Calculate the cross-sectional area of the ball with a plane spaced from the center of the ball at a distance of 4 cm, if the diameter of the ball is 10 cm.
The cross-section of a sphere by a plane has the shape of a circle, therefore, to calculate its area, we apply the formula:
S = πr ^ 2.
To do this, you need to find the radius of this section.
To do this, consider a triangle formed by the radius of the ball, the radius of the section and the distance from the center of the ball to the plane. We denote it by ΔAOH, where 6
ОА is the radius of the sphere;
HA – section radius;
OH is the distance between the plane and the center.
Since the diameter of the ball is 10 cm, the radius is equal to half of it:
ОА = 10/2 = 5 cm.
For the Pythagorean theorem, we find the radius of the section:
OA ^ 2 = OH ^ 2 + AH ^ 2;
AH ^ 2 = OA ^ 2 – OH ^ 2;
AH ^ 2 = 5 ^ 2 – 4 ^ 2 = 25 – 16 = 9;
AH = √9 = 3 cm.
Now let’s find the cross-sectional area:
S = 3.14 * 32 = 3.14 * 9 = 28.26 cm2.
Answer: the cross-sectional area is 28.26 cm2.