introduced as total entropy = black hole entropy + outside entropy

The laws of black hole mechanics suggest that one should identify the surface gravity of a black hole with temperature and the area of the event horizon with entropy, up to some multiplicative constants.

when quantum mechanical effects are taken into account, one finds that black holes emit thermal radiation (Hawking radiation) at temperature:

T_H = kappa/(2 Pi)

Where, kappa is the surface gravity.

The Bekenstein-Hawking entropy is given by :

S_BH = A/4

Relation with Themperature:

Any body with a temperature above absolute zero will radiate energy. And we have just seen that a black hole has a non-zero temperature. Thus thermodynamics says it will radiate energy and evaporate. We can calculate the rate of radiation for a given temperature from classical thermodynamics. We can also use the following formula to calculate the black hole's temperature:

T = hbar c^3 /[ 8 Pi k_B G M]

Hence the higher the mass of the black hole, the smaller the temperature. But the higher the mass, the more compact is the black hole. Hence, the higher is the mass, the smaller the black hole and the smaller the temperature.

"Please expalin correlation between size of black hole to temperture Probabaly smaller the black hole more the temperture Please also tell me is there any mistake in understanding"

You are right.

Cheers.

You're right - black holes are black body radiators, with a temperature inversely proportional to their mass. I can't explain it though!