Context. The structure and evolution of the disk of the Small Magellanic Cloud (SMC) are traced by studying the Cepheids. Aims. We aim to estimate the orientation measurements of the disk, such as the inclination, i, and the position angle of the line of nodes (PAlon), φ, and the depth of the disk. We also derive the age of the Cepheids and hence the age distribution of the SMC Cepheids. Methods. We used the V and I band photometric data of the fundamental and first-overtone Cepheids from the Optical Gravitational Lensing Experiment survey. The period-luminosity (PL) relations were used to estimate the relative distance and reddening of each Cepheid. The right ascension, declination, and relative distance from the centroid of each Cepheid were converted into x, y, and z Cartesian coordinates. A weighted least-square plane fitting method was then applied to estimate the structural parameters. The line-of-sight depth and then the orientation corrected depth or thickness of the disk were estimated from the relative distance measurements. The period-age-colour relation of Cepheids were used to derive the age of the Cepheids. Results. A break in the PL relations of both the fundamental mode and first-overtone Cepheids at P ~ 2.95 days and P ~ 1 day are observed. An inclination of 64.̊4±0.̊7 and a PAlon 155°.3 ± 6°.3 are obtained from the full sample. A reddening map of the SMC disk is also presented. The orientation-corrected depth or thickness of the SMC disk is found to be 1.76±0.6 kpc. The scale height is estimated to be 0.82±0.3 kpc. The age distribution of Cepheids matches the SMC cluster age distribution. Conclusions. The radial variation of the disk parameters mildly indicate structures and disturbances in the inner SMC (0.5