Area of Hemisphere (Curved and Total Surface Area)| Formulas

Area Of Hemisphere

Edited By Team Careers360 | Updated on Jul 02, 2025 05:19 PM IST

The total surface area of a hemisphere is another name for a hemisphere's surface area. The sum of all the faces makes up a hemisphere's total surface area (curved and base area). When a sphere is split into two equally sized parts by a plane, a hemisphere is created. In geometry, a hemisphere is the exact opposite of a sphere, to put it another way. Hemi, which means half, and sphere, a 3D mathematical shape, make up this object. Two equal hemispheres are produced when a sphere is split down the middle along its diameter.

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Hemisphere
Total Surface Area Of Hemisphere
Hollow Hemisphere

Hemisphere

A sphere is a three-dimensional solid in which the distances between all points on its surface and its center are equal. When the sphere is divided into two equal halves by the plane, a hemisphere is created. The two hemispheres come together to form a sphere.

Curved Surface Area Of Hemisphere

A hemisphere's curved surface area is the area that is covered by its curved surface. It has a surface area that is exactly half that of a sphere. The formula below can be used to determine the curved surface area of a hemisphere with radius r.

Curved surface area of a hemisphere = 12(curved surface area of a sphere) = 1/2(4πr2)=2πr2

Where, r is the radius of the hemisphere.

Total Surface Area Of Hemisphere

The combined space filled by the curved surface and the base surface of the hemisphere is referred to as the hemisphere's total surface area. By adding the areas of a hemisphere's base surface and curved surface, the total surface area of a hemisphere can be determined. It should be noticed that a hemisphere has a circle for its base. If a hemisphere's radius is known, the following formula can be used to determine its total surface area:

Total surface area of a hemisphere = Curved Surface Area + Base Area = 2πr2 + πr2 = 3πr2

Hollow Hemisphere

A hollow sphere can be defined as the sphere that has been thinned out to the point that a wall of equal thickness produces an interior ball with radius ‘r’ inside the exterior ball with the radius ‘R’.

Surface Area Of The Hollow Hemisphere

The formula for a hollow hemisphere's surface area is TSA of hollow hemisphere = 3R2 + r2.

It should be noted that a hollow hemisphere has two diameters for its circular bases: one for the inner base (the hollow region) and one for the outer base. The following formulas can be derived if "r" is the radius of the internal hemisphere and "R" is the radius of the external hemisphere.

The outer hemisphere's curved surface area is equal to 2R2.
The inner hemisphere's curved surface area is equal to 2r2.
If we observe the figure given above, we can see that a ring is formed. The Area of the formed ring = π(R2 - r2)
The Total surface area of the hollow hemisphere can be calculated if we add the formulas given above, that is, TSA of a hollow hemisphere = 2πR2 + 2πr2 + π(R2 - r2). This can be written as, TSA of hollow hemisphere = 2π(R2 + r2) + π(R2 - r2). And if we want to simplify it further, it becomes 3πR2 + πr2

Frequently Asked Questions (FAQs)

1. In mathematics, what is a hemisphere shape?

A hemisphere can be defined as the 3-Dimensional figure which is obtained by cutting a sphere into two equal halves through its diameter. Some of the real-life examples of a hemisphere are a cup, cake, hat etc.

2. Define Hollow Hemisphere?

A hemisphere is said to be hollow if the interior space is hollow. Two radii make up a hollow hemisphere: an internal radius for the hollow inner circle and an exterior radius for the hollow outer circle.

3. What is the curved surface area of a hemispherical shell?

The surface area of a hemisphere shell is equal to the surface area of a solid hemisphere. Therefore, the surface area of the hemispherical shell is 3πr2.

4. What Is a Total Surface Area Hemispherical Shell?

The formula for the total surface area of a hollow hemisphere can be used to get the total surface area of a hemispherical shell because it is a hollow hemisphere. The radius of the internal hemispherical shell, r, and the radius of the external hemispherical shell, R, can be used to compute the Total Surface Area of a hollow hemisphere using the formula Total Surface Area of a hollow hemisphere = 3R2 + r2.

5. What is a hemisphere and how does it relate to a sphere?

A hemisphere is half of a sphere, created by cutting a sphere in half along its center. It has a circular base and a curved surface that forms exactly half of the original sphere's surface area.

6. What's the difference between the surface area and the volume of a hemisphere?

The surface area of a hemisphere is a measure of the total area covering its surface (curved part and base), while the volume is the amount of space it encloses. They have different formulas and units: surface area is measured in square units (e.g., cm²), while volume is in cubic units (e.g., cm³).

7. How does the surface area to volume ratio of a hemisphere change as its size increases?

As the size of a hemisphere increases, its surface area to volume ratio decreases. This is because the surface area increases with the square of the radius (r²), while the volume increases with the cube of the radius (r³). Larger hemispheres have relatively less surface area compared to their volume.

8. How does the ratio of the surface area to the volume of a hemisphere compare to that of a cube?

A hemisphere generally has a lower surface area to volume ratio compared to a cube of similar size. This means that for a given volume, a hemisphere requires less surface area to enclose it, making hemispheres more efficient in terms of minimizing surface exposure for a given volume.

9. How does the surface area to volume ratio of a hemisphere compare to that of a full sphere?

A hemisphere has a higher surface area to volume ratio compared to a full sphere of the same radius. This is because while a hemisphere has 3/4 of the surface area of a full sphere, it only has 1/2 of the volume. This makes hemispheres less efficient than full spheres in minimizing surface exposure for a given volume.

10. How does the concept of surface area of a hemisphere apply in architecture?

In architecture, understanding the surface area of hemispheres is crucial for designing and constructing domes, geodesic structures, and other curved surfaces. It helps in calculating material needs, assessing structural loads, and planning for insulation or decoration of these surfaces.

11. How does the concept of surface area of a hemisphere relate to the field of optics?

In optics, the surface area of hemispheres is important in designing lenses, mirrors, and other optical components. Hemispherical mirrors, for instance, are used in wide-angle security cameras and solar collectors. Understanding their surface area helps in calculating reflective properties and light distribution.

12. How does the concept of surface area apply to real-world hemispheres?

Understanding the surface area of hemispheres is crucial in many real-world applications, such as calculating the amount of paint needed to cover a dome, determining the material required for a hemispherical tent, or estimating heat loss through a rounded roof.

13. If you were to "unwrap" the curved surface of a hemisphere and lay it flat, what shape would you get?

If you could theoretically "unwrap" the curved surface of a hemisphere and lay it flat, you would get a circular sector (like a slice of pizza). The radius of this sector would be equal to the radius of the hemisphere, and its arc length would be equal to the circumference of the hemisphere's base.

14. If you have a sphere and cut off a cap that's not exactly half, how would this affect the surface area calculation?

If you cut off a cap that's not exactly half a sphere, you're dealing with a spherical segment rather than a hemisphere. The surface area calculation becomes more complex, involving the height of the segment and requiring more advanced formulas or calculus to solve accurately.

15. How is the surface area of a hemisphere different from that of a full sphere?

The surface area of a hemisphere includes two parts: half of the curved surface area of a sphere, plus the area of the circular base. In contrast, a full sphere only has the curved surface area without any flat circular part.

16. What is the formula for calculating the surface area of a hemisphere?

The formula for the surface area of a hemisphere is: A = 2πr² + πr², where r is the radius of the hemisphere. This can be simplified to A = 3πr².

17. Why do we add πr² to 2πr² when calculating the surface area of a hemisphere?

We add πr² to 2πr² because the surface area of a hemisphere includes half of the sphere's curved surface (2πr²) plus the area of the circular base (πr²). This accounts for both the curved and flat portions of the hemisphere.

18. How does the surface area of a hemisphere compare to that of a full sphere with the same radius?

The surface area of a hemisphere is 3/4 or 75% of the surface area of a full sphere with the same radius. This is because a hemisphere has half of the curved surface area plus its circular base, which equals 3/4 of the full sphere's surface area.

19. Can you explain why the surface area of a hemisphere isn't exactly half that of a sphere?

The surface area of a hemisphere isn't exactly half that of a sphere because it includes the circular base in addition to half of the sphere's curved surface. The circular base adds extra area, making the total surface area more than just half of the sphere's surface area.

20. How does the surface area of a hemisphere relate to its volume?

While both surface area and volume increase as the radius increases, they don't increase at the same rate. The surface area is proportional to r², while the volume is proportional to r³. This means that as a hemisphere gets larger, its volume increases faster than its surface area.

21. Can the surface area of a hemisphere ever be equal to its volume?

No, the surface area and volume of a hemisphere can never be equal because they are measured in different units (square units vs. cubic units). However, for a specific radius, the numerical values might coincidentally be close.

22. How is the surface area of a hemisphere related to the concept of packing efficiency?

The surface area of a hemisphere relates to packing efficiency because it determines how much material is needed to enclose a given volume. Hemispheres, like spheres, have a lower surface area to volume ratio compared to many other shapes, making them efficient for minimizing material use while maximizing volume.

23. If you cut a hemisphere into quarters (like cutting an orange), how does the surface area of one quarter relate to the whole?

The surface area of one quarter of a hemisphere is not simply 1/4 of the whole hemisphere's surface area. It includes 1/4 of the circular base, 1/4 of the curved surface, plus two new rectangular faces created by the cuts. This makes the total surface area of a quarter more complex to calculate.

24. How would you use the surface area of a hemisphere to estimate the surface area of an egg?

To estimate an egg's surface area using a hemisphere, you could approximate the egg as two hemispheres of slightly different sizes joined together. Calculate the surface areas of both hemispheres separately and sum them. This would give a rough estimate, though it wouldn't account for the egg's true oval shape.

25. How does the surface area of a hemisphere relate to the concept of "minimal surfaces" in mathematics?

While a hemisphere isn't a minimal surface itself, it's related to the concept. Minimal surfaces have the least surface area possible for given boundary conditions. Spheres (and by extension, hemispheres) minimize surface area for a given volume, which is why soap bubbles form spherical shapes.

26. In what way does the surface area of a hemisphere relate to the concept of "packing spheres" in geometry?

The surface area of a hemisphere relates to sphere packing in that it represents the contact area when spheres are packed together. In a close-packed arrangement of spheres, each sphere is in contact with others at points that form hemispheres. Understanding these contact areas is crucial for studying the efficiency and properties of sphere packing.

27. How does the surface area of a hemisphere change if you slice off the top parallel to the base?

If you slice off the top of a hemisphere parallel to the base, you create a spherical segment. The surface area will decrease, but not linearly. The new surface area will include the remaining curved surface (which is less than 2πr²), the original circular base, and the new circular cut at the top.

28. How does the concept of surface area of a hemisphere apply in the field of meteorology?

In meteorology, understanding the surface area of hemispheres is important for modeling atmospheric phenomena over large scales. The Earth itself is often approximated as a hemisphere for calculations involving one hemisphere (like the Northern or Southern Hemisphere), affecting models of air circulation, heat distribution, and climate patterns.

29. How does the surface area of a hemisphere relate to the concept of "solid angle" in physics?

The surface area of a hemisphere is directly related to the concept of solid angle. A hemisphere subtends a solid angle of 2π steradians at its center, which is half of the full 4π steradians of a complete sphere. This relationship is important in fields like radiometry and photometry.

30. How does changing the radius affect the surface area of a hemisphere?

The surface area of a hemisphere is proportional to the square of its radius. If you double the radius, the surface area increases by a factor of 4. This is because the formula (3πr²) contains r², so any change in r is squared in the final result.

31. How can you visualize the surface area of a hemisphere in real life?

Imagine peeling the skin off half an orange. The peel you remove, including the flat circular bottom, represents the surface area of the hemisphere. This includes both the curved part and the flat circular base.

32. Why is π (pi) used in the formula for the surface area of a hemisphere?

π (pi) is used because it's fundamental in calculating the area of circles and curved surfaces. Since a hemisphere is based on a circular shape, π is necessary for accurately determining its surface area.

33. How would you find the radius of a hemisphere if you know its surface area?

To find the radius given the surface area, you would use the formula A = 3πr² and solve for r. Rearranging the equation gives you r = √(A / 3π). This allows you to calculate the radius when you know the surface area.

34. What's the relationship between the surface area of a hemisphere and its circular base?

The surface area of a hemisphere is always three times the area of its circular base. This is because the total surface area (3πr²) is composed of two parts: the curved surface (2πr²) and the circular base (πr²).

35. How does the surface area of a hemisphere compare to that of a cylinder with the same base and height?

The surface area of a hemisphere is always less than that of a cylinder with the same base radius and height equal to the radius. This is because the curved surface of the hemisphere is more compact than the rectangular side of the cylinder.

36. Why isn't the curved surface area of a hemisphere simply half the surface area of a sphere?

The curved surface area of a hemisphere is indeed half the surface area of a sphere. However, when we talk about the total surface area of a hemisphere, we must also include its circular base, which makes it more than half the sphere's surface area.

37. What's the significance of the factor 3 in the simplified formula 3πr² for a hemisphere's surface area?

The factor 3 in 3πr² represents the sum of the two components of a hemisphere's surface area: 2 from the curved part (half of 4πr² for a full sphere) and 1 from the circular base (πr²). It shows that the total area is three times that of the circular base.

38. How would you explain the concept of a hemisphere's surface area to someone who doesn't know calculus?

You can explain it as the sum of two parts: the area of a circle (the base) plus the area of half a sphere's skin (the curved part). Imagine wrapping a flat circle and half of a hollow ball with paper – the total paper used represents the surface area.

39. If you double the diameter of a hemisphere, how does it affect its surface area?

Doubling the diameter means doubling the radius. Since the surface area is proportional to the square of the radius (3πr²), doubling the radius will increase the surface area by a factor of 4. So, the new surface area will be 4 times the original.

40. Can you derive the formula for the surface area of a hemisphere without calculus?

Yes, you can derive it without calculus. Start with the known formula for a sphere's surface area (4πr²). Take half of this for the curved part of the hemisphere (2πr²), then add the area of the circular base (πr²). This gives you 2πr² + πr² = 3πr².

41. What's the difference between the lateral surface area and the total surface area of a hemisphere?

The lateral surface area of a hemisphere refers only to its curved surface, which is 2πr². The total surface area includes both the curved surface and the circular base, which is 3πr². The difference between them is the area of the circular base, πr².

42. How does the surface area of a hemisphere compare to that of a cone with the same base radius and height?

A hemisphere has a larger surface area than a cone with the same base radius and height (equal to the radius). This is because the curved surface of a hemisphere bulges out more than the slanted surface of a cone, covering more area.

43. What's the relationship between the surface area of a hemisphere and the area of a great circle of the original sphere?

The surface area of a hemisphere (3πr²) is exactly three times the area of a great circle of the original sphere (πr²). This relationship highlights the efficiency of spherical shapes in enclosing volume with minimal surface area.

44. How would you explain why the surface area of a hemisphere isn't 2πr² + 2πr²?

The surface area isn't 2πr² + 2πr² because this would be counting the entire sphere's surface area (4πr²) plus an extra circular base (πr²). A hemisphere only has half of the sphere's curved surface (2πr²) plus one circular base (πr²), which gives us 3πr² in total.

45. In what scenarios might calculating the surface area of a hemisphere be more relevant than its volume?

Calculating the surface area of a hemisphere is more relevant in scenarios involving coverage or contact, such as determining the amount of paint needed for a dome, the material required for a hemispherical tent, or the heat transfer through a rounded surface. Volume would be more relevant for capacity-related problems.

46. Can you explain why the surface area of a hemisphere is always greater than the area of its "great circle"?

The surface area of a hemisphere (3πr²) is always greater than the area of its "great circle" (πr²) because it includes both the curved surface and the circular base. The curved surface alone (2πr²) is already twice the area of the great circle, and adding the base makes it three times as large.

47. What's the significance of the fact that a hemisphere's surface area is exactly 75% of a sphere's surface area?

This 75% relationship is significant because it shows that adding a flat base to half a sphere increases its surface area by 25% compared to half of the original sphere's surface. It demonstrates how the addition of the base contributes substantially to the total surface area.

48. Why is understanding the surface area of a hemisphere important in the field of heat transfer?

In heat transfer, surface area is crucial because it determines the area available for heat exchange. Understanding a hemisphere's surface area is important for designing efficient heat exchangers, cooling systems, or insulation for hemispherical structures. The ratio of surface area to volume also affects how quickly an object heats up or cools down.

49. How would you explain the concept of surface area of a hemisphere to a young student using everyday objects?

You could explain it using a hollow plastic ball cut in half and a circular piece of paper. Show that the surface area includes both the curved part of the half-ball (which you can't flatten) and the circular paper base. Emphasize that it's more than just half of the whole ball's surface because of the added base.

50. Can you explain why the ratio of the surface area of a hemisphere to that of its inscribed cube is always greater than 1?

The ratio of the surface area of a hemisphere to that of its inscribed cube is always greater than 1 because a hemisphere bulges out beyond the boundaries of the cube. While the cube's surface area is 6r², the hemisphere's is 3πr², which is always larger since π is greater than 2.

51. If you had a hemisphere made of a stretchy material and inflated it into a full sphere, how would the surface area change?

If you could stretch a hemisphere into a full sphere, its surface area would increase. The original hemisphere has a surface area of 3πr², while the full sphere would have 4πr². This represents an increase of πr², or about 33.3%. However, in reality, the material would need to stretch and thin out to achieve this.