Inscribed Solids Problems with Geometric Relationships
Inscribed solids are three-dimensional figures placed inside another solid so that specified vertices, edges or curved boundaries touch the outer surface. Common cases include a cube or cuboid inside a sphere and a cylinder or cone inside a sphere. Their dimensions are connected using the Pythagorean theorem and standard mensuration formulas.
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What Are Inscribed Solids?
The main method is to take a suitable cross-section and apply the Pythagorean theorem. For a cube of edge a inside a sphere of radius R, the cube's space diagonal is the sphere's diameter, so √3a = 2R. For a right circular cylinder of radius r and height h inside a sphere of radius R, r² + (h/2)² = R².
Inscribed Solids Video Lessons
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Inscribed Shapes and Volume Ratios
Learn how to analyse inscribed three-dimensional shapes and compare their volumes using geometric relationships and ratio-based methods.
Inscribed Solids Formula & Tricks
Important Formulas
The space diagonal of a cube with edge a equals the diameter of its circumscribed sphere.
For a cuboid with length l, breadth b and height h, its space diagonal equals the sphere's diameter.
A central cross-section forms a rectangle inside a circle. Here r is cylinder radius, h is height and R is sphere radius.
For a cone of height h and base radius r inscribed in a sphere of radius R, a cross-section through the axis gives this relation.
After finding an unknown dimension from the inscribed-solid relation, use the appropriate volume formula.
Quick Tricks
When a cube or cuboid is inscribed in a sphere, do not compare an edge with the sphere radius. Compare the space diagonal with the sphere diameter 2R.
In a central section of a cylinder inside a sphere, the horizontal leg is r, the vertical leg is h/2 and the hypotenuse is R.
Every length in an inscribed-solid equation must use the same unit. Convert diameters to radii or radii to diameters consistently before substitution.
Inscribed Solids Concepts
Cube Inscribed in a Sphere
A cube has space diagonal √(a² + a² + a²) = a√3. Therefore, if the edge is known, R = a√3/2. If the sphere radius is known, the cube edge is a = 2R/√3.
Cuboid Inscribed in a Sphere
The four space diagonals of a cuboid have the same length. If two dimensions and the sphere radius are known, the third dimension can be found by rearranging: h = √(4R² − l² − b²), provided the expression is non-negative.
Cylinder Inscribed in a Sphere
Take the cross-section through the cylinder axis. The cylinder's half-height h/2, radius r and sphere radius R form a right triangle. Hence r = √(R² − h²/4), and h = 2√(R² − r²).
Cone Inscribed in a Sphere
Place the cone's apex on the sphere and let its base plane be at distance R − h from the sphere's centre. The base radius satisfies r² = R² − (R − h)², which simplifies to r² = 2Rh − h².
Inscribed Solids Visual Guide & Data
Use the table and chart below to understand the data better.
Common Inscribed-Solid Relations
The defining dimension relation for standard solids inscribed in a sphere.
| Inscribed solid | Relation | Meaning |
|---|---|---|
| Cube | a√3 = 2R | Space diagonal equals sphere diameter |
| Cuboid | √(l² + b² + h²) = 2R | Space diagonal equals sphere diameter |
| Cylinder | r² + h²/4 = R² | Radius and half-height form a right triangle with R |
| Cone | r² = 2Rh − h² | Base radius follows from cone height and sphere radius |
Note: All lengths must be measured in the same unit.
Cube Inscribed in a Sphere
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Inscribed Solids Revision Points
Use the boundary relation first, then apply the required area or volume formula.
- For a cube in a sphere, a√3 = 2R.
- For a cuboid in a sphere, √(l² + b² + h²) = 2R.
- For a cylinder in a sphere, r² + h²/4 = R².
- For a cone in a sphere, r² = 2Rh − h².
- The sphere diameter, not radius, equals the space diagonal of an inscribed cube or cuboid.
- Use consistent units for all dimensions.
Inscribed Solids FAQs
What is the formula for a cube inscribed in a sphere?
If the cube edge is a and the sphere radius is R, then a√3 = 2R. Therefore, a = 2R/√3 and R = a√3/2.
A cube of edge 8 cm is inscribed in a sphere. What is the sphere radius?
The radius is a√3/2 = 8√3/2 = 4√3 cm.
What is the relation between the dimensions of a cuboid and its circumscribed sphere?
The cuboid's space diagonal equals the sphere diameter: √(l² + b² + h²) = 2R.
A cylinder of height 8 cm is inscribed in a sphere of radius 5 cm. Find its radius.
r² = 5² − (8/2)² = 25 − 16 = 9, so r = 3 cm.
What is the cone formula for a cone inscribed in a sphere?
For sphere radius R, cone height h and base radius r, r² = 2Rh − h².
Can the cylinder height be greater than the diameter of the sphere?
No. Since r² + h²/4 = R² and r² is non-negative, h/2 ≤ R. Therefore, h ≤ 2R.