Let W(A) be the numerical range of an n × n complex matrix A. Using algebraic geometry technique and a result of Kippenhahn, Anderson showed that if W(A) is contained in a circular disk, and W(A) contains more than n boundary points of the circular disk, then W(A) equals to the circular disk, and the center of the circular disk will be an eigenvalue of the matrix A. Many researchers have reproved and refined the result of Anderson. Very recently, Wu unified these results and proved the following statements for a matrix A of the form . 1. If W(A) is contained in a circular disk 𝒟 centered at α, and W(A) contains at least m + 1 boundary points of 𝒟, then W(A) = 𝒟. 2. If W(A) contains a circular disk 𝒟 centered at α, and the boundary of W(A) contains at least m + 1 boundary points of 𝒟, then the boundary of W(A) contains a circular arc which is the boundary of 𝒟. 3. If the boundary of W(A) contains at least 2m + 1 boundary points of a circular disk 𝒟 centered at α, then 𝒟 ⊆ W(A). Moreover, under any one of the three conditions, α is an eigenvalue of A with algebraic multiplicity larger than its geometric multiplicity. The proofs of Wu utilized the Bézout's theorem, Riesz-Fejér theorem, etc. In this note, short and elementary proofs using only simple properties of polynomials and continuous functions are given to Wu's results. Furthermore, the results are extended to the higher numerical ranges of matrices.