Frustum and Combined Solids

Also called

frustum of a cone · frustum · bucket volume · combination of solids · combined solids · ice-cream cone · cone and hemisphere · capsule · cylinder with hemispherical ends · tent · cylinder surmounted by a cone · slant height of a frustum · volume of a frustum · curved surface area · surface area and volume · छिन्नक · शंकु का छिन्नक · संयुक्त ठोस · अर्धगोला

Topics

• A frustum is what remains of a cone when its top is sliced off by a plane parallel to the base.
• A bucket is a frustum: its volume is ⅓πh(r₁² + r₁r₂ + r₂²), the big cone minus the small cone sliced off it.
• The slant height of a frustum is l = √(h² + (r₁ − r₂)²), from the right triangle in its cut.
• Curved surface of a frustum: π(r₁ + r₂)l; a bucket’s total metal adds the base circle πr₁².
• The volume of a combined solid is the sum of the volumes of its parts.
• An ice-cream cone with a scoop: ⅓πr²h for the cone plus ⅔πr³ for the hemisphere of the same radius.
• A capsule is a cylinder with a hemisphere on each end: πr²h plus a whole sphere, 4⁄3πr³.
• A tent of a cylinder topped by a cone: canvas needed is the curved surface 2πrh plus πrl, with no floor.
• When solids are joined, the surfaces that touch are not counted in the total surface area.

About this model

A GLB file of about 337 KB, shown in Lably's 3D viewer and, on a phone, placed in your room with AR. Credit: Model, science and animation: Lably (built procedurally with three.js; no downloaded meshes). | CC BY 4.0. Original geometry and animation made for Lably.

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