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Inscribed and Circumscribed Solids: When One Solid Sits Inside Another

A solid is inscribed in another when it sits inside and touches it at special points; the outer one is circumscribed. Sphere in a cube: r = a/2. Sphere around a cube: R = a√3/2. Sphere in a cylinder: h = 2r. Cylinder in a sphere: R² = r² + h²/4. Sphere in a cone: ρ = rh/(r + l). Draw a cross-section through the axis and each problem becomes a flat triangle, square or circle problem.

🎬 Step-by-step story

  1. A sphere sits inside a cube and touches all six faces. The sphere's diameter equals the cube's edge, so r = a/2.
  2. Now the sphere is around the cube and touches all eight corners. The diameter is the cube's long diagonal, so R = a√3 / 2.
  3. A sphere inside a cylinder touches both lids and the side. So the cylinder height equals the diameter: h = 2r. The sphere is 2/3 of the cylinder.
  4. A cylinder inside a sphere has both rims on the sphere. Slide the height: a right triangle gives R² = r² + (h/2)².
  5. A sphere inside a cone touches the base and the slanted side. Cut the cone along its axis: you see a circle inside a triangle. Then ρ = rh/(r + l).
  6. Your turn. Pick a pair, slide the size, and watch how the two radii depend on each other.

Tip: drag the 3D scene to turn it. Use two fingers to zoom.

🤔 Common doubts, cleared

Why is the diameter equal to the cube's edge for the sphere inside?

The sphere touches two opposite faces. The shortest distance between these faces is the edge a, and it passes through the centre, so it is a diameter. See the red line.

Why is the diagonal a diameter for the sphere around a cube?

The sphere's centre is the cube's centre, and a corner on the sphere is one radius away. The line to the opposite corner is two radii: a diameter. That line is the space diagonal.

Why must the cylinder height equal the diameter?

The sphere touches both lids, so the lids are one diameter apart. The side is touched too, so the cylinder radius is r.

Where does the right triangle come from in a cylinder inside a sphere?

Join the sphere's centre to a point on the rim. One leg goes along half the height, the other along the cylinder's radius. The hypotenuse is R. Slide the height to see r change.

Why do we use the triangle's incircle for a sphere in a cone?

Cut through the axis and the sphere becomes a circle touching all three sides of the triangle: that is the incircle. Its radius is area ÷ semi-perimeter.

Do the same rules work for every size?

Yes. Change the size in the free-play step: each radius rule stays the same because the shapes only scale up or down.

What do inscribed and circumscribed mean?

When a solid sits inside another and touches it at special points, it is inscribed in it. The outer one is circumscribed about the inner. For example, a sphere inside a cube is inscribed in the cube, and the cube is circumscribed about the sphere. Touching can happen on faces (tangent), along edges or at corners. A tangent plane touches a sphere at exactly one point.

The best trick: cut through the axis and look at the flat shape, a cross-section. A sphere in a cube becomes a circle in a square. A cone with a sphere becomes a circle in a triangle.

Sphere and cube

Sphere inside a cube (touches 6 faces). The sphere's diameter is the cube's edge: 2r = a, so r = a/2. Volume ratio sphere : cube = π/6 ≈ 0.52.

Sphere around a cube (touches 8 corners). The longest line in the cube, the space diagonal, goes through the centre and is a diameter. The space diagonal is a√3 (Pythagoras twice). So 2R = a√3 and R = a√3/2. Volume ratio cube : sphere = 2/(π√3) ≈ 0.37.

Between these two there is a third sphere that touches the 12 edges: its radius is a/√2. And R : r = √3 for the first two spheres around the same cube.

Sphere and cylinder

Sphere inside a cylinder touching both lids and the curved side: the cylinder has the same radius r and its height equals the diameter, h = 2r. Then sphere volume = (4/3)πr³ and cylinder volume = 2πr³: ratio 2 : 3. Surprise: the sphere's surface area 4πr² to the cylinder's total surface area 6πr² is also 2 : 3. This was Archimedes' favourite result.

Cylinder inside a sphere (both rims on the sphere): cut through the axis. The centre of the sphere is the middle of the cylinder's axis. The radius to a rim point is the hypotenuse of a right triangle with legs r (cylinder radius) and h/2:

R² = r² + (h/2)².

Sphere and cone

Sphere inside a cone (touches the base and the slanted side). Cut along the axis: the cone becomes an isosceles triangle with base 2r, height h and equal sides l = √(r² + h²). The sphere becomes the incircle of that triangle. The incircle radius = area ÷ semi-perimeter:

ρ = (½ · 2r · h) / ((2l + 2r)/2) = rh/(r + l).

Cone inside a sphere (apex and base rim on the sphere, base radius r, height h): R = (r² + h²)/(2h). In the cross-section, the apex and the two ends of the base lie on a circle. A regular tetrahedron of edge a has an inscribed sphere of radius a√6/12 and a circumscribed sphere of radius a√6/4, so R = 3r.

Solving a problem: a 4-step plan

  1. Name which solid is inside (inscribed) and which is outside.
  2. Decide what touches what: faces, rim, corners, slanted side.
  3. Cut along the axis and draw the flat cross-section.
  4. Use a right triangle (Pythagoras), diagonal, or incircle rule. Then find the volume or area asked.

Always label which radius is which (r for the cylinder/cone, R for the outer sphere, ρ for an inner sphere) before you start.

Try it: ball in a box

Take a ball (or a round fruit) and a box that holds it snugly. Measure the ball's width with a ruler and compare with the box's edge. They match: that is r = a/2. Next, try to imagine the ball with the box's corners touching it from outside: guess what the ball's diameter would be compared to the box's diagonal, then check with the 3D.

Key formulas and definitions

Worked examples

1. A sphere is inscribed in a cube of edge 10 cm. Find the sphere's volume.

r = a/2 = 5 cm. V = (4/3)π × 125 = 500π/3 ≈ 523.6 cm³.

2. A cube of edge 6 cm is inscribed in a sphere. Find the sphere's radius and surface area.

R = a√3/2 = 3√3 ≈ 5.20 cm. Surface area = 4πR² = 4π × 27 = 108π ≈ 339.3 cm².

3. A sphere of radius 4 cm fits exactly inside a cylinder. Find the cylinder's volume and the fraction of it taken by the sphere.

h = 2r = 8 cm. Cylinder: π × 16 × 8 = 128π cm³. Sphere: (4/3)π × 64 = 256π/3 cm³. Fraction = (256/3) ÷ 128 = 2/3.

4. A cylinder of radius 3 cm and height 8 cm is inscribed in a sphere. Find the sphere's radius.

R² = r² + (h/2)² = 9 + 16 = 25, so R = 5 cm.

5. A cone has base radius 3 cm and height 4 cm. Find the radius of the largest sphere that fits inside it.

l = √(9 + 16) = 5 cm. ρ = rh/(r + l) = (3 × 4)/(3 + 5) = 12/8 = 1.5 cm.

6. A cone with base radius 4 cm and height 8 cm is inscribed in a sphere. Find the sphere's radius.

R = (r² + h²)/(2h) = (16 + 64)/16 = 5 cm.

7. A sphere is inscribed in a cube, and the cube is inscribed in a larger sphere. Find the ratio of the surface areas of the larger sphere to the smaller sphere.

Small sphere r = a/2. Large sphere R = a√3/2. R/r = √3. Surface area ratio = (R/r)² = 3. So the ratio is 3 : 1.

Common mistakes

Practice quiz

1. A sphere is inscribed in a cube of edge a. Its radius is:
2. A cube is inscribed in a sphere. The sphere's diameter equals the cube's:
3. A sphere is inscribed in a cylinder. The ratio of the volumes (sphere : cylinder) is:
4. A cylinder of r = 6 and h = 16 is inscribed in a sphere. The sphere's radius is:
5. The cross-section of a sphere inscribed in a cone, cut along the axis, is:

Practice: answer these yourself

Type or choose your answer, then press Check. Use a hint if you are stuck; the full solution appears after you answer.

Frequently asked questions

What is the difference between inscribed and circumscribed?

The inner solid is inscribed in the outer one; the outer one is circumscribed about the inner one. They are two ways to say the same picture.

Why do we cut through the axis?

It turns a hard 3D picture into a flat one: circles, triangles and rectangles. Then simple rules (Pythagoras, incircle) give the radius.

Does the sphere in a cone always touch the base?

For the largest sphere inside a cone, yes: it touches the base and the slanted side all around. A smaller sphere floating inside is not an inscribed sphere.

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