Chebyshev-type quadrature for the weight functions \[ {w_a}(t) = \frac {{1 - at}}{{\pi \sqrt {1 - {t^2}} }},\quad - 1 < t < 1,\quad - 1 < a < 1,\] is related to a problem concerning partial sums of the exponential series, namely the problem to extend the nth partial sum to a polynomial of degree 2N having all zeros on the circle $|z| = |a|N$. Using this connection, we show that the minimal number N of nodes needed for Chebyshev-type quadrature of degree n for ${w_a}(t)$ satisfies an inequality ${C_1}n \leq N \leq {C_2}n$ with positive constants ${C_1},{C_2}$. As an application we prove that the minimal number N of nodes for Chebyshev-type quadrature of degree n on a torus embedded in ${{\mathbf {R}}^3}$ satisfies an inequality ${C_1}{n^2} \leq N \leq {C_2}{n^2}$.