Rotate a sector once and get a solid where a cone and a sphere meet
If a sector with radius r and central angle θ is rotated once about one of its radii, the other radius traces the lateral surface of a cone, while the arc traces part of a sphere. The resulting solid is called a spherical sector.
In sector OAB, take radius OA as the axis of rotation and rotate it through 360°. The path traced by radius OB becomes the lateral surface of a cone, while the path traced by arc AB becomes part of a spherical surface.
The lateral surface of a cone can be flattened, so we can draw its net and use it to find the surface area. A spherical surface, however, cannot be flattened no matter how it is cut, so we calculate its area directly with a formula instead of a net.
For the volume, split this solid into a cone and a part of a sphere (spherical segment), find each volume, and combine them. However, when the central angle is greater than 90°, the cone portion extends inward, so its volume must be subtracted.
Use the sliders to change radius r and central angle θ. See how the solid on the left and the net of the cone’s lateral surface on the right change. The spherical part cannot be flattened, so its area is calculated numerically.
Cone lateral surfacePart of spherical surface
Radius r6cm
Central angle θ60°
Use the same sliders to split the solid into a cone (coral) and a spherical segment (blue), then add their volumes to check that they match the total volume.