This paper presents two constructions of hyperkähler manifolds: one based on a Legendre transform and the other on a symplectic quotient. These methods, originally developed in the context of supersymmetric nonlinear σ-models, are described entirely geometrically. The paper aims to clarify the relationship between supersymmetry and modern differential geometry, making these concepts accessible to a new audience. It reviews fundamental ideas in Kähler and hyperkähler geometry and establishes notation. The first construction relates the Kähler potentials of certain hyperkähler manifolds to a linear space via a Legendre transform. The second construction is based on a symplectic quotient of a hyperkähler manifold. Examples are provided, and the applicability of both constructions is discussed. Section 3 provides background on quotients, symplectic and Kähler quotients, and twistor theory, explaining the geometric meaning of the constructions. Section 4 describes nonlinear σ-models and related material needed for subsequent sections. Section 5 covers essential aspects of supersymmetry, and Section 6 uses supersymmetry to derive the constructions. The paper also discusses the use of various index types. On a 2n-dimensional Kähler manifold, holomorphic coordinates are chosen, and the complex structure is defined. The paper provides a detailed description of these constructions and their geometric interpretations.This paper presents two constructions of hyperkähler manifolds: one based on a Legendre transform and the other on a symplectic quotient. These methods, originally developed in the context of supersymmetric nonlinear σ-models, are described entirely geometrically. The paper aims to clarify the relationship between supersymmetry and modern differential geometry, making these concepts accessible to a new audience. It reviews fundamental ideas in Kähler and hyperkähler geometry and establishes notation. The first construction relates the Kähler potentials of certain hyperkähler manifolds to a linear space via a Legendre transform. The second construction is based on a symplectic quotient of a hyperkähler manifold. Examples are provided, and the applicability of both constructions is discussed. Section 3 provides background on quotients, symplectic and Kähler quotients, and twistor theory, explaining the geometric meaning of the constructions. Section 4 describes nonlinear σ-models and related material needed for subsequent sections. Section 5 covers essential aspects of supersymmetry, and Section 6 uses supersymmetry to derive the constructions. The paper also discusses the use of various index types. On a 2n-dimensional Kähler manifold, holomorphic coordinates are chosen, and the complex structure is defined. The paper provides a detailed description of these constructions and their geometric interpretations.