Angles in Circles
Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made perpendicular to the radius through its endpoint placed on the circle as well as the line drawn at right angles to a tangent across the point of contact when the circle passes through the center of the circle. The chord is a line segment with its endpoints on the circle. A secant line or secant is the infinite extension of the chord.
Arcs in Circles
A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.
(a) Find a parametric representation for the torus obtained by rotating about the z-axis the circle in the xz-plane with center (b, 0, 0) and radius a < b. [Hint: Take as parameters the angles θ and α shown in the figure.]
(b) Use the parametric equations found in part (a) to graph the torus for several values of a and b.
(c) Use the parametric representation from part (a) to find the surface area of the torus.
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