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A sphere is a three-dimensional geometric object of a two-dimensional shape, i.e., circle. On the one hand, a circle is a closed figure having a fixed center drawn-out keeping a constant length, but on the other hand, a sphere shares its resemblance with a round ball. Although both, sphere and circle are round in shape measured using their respective radius.
Further, any vertices or edges cannot be found in a sphere as similar is the case with other 3D shapes. The focuses on the sphere's outside layer are equidistant from the center. As a result, the distance between the sphere's center and outer layers is always equal.
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Key Takeaways: Radius, Diameter, Circumference, Equation of Sphere, Surface Area of Sphere, Volume of Sphere.
Parts of Sphere
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The various parts of a sphere are:
- The sphere is characterized by three respective axis known as x, y, and z.
- The distance between the surface of the sphere and its center is known as radius.
- The line that runs across the center of the sphere from one end to the other is known as diameter.
- The distance traveled around the sphere is known as the circumference.
- The area occupied by the outer layer of the sphere is known as the surface area.
- The amount of space a spherical object has is known as volume.
The video below explains this:
Surface Area and Volume Detailed Video Explanation:
Deriving the Equation of Sphere
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The formula of a circle having a center at the (0,0) origin is given by: x2 + y2 = r2
When the point (x,y) is located on the circle and the right triangle has sides of lengths |x| and |y| and a hypotenuse length |r|, then it is also comparable to the mathematical approach used to initiate the Pythagorean Theorem as given below:
However, if the circle's focal point is not the start of the coordinate axis, being represented by random points, say (h,k), the equation of the circle is: (x-h)2 + (y-k)2 = r2
Although, it required to apply the Pythagorean Theorem twice to drive at the equation of the sphere.
In the diagram below, O represents the beginning, and P(x,y,z) represents a three-dimensional point. P is a point on a sphere having a radius r only if the distance between O and P is r.
As OAB here is the right triangle, i.e, x2 + y2 = s2 and OBP is another right triangle, i.e., s2 + z2 = r2. As a result, the distance between the points O and P can be conveyed by: x2 + y2 + z2 = |OP|2
As a result, when x2 + y2 + z2 = r2 only then it can be concluded that (x,y,z) lies with radius r on the sphere.
Moreover, x2 + y2 + z2 = r2 is the general equation of a sphere or the equations of a sphere most commonly known as the equation of the sphere as well.
If (a, b, c) is the sphere's center, r is the radius, and x, y, and z are the coordinates of points on the sphere's surface, then the general equation mathematically forms is (x – a)2+(y – b)2+(z – c)2 = r2
Read More: Difference between Area and Volume
Deriving The Surface Area Of Sphere
The measure of the absolute area occupied by the solid object is its surface area.
The surface area of a sphere can be calculated by the formula: A= 4πr2, where the radius of the sphere is denoted by r.
Also Read: Surface Areas and Volumes Revision Notes
Deriving The Volume Of The Sphere
The amount of space occupied by any three-dimensional figure is its volume.
The volume of a can be calculated by the formula: V= 4/3πr3, where the radius of the sphere is denoted by r.
Also read - Volume of Sphere: Formula, Derivation & Solved Examples
Things To Remember
- The distance between the sphere's center and outer layers is always equal.
- Diameter of the sphere can be calculated through this formula: D = 2r
- The formula to calculate circumference of the sphere: 2πr
- The equation of sphere can be calculated by applying this formula: x2 + y2 + z2 = r2
- To the find the surface area of sphere this formula can be applied: A= 4πr2
- When there is need to find the volume of the sphere, this formula is used: V= 4/3πr3
Read More:
Sample Questions
Ques: The radius of the sphere is 6 cm. Find the surface area and volume of a sphere? (3 Marks)
Ans: Given, r = 6 cm
Surface Area of sphere =\(4 \pi r^{2}\)
=\(4 \pi 6^{2}\)
= \(4 \times 3.14 \times 36\)
= 452.60cm²
Volume of a sphere = \(\frac{4}{3}\pi r^{3}\)
= \(\frac{4}{3}\pi 6^{3}\)
= \(\frac{4}{3} \times 3.14 \times 256\)
= 904.32 cm³
Ques: What would be the equation of a sphere through a circle in the standard form whose center and radius are given by (-2,-3,4) and 5 cm respectively? (3 Marks)
Ans: (-2,-3,4)
These are the values of a,b,c respectively.
Radius= 5 cm
The general equation of a sphere: (x-a)2 + (y-b)2+ (z-c)2= r2
Now, substituting the values of x,y,z and r by -2,-3 and 4 and 5 respectively:
(x-(-2)) 2 + (y-(-3)) 2 + (z-4) 2 = 52
(x+2)2 + (y+3)2 + (z-4)2 = 25
Ques: A spherical ball has a surface area of 3000 cm2 . Find the radius of the ball, correct to 2 decimal places, using π = 3.142. (3 Marks)
Ans: Surface area = 4 × π × r2
In order to find r, we need to isolate it from the equation above:
r2 = surface area / (4π)
r2 = 3000 / (4 × π)
r2 = 238.85
r = √238.85
r = 15.45cm
Ques: A stunt pilot is flying an airplane around the center of a sphere of radius 40 feet and forms a circle. Determine the surface area of the sphere. (Use π = 3.14). (2 Marks)
Ans: Given, the radius r of the sphere is 40 feet.
The surface area of the sphere = 4πr2 = 4 × π × 402 = 20096 feet2
Therefore, the surface area of the sphere is 20096 feet2
Ques: The cross-section of a rubber ball has an outer diameter of 25 inches. The thickness of the rubber is 0.10 inches. What is the area of the inside surface of the ball to the nearest sq. inches? (Use π = 3.14). (5 Marks)
Ans: Given the outer diameter = 25 inches and thickness = 0.10 inch Thus, outer radius = 25/2 inches = 12.5 inches
Inner radius = Outer radius - Thickness = 12.5 - 0.5 = 12 inches
Inner surface area of the ball = 4πr2 = 4 × π × (12) 2 = 4 × π × 144 = 1808.64 inch2 Therefore, the area of the inside surface of the ball is 1808.64 square inches.
Ques: The surface area of a sphere of radius 10 cm is five times the area of the curved surface of a cone of radius 8 cm. Find the height and the volume of the cone (taking π = 22/7). (4 Marks)
Ans: Surface area of the sphere = 4π × 10 × 10 cm2.
Curved surface area of the cone = π × 8 × l cm2,
where l is the slant height of the cone.
According to the statement: 4π × 10 × 10=5 × π × 8 × l or l = 10 cm.
Now, l2 = h2+ r2
Therefore, (10)2 = h2+ (8)2
where h is the height of the cone
or (10)2– (8)2= h2
or (10 + 8) (10 – 2) = h2
or 36 = h2
or h = 6 cm
Volume of Cone =1/3 πr2 = 1/3 × 22/7 × 8 × 8 × 6 cm3 = 22 × 128/ 7 cm3 = 2816/7 cm3 = 402.28 cm3 (approximately)
Ques: The radius of a sphere is increased by 10%. Prove that the volume will be increased by 33.1% approximately. (3 Marks)
Ans: The volume of a sphere = 43 πr2
10% increase in radius = 10% r
Increased radius = r + 1/10r = 11/10r
The volume of the sphere now becomes= 4/3π (11/10r)3 = 4/3π × 1331/1000r3 = 4/3π × 1.331r3
Increase in volume = 4/3π × 1.331 r3 - 4/3πr3 = 4/3πr3 (1.331-1) = 4/3 πr3 × .331
Percentage increase in volume= 4/3 πr3 × .331/ 4/3 πr3 × 100 = 33.1
Ques: Write the equation of the sphere in the standard form where the center and radius of the sphere are given as (12, 9, -6) and 10 cm respectively. (3 Marks)
Ans: Given: Center = (12, 9, -6) = (a, b, c) and Radius = 10 cm
The equation of the sphere in the standard form: (x-a)2 + (y-b) 2 + (z-c) 2 = r2 Substituting the given values:
(x-12) 2 + (y-9) 2 + (z -(-6)) 2 =102
(x-12) 2 + (y-9) 2 + (z +6) 2 = 100
Thus, the equation of the sphere is (x-12) 2 + (y-9) 2 + (z +6) 2 = 100
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