Multiple choice

A cone is made from a circular sector of radius 2 units. If total surface area of cone (including base) is $\displaystyle\pi $ sq. units, find the area of the sector.

  1. $\displaystyle \left ( 3-2\sqrt{2} \right )\pi$
  2. $\displaystyle \left ( 2\sqrt{2}-2\right )\pi$
  3. $\displaystyle \sqrt{2}\pi$
  4. $\displaystyle \sqrt{3}\pi$
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B Correct answer
Explanation

Let r be the radius of the sector (2) and R be the radius of the cone base. Arc length of sector = 2*pi*R = theta*r. If the sector radius is 2, the slant height of the cone is 2. Total surface area = pi*R^2 + pi*R*slant = pi*R^2 + 2*pi*R = pi. So R^2 + 2R - 1 = 0. R = (-2 + sqrt(4 + 4))/2 = sqrt(2) - 1. Area of sector = (1/2)r^2*theta = (1/2)*r(r*theta) = (1/2)2(2*pi*R) = 2*pi*R = 2*pi*(sqrt(2) - 1).

AI explanation

Using the total surface area formula for a cone, pi * R * (R + l) = pi, and substituting the given slant height l = 2, we find R^2 + 2R = 1. Solving this quadratic equation gives R = (sqrt(2) - 1). The area of the sector used to form the cone equals the lateral surface area of the cone, which is pi * R * l. Multiplying 2 by pi and by (sqrt(2) - 1) gives the sector area as (2 * sqrt(2) - 2) * pi.

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