Multiple choice

A chord, whose length is equal to the radius of a circle, is drawn to divide the circle into two parts. If the radius of the circle is 42 cm, then what is the area of the smaller part (in cm2)?

  1. 21 2 ( π 3 − 3 2 )

  2. 42 2 ( π 2 − 3 4 )

  3. 42 2 ( π 6 − 3 4 )

  4. 21 2 ( π 6 − 3 4 )
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

A chord equal to the radius forms an equilateral triangle with the center. The area of the sector is (60/360) * pi * r^2 = (1/6) * pi * r^2. The area of the triangle is (sqrt(3)/4) * r^2. The area of the smaller segment is (pi/6 - sqrt(3)/4) * r^2. With r = 42, area = 42^2 * (pi/6 - sqrt(3)/4).

AI explanation

A chord equal to the radius of 42 cm forms an equilateral triangle with two radii, creating a sector with a central angle of 60 degrees, which is pi/3 radians. The area of this sector is (1/2) * r^2 * theta, which calculates to (1/2) * 42^2 * (pi/3) = 294 pi. Subtracting the area of the equilateral triangle, which is (sqrt(3)/4) * 42^2 = 441 * sqrt(3), gives the segment area as 462 pi - 441 * sqrt(3). This factors to 42^2 * (pi/6 - sqrt(3)/4).

Keep practicing — related questions