A spherical wooden ball of radius \(r\) is to be divided into eight identical parts by cutting by planes passing through the same diameter. What is the surface area of each final piece?
Understanding the Spherical Ball Division
The problem asks us to find the surface area of one piece when a spherical wooden ball of radius \(r\) is divided into eight identical parts. The division is done using planes that pass through the same diameter.
Imagine a sphere. To divide it into eight identical parts using planes passing through a single diameter, we can visualize four planes intersecting along that chosen diameter. These planes must be equally spaced angularly around the diameter. This method creates eight identical wedge-shaped pieces, much like slicing an orange into eight equal wedges where the cuts all meet at the center along the stem-to-base line (the diameter).
Calculating Surface Area Components
The surface area of each resulting piece consists of two types of surfaces:
- The curved outer surface, which is a fraction of the original sphere's surface.
- The flat inner surfaces created by the cuts.
Calculating the Curved Surface Area
The original sphere has a total surface area given by the formula:
\(A_{sphere} = 4\pi r^2\)Since the sphere is divided into eight identical parts, the curved surface area of one piece is simply one-eighth of the total surface area of the sphere:
\(A_{curved} = \frac{1}{8} \times A_{sphere}\) \(A_{curved} = \frac{1}{8} \times (4\pi r^2)\) \(A_{curved} = \frac{4\pi r^2}{8}\) \(A_{curved} = \frac{\pi r^2}{2}\)Calculating the Flat Surface Area
Each of the eight identical pieces is bounded by two flat surfaces. These flat surfaces are formed where the cutting planes intersect the sphere. Because the planes pass through a diameter, their intersection with the sphere forms a great circle. With four planes passing through the same diameter, each flat surface created is a semi-circle with radius \(r\).
The area of a full circle with radius \(r\) is \(\pi r^2\). Therefore, the area of a semi-circle is:
\(A_{semi-circle} = \frac{1}{2} \pi r^2\)Since each piece has two such flat semi-circular surfaces:
\(A_{flat} = 2 \times A_{semi-circle}\) \(A_{flat} = 2 \times \left(\frac{1}{2} \pi r^2\right)\) \(A_{flat} = \pi r^2\)Total Surface Area of One Piece
To find the total surface area of one final piece, we add the area of the curved part and the area of the two flat parts:
\(A_{total} = A_{curved} + A_{flat}\) \(A_{total} = \frac{\pi r^2}{2} + \pi r^2\)To sum these, we find a common denominator:
\(A_{total} = \frac{\pi r^2}{2} + \frac{2\pi r^2}{2}\) \(A_{total} = \frac{\pi r^2 + 2\pi r^2}{2}\) \(A_{total} = \frac{3\pi r^2}{2}\)Thus, the surface area of each final piece is \(\frac{3\pi r^2}{2}\).
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