Sphere:- Definition, Properties, Surface Area and Volume

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A sphere is a three-dimensional geometric surface where every point on it is the same distance from a fixed center.

Think of a ball: it looks like a sphere, though a real ball is only an approximation of a perfect mathematical sphere because it may have tiny imperfections.

What Is a Sphere?

A sphere is a three-dimensional shape where all points on its surface are the same distance from a fixed point called the center. This fixed distance is called the radius.

Mathematically, a sphere is the set of all points in three-dimensional space that are equidistant from a given point. This means if you pick any point on the surface of the sphere, its distance from the center is always the same.

Examples include a football, a globe, a bubble, a small water droplet, and a planet like Earth. These are approximations of perfect spheres, since real objects may have slight irregularities.

what is Sphere

Parts of a Sphere

Center: The center is the fixed point from which all points on the sphere’s surface are equidistant.

Radius: The radius is the distance from the center to any point on the surface of the sphere. It is usually denoted by r.

Diameter: The diameter is a straight line passing through the center that connects two opposite points on the surface. Its length is twice the radius, so d = 2r.

Surface and interior: The surface is the outer boundary of the sphere, while the interior is the space enclosed inside it. A sphere is technically just the surface, while the solid object including the interior is called a ball.

Great circle and hemisphere: A great circle is the largest circle that can be drawn on a sphere, passing through the center. A hemisphere is half of a sphere, divided by a plane passing through the center.

Sphere vs Circle vs Solid Ball

FeatureCircleSphereSolid ball
Dimensions2D3D surface3D solid
Defining propertyAll points equidistant from center in a planeAll points on surface equidistant from center in 3DSphere plus all points inside it
ExampleA drawn circle on paperThe surface of a footballThe entire football including inside

A circle is flat and two-dimensional, a sphere is a curved three-dimensional surface, and a ball is the solid object including the space inside.

Properties of a Sphere

A sphere has several important properties. It is perfectly symmetrical around its center. It has a continuous curved surface with no edges or vertices. Any cross-section through the center is a great circle. The radius from the center to any point on the surface is always the same.

Formulas of a Sphere

Diameter and radius

d=2rorr=d2d = 2r \quad \text{or} \quad r = \frac{d}{2}

Surface area

A=4πr2A = 4\pi r^2

Volume of the enclosed solid

V=43πr3V = \frac{4}{3}\pi r^3

Circumference of a great circle

C=2πrC = 2\pi r

Solved Examples on a Sphere

Finding the radius from the diameter

A sphere has a diameter of 10 cm. Find its radius.

r=d2=102=5 cmr = \frac{d}{2} = \frac{10}{2} = 5\text{ cm}

Calculating surface area

A sphere has a radius of 7 cm. Find its surface area.

A=4πr2=4π(72)=4π(49)=196π≈615.75 cm2A = 4\pi r^2 = 4\pi(7^2) = 4\pi(49) = 196\pi \approx 615.75\text{ cm}^2

Calculating volume

A sphere has a radius of 3 cm. Find its volume.

V=43πr3=43π(33)=43π(27)=36π≈113.10 cm3V = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi(3^3) = \frac{4}{3}\pi(27) = 36\pi \approx 113.10\text{ cm}^3

Finding the radius from the volume

A sphere has a volume of 36\pi cm³. Find its radius.

43πr3=36π\frac{4}{3}\pi r^3 = 36\pi
r3=36π×34π=27r^3 = \frac{36\pi \times 3}{4\pi} = 27
r=3 cmr = 3\text{ cm}

Comparing two spheres

Sphere A has radius 2 cm. Sphere B has radius 4 cm. Compare their volumes.

VA=43π(23)=43π(8)=32π3V_A = \frac{4}{3}\pi(2^3) = \frac{4}{3}\pi(8) = \frac{32\pi}{3}
VB=43π(43)=43π(64)=256π3V_B = \frac{4}{3}\pi(4^3) = \frac{4}{3}\pi(64) = \frac{256\pi}{3}
VBVA=256/332/3=8\frac{V_B}{V_A} = \frac{256/3}{32/3} = 8

Answer: Sphere B has 8 times the volume of Sphere A, because volume depends on the cube of the radius.

Equation of a Sphere (Class 12 Extension)

This section is a Class 12 extension and may be beyond some students’ prescribed syllabus.

Sphere centered at the origin

x2+y2+z2=r2x^2 + y^2 + z^2 = r^2

Sphere with a general center

(x−h)2+(y−k)2+(z−l)2=r2(x-h)^2 + (y-k)^2 + (z-l)^2 = r^2

Here, (h, k, l) is the center and r is the radius.

Worked coordinate-geometry example

Find the equation of a sphere with center at (2, 3, 4) and radius 5.

(x−2)2+(y−3)2+(z−4)2=25(x-2)^2 + (y-3)^2 + (z-4)^2 = 25

Real-Life Applications of Spheres

Spheres appear in sports equipment like footballs and basketballs, in ball bearings used in machines, in spherical tanks for storing gases or liquids, in astronomy where planets and stars are modeled as spheres, and in mathematical models of Earth and other celestial bodies.

Common Mistakes When Solving Sphere Problems

Confusing radius with diameter is a common error. Always check which one is given before using a formula. Mixing up area and volume formulas is another mistake. Surface area uses r2, while volume uses r3.

Using incorrect units, like giving volume in square units instead of cubic units, is also common. Finally, confusing a sphere with a hemisphere can lead to using the wrong formula, since a hemisphere is only half a sphere.

Frequently Asked Questions

Q: What is a sphere in simple words?

A sphere is a perfectly round three-dimensional shape where every point on the surface is the same distance from the center.

Q: What is the difference between a sphere and a circle?

A circle is flat and two-dimensional, while a sphere is a curved three-dimensional surface.

Q: What is the formula for the volume of a sphere?

The formula is

V=43πr3.V = \frac{4}{3}\pi r^3.

Q: What is the formula for the surface area of a sphere?

The formula is

A=4πr2.A = 4\pi r^2.

Q: How do you find the radius if you know the diameter?

Divide the diameter by 2, so

r=d2.r = \frac{d}{2}.

Q: What is a hemisphere?

A hemisphere is half of a sphere, divided by a plane passing through the center.

Quick Revision and Practice Questions

A sphere has a radius of 6 cm. What is its diameter?

Answer: 12 cm.

Find the surface area of a sphere with radius 5 cm.

Answer:

100π≈314.16cm2.100\pi \approx 314.16 cm².

Find the volume of a sphere with radius 2 cm.

Answer:

32π3≈33.51cm3\frac{32\pi}{3} \approx 33.51 cm³

A sphere has a volume of 288\pi cm³. Find its radius.

Answer: 6 cm.

What is the equation of a sphere centered at the origin with radius 4?

Answer:

x2+y2+z2=16.x^2 + y^2 + z^2 = 16.

References

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