Ice-sheet surfaces have scales of fluctuation that arc similar to the diameter of the area illuminated by a satellite radar altimeter. The present theory of altimeter, developed to describe scattering from the ocean surface, does not deal properly with the geometry of ice-sheet surfaces. In this paper, the theory of altimeter is extended to cover this geometry. A general relation for the altimeter echo from a surface of unknown geometry is developed, including the effects of the penetration of the surface by the radar waves. This expression is linearized, using the characteristic operating geometry of satellite altimeters and the gentle nature of ice-sheet gradients. From this expression, an integral equation is derived, from whose solution the spatial average of the height of the surface relative to a spherical datum can be determined. The integral equation is of a Volterra type, which permits the uniqueness of the solution for the average height to be investigated simply. The method is extended to provide a solution for the spatial average of the height of a local region of the ice sheet, provided the region remains large in comparison with the area illuminated by the altimeter, and to deal with variations in the antenna bore-sight alignment. The results have a number of implications for the collection and reduction of echoes in an experiment to determine the average height of an ice sheet. The unique determination of the average height requires the echo to be known over a time interval that depends on the extrema of the surface, which therefore must be known a priori. The average height itself can be determined by the operation on the echo of a linear operator whose kernel is derived from the solution of the Volterra-type equation. This marks a change from the procedures currently used in practice to reduce echoes from ice sheets.