determine matplotlib axis size in pixels

This gives the width and height in inches.

bbox = ax.get_window_extent().transformed(fig.dpi_scale_trans.inverted())
width, height = bbox.width, bbox.height

That probably suffices for your purpose, but to get pixels, you can multiply by fig.dpi:

width *= fig.dpi
height *= fig.dpi

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For example,

import matplotlib.pyplot as plt

def get_ax_size(ax):
    bbox = ax.get_window_extent().transformed(fig.dpi_scale_trans.inverted())
    width, height = bbox.width, bbox.height
    width *= fig.dpi
    height *= fig.dpi
    return width, height

fig, ax = plt.subplots()
print(get_ax_size(ax))
#(496.0, 384.00000000000006)

ax2 = plt.axes([0.3, 0.3, 0.7, 0.7])
print(get_ax_size(ax2))
# (448.0, 336.0)

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To make an image of exactly that figure size, you have to remove whitespace between the figure and the axis:

import numpy as np
import matplotlib.pyplot as plt

def get_ax_size(ax):
    bbox = ax.get_window_extent().transformed(fig.dpi_scale_trans.inverted())
    width, height = bbox.width, bbox.height
    width *= fig.dpi
    height *= fig.dpi
    return width, height

data = np.arange(9).reshape((3, 3))
fig = plt.figure(figsize=(8,6), dpi=80)
ax = plt.Axes(fig, [0., 0., 1., 1.])
ax.set_axis_off()
fig.add_axes(ax)
ax.imshow(data, aspect="equal")
print(get_ax_size(ax))
# (640.0, 480.0)
plt.savefig('/tmp/test.png', dpi=80)

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% identify /tmp/test.png
/tmp/test.png PNG 640x480 640x480+0+0 8-bit DirectClass 50.5KB 0.020u 0:00.020