Planar functions, introduced by Dembowski and Ostrom, are functions from a finite field to itself that give rise to finite projective planes. They exist, however, only for finite fields of odd characteristics. They have attracted much attention in the last decade thanks to their interest in theory and those deep and various applications in many fields. This paper focuses on planar trinomials over cubic and quartic extensions of finite fields. Our achievements are obtained using connections with quadratic forms and classical algebraic tools over finite fields. Furthermore, given the generality of our approach, the methodology presented could be employed to drive more planar functions on some finite extension fields.