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Area of Sphere is the area occupied by the surface of a spherical object in a three-dimensional space. The area of the sphere is also referred to as the total surface area of the sphere. Sphere is a solid shape bounded by a curved surface with every point on the surface being the same distance from the center. As a sphere is a completely curved shape, thus, the curved surface area is equal to the total area of the sphere.
The surface area of sphere is 4πr2, where r refers to the radius of the sphere. The radius of a sphere is the distance between the center of the sphere to the outermost surface. In terms of diameter, the surface area of the sphere is 4π(d/2)2 where d is the diameter of the sphere.
Read More: NCERT Solutions for Class 9 Maths Surface Areas And Volumes
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Key Terms: Sphere, Surface Area of Sphere, Radius, Diameter, Three-dimensional Space, Total Surface Area, Curved Surface Area
What is Sphere?
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A sphere is a perfectly round geometrical shape in three-dimensional space. Mathematically, the sphere is defined as a collection of points in three-dimensional space that are all at the same distance from a single point which is the center.
- The radius of a sphere refers to the distance between its center and its outermost surface.
- The diameter of the sphere is the line that connects two points on the sphere and is twice the length of the radius. Diameter = 2 x Radius
Sphere
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| Related Topics | ||
|---|---|---|
| Equation Of Sphere | Introduction to Three-Dimensional Geometry | Geometry |
| Rectangular Prism | Surface Area of a Pyramid | Surface Area Of A Prism Formula |
Properties of a Sphere
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Here are the important properties of a sphere:
- A sphere has no faces or edges.
- A sphere is perfectly symmetrical in every way.
- A sphere is not a polyhedron at all.
- The points on the surface of the sphere are all equidistant from the center.
- The sphere has just one surface (not face).
Read More: Surface Areas and Volumes MCQs
Surface Area of Sphere
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A sphere is a three-dimensional solid shape whose total surface area is equal to the curved surface area.
- The curved surface area refers to the region of the surface that is only covered by the curved component. The circular base is not taken into account in the formula.
- The total surface area is made up of the curved surface area as well as the base area.
Since the sphere is a completely curved shape, thus, the curved surface area and the total surface area of sphere are equal.
Surface area (TSA) of Sphere = CSA = 4πr2square units
Read More: Difference Between Area and Perimeter
How to Calculate the Area of Sphere?
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The surface area of a sphere refers to the space occupied by its surface. The surface area of the sphere can be calculated using the area of sphere formula with the help of the given steps:
- First, find out the radius of the sphere.
- Substitute the value of radius in the surface area formula of the sphere- 4πr2
- Solve the expression to find out the area of sphere.
Solved Examples on Area of Sphere
Example 1: Calculate the surface area of a spherical ball with a radius of 9 inches.
Ans. In order to find the area of the spherical ball, we need to follow the given steps:
Radius of Spherical Ball (r) = 9 inches
Surface Area of Sphere = 4πr2
Substituting the value of r = 9, we get,
4 × 3.14 × 92 = 4 × 3.14 × 81 = 1017.36
Thus, the surface area of the spherical ball is 1017.36 in2.
Example 2: The radius of a sphere is 15 cm. Calculate its surface area. (3 Marks)
Ans. According to the question,
Radius of the Sphere = 15 cm
Surface Area of Sphere = 4πr2
Putting the value of r in the formula, we get
Surface area of sphere = = 4 × π × 152 = 2828.57 cm2
Thus, the surface area of the given sphere is 2828.57 cm2
Read More: Surface Areas and Volumes Important Questions
Area of Hemisphere
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Hemisphere is another three-dimensional shape that is exactly half of the sphere. When a plane cuts a sphere into two equal halves, a hemisphere is formed.
The total surface area of hemisphere is equal to the sum of its curved surface area (Half the area of sphere) and base area (circular base).
TSA of Hemisphere = Half of Area of Sphere + Base Area
TSA = 2πr2 + πr2
TSA = 3πr2
Volume of Sphere
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Volume of sphere refers to the capacity of a sphere. It refers to the number of cubic units needed to fill a sphere. Volume of Sphere is calculated using the given formula:
Volume = 4/3 πr3
Sphere Formulas
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| Chapter-Related Topics | ||
|---|---|---|
| Surface Area of Cone | Surface Area of a Cylinder Formula | Area of Hollow Cylinder |
| Surface Area of Right Circular Cone | Surface Area of Cuboid | Lateral Surface Area |
Things to Remember
- Sphere is a three-dimensional round object with no vertices or edges.
- Since the sphere is an entirely curved shape, the total surface area and the curved surface area of sphere are equal.
- Radius of a sphere refers to the distance of a point from the center to the outermost surface.
- Diameter of a sphere refers to the line that connects two points on the sphere. It is twice the length of the radius.
- Surface Area of Sphere = 4πr2
- Volume of Sphere = 4/3 πr3
Sample Questions
Ques. Calculate the surface area of a sphere if its radius is given as 6 units. (3 Marks)
Ans. Given that,
Radius of Sphere (r) = 6 units.
Using the surface area of sphere formula,
Surface area of the sphere = 4πr2
= 4 × π × 62 = 4 × 3.14 × 36
= 452.16 unit2
Thus, the surface area of the sphere is 452.16 unit2.
Ques. A boy has a spherical ball with a radius of 12 feet. What will be the surface area of the given ball? (3 Marks)
Ans. Given that,
Radius of the Spherical Ball = 12 feet
Surface area of sphere = 4πr2
= 4 x 3.14 x 12 x 12
= 1808.64
Thus, the area of the spherical ball is 1808.64 feet2.
Ques. What is the area and volume of a sphere? (2 Marks)
Ans. Surface area of a sphere refers to the region occupied by the surface of a sphere in the three-dimensional space. The surface area of sphere is calculated as 4πr2.
The volume of a sphere refers to the number of cubic units needed to fill a sphere. The volume of sphere is given as 4/3 πr3.
Ques. Given that the radius of a sphere is 20 feet, what will be its surface area? (3 Marks)
Ans. Given that.
Radius ('r') of the Sphere = 20 feet
Surface Area of Sphere = 4πr2
Substituting the values,
Surface area of sphere = = 4 × π × 202 = 5024 feet2
Thus, the surface area of the sphere is 5024 feet2
Ques. What will be the cost to paint a football which is in the shape of a sphere with a radius of 7 cm? The cost of painting the football is INR 2.5/square cm. ( π = 22/7) (3 Marks)
Ans. Given that,
Radius of Ball = 7 cm
Surface area of sphere = 4πr2
= 4 × (22/7) × 7 × 7
= 616 cm2
Therefore, the cost of painting the container will be
2.5 × 616 = Rs. 1540
Thus, Rs. 1540 would be spent on the ball in order to be painted.
Ques. What is the Surface Area of a Sphere if the radius is tripled? (2 Marks)
Ans. The surface area of the sphere will be 36πr2 when the radius is tripled because 'r' becomes 3r'. The surface area of a sphere is given as 4πr2, so if we replace 'r' with 3r, the surface area will be 4π(3r)2 = 4π × 9r2 = 36πr2
Ques. What is the Curved Surface Area of a Sphere? (2 Marks)
Ans. A sphere has just one surface which is curved. As there is no flat surface in a sphere, thus, the curved surface area of a sphere is equal to its total surface area of the sphere which is 4πr2.
Ques. What will be the curved surface area of a sphere having a radius equal to 3.5 cm? (π= 22/7) (2 Marks)
Ans. Radius is given as 3.5 cm.
We know that the total surface area of a sphere is equal to the curved surface area of a sphere, so
Curved Surface Area of Sphere = Total Surface Area of Sphere
Curved Surface Area of Sphere = 4πr2
= 4 × (22/7) × 3.5 × 3.5
Therefore, the curved surface area of the given sphere is 154 cm2.
Ques. What is the Surface Area of a Sphere in terms of diameter? (1 Mark)
Ans. In terms of diameter, the surface area of a sphere is given as:
S = 4π(d/2)2
Where d refers to the diameter of the sphere.
Ques. What is the surface area of the sphere if the radius is halved? (2 Marks)
Ans. The surface area of the sphere is one-fourth when the radius is halved as r becomes r/2. The surface area of a sphere is 4πr2, so, if we replace 'r' with r/2, the formula becomes 4π(r/2)2 = πr2 which is one-fourth of the total surface area. Hence, the surface area of the sphere gets one-fourth as its radius gets halved.
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