An exchangeable tool segment (500) for use in a
grinding disc unit of a lateral
surface grinding machine has a
cutting means support (510) which has mounting structures for fastening the
cutting means support in a mounting region provided at the circumferential region of the main body of the
grinding disc unit. The main body has, on its circumference, many mounting regions each for mounting one tool segment. On a
cutting layer side (515), the cutting means support has, in the operationally ready state, a cutting layer (520) which forms an
abrasive effective surface of the tool segment. The
abrasive effective surfaces of the cutting
layers are, in an operationally ready state of the cutting disc unit, arranged in a side surface of the cutting disc unit within an annular effective region with a radial distance to the disc
rotational axis, said side surface being oriented perpendicularly to the disc
rotational axis. The cutting means support (510) defines a longitudinal direction (L), a width direction (B) perpendicular thereto and a height direction (H) perpendicular to the longitudinal direction and the width direction, and can be mounted on the main support such that, in the mounted state, the longitudinal direction (L) is oriented tangentially to the disc
rotational axis, the width direction is oriented substantially in the radial direction and the height direction is oriented parallel to the disc rotational axis. The cutting means support furthermore has a bottom side (512), an outer side (513), an inner side (514) opposite thereto in the width direction (B) and a cutting layer side (515) opposite the bottom side in the height direction and end faces oriented transverse to the longitudinal direction. In the region of the end faces, a bevel is in each case formed such that the cutting means support has, in the section along a
sectional plane oriented perpendicular to the height direction between the bevels, a trapezoid shape such that the cutting means support is longer on the inner side (514) than on the outer side.