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PRERNA FOR IAS
FRUSTUM OF A CONE
1. Definition of a Frustum of a Cone
A frustum of a cone is the portion of a cone remaining after its top is cut off by a plane parallel to the base. It has two circular bases of different sizes connected by a curved surface. Frustums are commonly seen in buckets, lampshades, and containers.
2. Two Circular Bases
A frustum has two circular bases that are parallel but not equal in size. The larger base has radius R, while the smaller top base has radius r. These bases define the shape and are used in calculating surface area and volume.
3. Curved Surface
The curved surface connects the two circular bases smoothly. Unlike flat surfaces, this surface is slanted and wraps around the frustum. The curved surface area is an important measurement in geometry and is used in practical applications such as manufacturing and design.
4. No Vertices
A frustum of a cone has no vertices because it does not have any sharp corners or points. The apex of the original cone is removed during the cutting process. As a result, the shape consists only of curved and circular surfaces.
5. No Edges
A frustum has no straight edges like polygons. Instead, it has circular boundaries where the curved surface meets the top and bottom bases. These circular boundaries are not considered edges in the same way as those of polyhedra.
6. Bases are Parallel
The two circular bases of a frustum are always parallel to each other. This occurs because the cone is cut by a plane parallel to its original base. Parallel bases ensure uniform geometry and help derive formulas for area and volume.
7. Axis of a Frustum
The axis is the line joining the centers of the two circular bases. It is perpendicular to both bases and passes through the center of the frustum. The axis helps define symmetry and is important in geometric calculations.
8. Height of a Frustum
The height of a frustum, represented by h, is the perpendicular distance between the two circular bases. It is measured along the axis. Height is a key dimension used in calculating both the volume and slant height of the frustum.
9. Slant Height
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Learn the definition, properties, and formulas for frustum of a cone. Understand slant height, surface area, volume calculations with detailed explanations.
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