How do you find the surface area of a combination of solids?

For combined solids, add up only the exposed (visible) surface areas. Do NOT include the faces where two solids join. For example, a hemisphere on a cylinder: the flat circle at the top of the cylinder where the hemisphere sits is not exposed, so exclude it. Include: CSA of hemisphere (2πr²) + CSA of cylinder (2πrh) + base of cylinder (πr²). The key is to visualise which surfaces are visible from outside.

What is a frustum of a cone?

A frustum is formed when a cone is cut by a plane parallel to its base, removing the top portion. It has two circular faces of different radii (r₁ at base and r₂ at top) and a curved lateral surface. Frustum formulas: Slant height l = √[h² + (r₁−r₂)²]; CSA = π(r₁+r₂)l; Volume = (πh/3)(r₁² + r₁r₂ + r₂²). Common examples: bucket, tumbler/glass, funnel.

What happens to volume when a solid is melted and recast?

When a solid is melted and recast into another shape, the volume remains constant — no material is gained or lost (ignoring minor losses). Only the shape changes. So: Volume of original solid = Volume of new solid. This allows you to find unknown dimensions. Example: melting 3 spheres of radius r₁ and recasting as one large sphere — (3 × (4/3)πr₁³) = (4/3)πR³, giving R = ³√3 × r₁.

Combination of Solids

Surface Areas and Volumes • Class 10 Mathematics • NCERT • CBSE

Class 10 extends Class 9 solid geometry to combinations. Volume of solid = sum of volumes of individual parts. Surface area of combined solid = sum of exposed surface areas (not counting joined faces). Frustum of cone: V = (πh/3)(r₁² + r₁r₂ + r₂²), CSA = π(r₁+r₂)l.

Key Formulas

Frustum CSA = π(r₁+r₂)l
Frustum TSA = π(r₁+r₂)l + π(r₁²+r₂²)
Frustum V = (πh/3)(r₁² + r₁r₂ + r₂²)
Frustum l = √[h² + (r₁−r₂)²]

Frequently Asked Questions

How do you find the surface area of a combination of solids?: For combined solids, add up only the exposed (visible) surface areas. Do NOT include the faces where two solids join. For example, a hemisphere on a cylinder: the flat circle at the top of the cylinder where the hemisphere sits is not exposed, so exclude it. Include: CSA of hemisphere (2πr²) + CSA of cylinder (2πrh) + base of cylinder (πr²). The key is to visualise which surfaces are visible from outside.
What is a frustum of a cone?: A frustum is formed when a cone is cut by a plane parallel to its base, removing the top portion. It has two circular faces of different radii (r₁ at base and r₂ at top) and a curved lateral surface. Frustum formulas: Slant height l = √[h² + (r₁−r₂)²]; CSA = π(r₁+r₂)l; Volume = (πh/3)(r₁² + r₁r₂ + r₂²). Common examples: bucket, tumbler/glass, funnel.
What happens to volume when a solid is melted and recast?: When a solid is melted and recast into another shape, the volume remains constant — no material is gained or lost (ignoring minor losses). Only the shape changes. So: Volume of original solid = Volume of new solid. This allows you to find unknown dimensions. Example: melting 3 spheres of radius r₁ and recasting as one large sphere — (3 × (4/3)πr₁³) = (4/3)πR³, giving R = ³√3 × r₁.

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