Question

A sphere passes through the eight corners of a cube of side
10 cm. Find the volume of the sphere.​

Answer 1

Side of cube, a = 10

Main diagonal of cube, d = 10(root3)

Diameter of sphere is equal to the diagonal of cube

Hence, radius is

$\frac{10 \sqrt{3} }{2} = 5 \sqrt{3}$

Volume of sphere

$V_s = \frac{4}{3} \pi {r}^{3} \\ = \frac{4}{3} \times \frac{22}{7} \times {(5 \sqrt{3} )}^{3} \\ = \frac{88 \times 125 \times 3 \sqrt{3} }{21} \\ = 523.81 \times 3 \sqrt{3} = 2721.80$

Volume of sphere is 2721.80 cm^3

Answer 2

Answer:

The volume of the sphere = 2719.24 cm³

Step-by-step explanation:

Given,

A sphere passes through the eight corners of a cube

To find,

The volume of the sphere

Recall the formula

The body diagonal or the diagonal of the cube passes through the geometrical centre of the cube  is given by the formula

√3a, where 'a' is the side of the cube

The volume of the sphere = $\frac{4}{3}\pi r^3$

Solution:

Since side of the cube is 10cm, the length body diagonal of the cube

= √3a

= √3×10

= 10√3cm

Since the sphere passes through all the vertices of the cube, the centre of the sphere will be equal to the geometrical centre of the cube

and also, the diameter of the sphere = length of the body diagonal of the cube

Hence, the diameter of the sphere =10√3cm

Radius of the sphere = 5√3cm

Volume of the sphere = $\frac{4}{3}\pi r^3$

= $\frac{4}{3}$π( 5√3)³

= $\frac{4}{3}$π( 5√3)³

= $\frac{4}{3}$×π × 125×3√3

= 4 ×π × 125×√3

= 500×3.14 ×1.732

= 2719.24 cm³

∴ The volume of the sphere = 2719.24 cm³

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