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Surface Area of a Hemisphere, Definition, Formula

Surface Area of a Hemisphere refers to the total area of its outer surface, including both the curved surface and the flat circular base. It is essential for calculating the material required for construction or the space it occupies.

Shruti Dutta16 Dec, 2024

The surface area of a hemisphere is an essential concept in geometry, especially when dealing with three-dimensional shapes. A hemisphere is a half of a sphere, created when a sphere is divided by a plane passing through its center.

It consists of two parts: the curved surface, which is the outer, rounded part, and the flat circular base. Calculating the surface area of a hemisphere involves determining both these components. The surface area is important in various real-world applications, such as determining the material required to cover a hemispherical object, designing domes, or calculating the outer area of objects in engineering and architecture. This article will explore the formula for calculating the surface area of a hemisphere and provide practical examples for better understanding. [video width="1920" height="1080" mp4="https://www.pw.live/exams/wp-content/uploads/2024/12/Copy-of-Corurious-Jr-Reel-2-Landscape-1-2-1.mp4"][/video]

What is the Surface Area of Hemisphere?

The surface area of a hemisphere refers to the total area covering all its surfaces. A hemisphere is a three-dimensional shape formed when a sphere is divided by a plane passing through its center, effectively creating half of the sphere. A hemisphere can either be hollow or solid. The surface area is calculated in square units and includes both the curved surface and the flat circular base. This measurement is essential for various practical applications where knowing the surface coverage of a hemispherical shape is needed. The total surface area helps estimate the material required for manufacturing, construction, or other purposes involving hemispherical objects. surface area of hemisphere

Also Check: AM and PM

Surface Area of a Hemisphere Formula

To calculate the surface area of a hemisphere, we need to consider both the curved surface area and the flat circular base. Here's a step-by-step detailed guide to finding the surface area:

Step-by-Step Process:

Step 1: Understand the Formula for Surface Area:

The total surface area of a hemisphere consists of two parts:

The curved surface area (the outside surface of the hemisphere)
The area of the circular base (the flat surface at the bottom)
The formula for the surface area of a hemisphere is:

Surface Area=3πr2 Where : 𝑟 is the radius of the hemisphere. 𝜋(pi) is a constant approximately equal to 3.14159.

Step 2: Find the Radius (r):

Identify or measure the radius (r) of the hemisphere. The radius is the distance from the center of the hemisphere to any point on its curved surface. If you're given the diameter (the distance across the flat circular surface), divide it by 2 to find the radius: 𝑟=Diameter2r= Step 3: Square the Radius (r) Once you have the radius, the next step is to square it (multiply the radius by itself): 𝑟2=𝑟×𝑟

Step 4: Multiply by Pi (𝜋:Multiply the squared radius by pi (𝜋):𝜋𝑟2

This gives the area of the flat circular base of the hemisphere.

Step 5: Multiply by 3:

To find the total surface area of the hemisphere, multiply the result from step 4 by 3. This accounts for both the curved surface area and the circular base area. Surface Area =3πr2 Step 6: Calculate the Final Value: Now, you can calculate the final surface area. Use 𝜋≈3.14159 to get the numerical result. Example Problem : Let’s go through a sample calculation to understand the process better. Given : Radius 𝑟=5cm Step 1 : Square the Radius:𝑟2=5×5=25 Step 2 : Multiply by Pi:𝜋𝑟2=3.14159×25≈78.53975 Step 3 : Multiply by 3: Surface Area=3×78.53975≈235.62cm2 Final Answer : The surface area of the hemisphere is approximately 235.62 cm². [video width="1920" height="1080" mp4="https://www.pw.live/exams/wp-content/uploads/2024/12/Curious-Jr-Ad.mp4"][/video] Curved Surface Area of a Hemisphere Formula The formula for the curved surface area of a hemisphere is: Curved Surface Area=2πr Where : 𝑟 is the radius of the hemisphere. 𝜋(pi) is a constant approximately equal to 3.14159. Explanation : The curved surface area refers to the outer surface of the hemisphere, excluding the flat circular base. To find the curved surface area, you simply multiply 2 times pi times the square of the radius.

Also Check: Area of Rectangle

Steps to Calculate Curved Surface Area:

Find the radius (𝑟) of the hemisphere.
Square the radius (𝑟2).
Multiply by 𝜋(approximately 3.14159).
Multiply by 2 to account for the curved surface.

Example : If the radius of the hemisphere is 5 cm: Curved Surface Area = 2π(5) 2=2×3.14159×25≈157.08cm So, the curved surface area of the hemisphere is approximately 157.08 cm².

Total Surface Area of a Hemisphere Formula

The total surface area of a hemisphere refers to the combined area of its curved surface and its flat circular base. A hemisphere is a three-dimensional shape obtained by cutting a sphere in half, with one half being the curved surface and the other being the flat base. The formula for calculating the total surface area is 3𝜋𝑟2, where 𝑟 is the radius of the hemisphere. This formula accounts for both the curved surface area, which is 2πr and the area of the circular base, which is 𝜋𝑟2 By adding these two components together, we obtain the total surface area. The total surface area is important for practical applications such as determining the material needed to cover a hemisphere or calculating the volume in certain designs and constructions.

Surface Area of Hollow Hemisphere

The surface area of a hollow hemisphere refers to the combined area of its outer and inner curved surfaces, as well as its circular bases. A hollow hemisphere consists of two hemispheres, one inside the other, and it typically has an inner and outer radius. The surface area is calculated by considering both the outer and inner parts separately and then adding them together.

Formula for Surface Area of a Hollow Hemisphere:

The total surface area of a hollow hemisphere is the sum of the curved surface areas of the outer and inner hemispheres and the areas of the two circular bases (outer and inner).

Surface Area of Hollow Hemisphere =2πR

Where :

𝑅 is the outer radius of the hemisphere.
𝑟 is the inner radius of the hemisphere.
𝜋 is a constant approximately equal to 3.14159.

This formula consists of : 2πR: The curved surface area of the outer hemisphere.

𝜋𝑟2:The curved surface area of the inner hemisphere.
𝜋𝑅2:The area of the outer circular base.
πr2: The area of the inner circular base.

[video width="1920" height="1080" mp4="https://www.pw.live/exams/wp-content/uploads/2024/12/Courios-jr-Video.mp4"][/video] Simplified Formula :

Surface Area of Hollow Hemisphere=3πR 2+3πr

Explanation:

The outer curved surface area and the outer base area contribute to the total surface area for the external part of the hemisphere.
The inner curved surface area and the inner base area contribute to the hollow hemisphere's internal surface area.

Example Calculation : If the outer radius of the hollow hemisphere is 6 cm and the inner radius is 4 cm, the surface area would be calculated as follows:

Surface Area of Hollow Hemisphere=3π(6)2+3π(4) 2=3π(36)+3π(16)=108π+48π=156πcm 2

Approximating: Surface Area of Hollow Hemisphere≈156×3.14159≈489.74cm2. Thus, the surface area of the hollow hemisphere is approximately 489.74 cm².

Related Articles
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Surface Area of a Hemisphere FAQs

Why is the total surface area of a hemisphere different from a sphere?

The total surface area of a hemisphere is only half of the sphere's curved surface area plus the area of the flat base. For a sphere, there is no base, so the surface area is calculated using the formula 4πr2.

How is the surface area of a hemisphere used in real life?

The surface area of a hemisphere is important for applications such as determining the material needed to cover a dome or hemispherical structure, calculating the outer area of various objects in engineering and design, and in scientific contexts like calculating the surface area of planets or spherical vessels.

Can the surface area of a hemisphere be calculated for irregular shapes?

The surface area formula applies to a perfect hemisphere with a consistent radius. For irregular hemispherical shapes, more complex methods or approximations might be required.

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What is the Surface Area of Hemisphere?

Also Check: AM and PM

Surface Area of a Hemisphere Formula

To calculate the surface area of a hemisphere, we need to consider both the curved surface area and the flat circular base. Here's a step-by-step detailed guide to finding the surface area:

Step-by-Step Process:

Step 1: Understand the Formula for Surface Area:

The total surface area of a hemisphere consists of two parts:

The curved surface area (the outside surface of the hemisphere)
The area of the circular base (the flat surface at the bottom)
The formula for the surface area of a hemisphere is:

Surface Area=3πr2 Where : 𝑟 is the radius of the hemisphere. 𝜋(pi) is a constant approximately equal to 3.14159.

Step 2: Find the Radius (r):

Step 4: Multiply by Pi (𝜋:Multiply the squared radius by pi (𝜋):𝜋𝑟2

This gives the area of the flat circular base of the hemisphere.

Step 5: Multiply by 3:

Also Check: Area of Rectangle

Steps to Calculate Curved Surface Area:

Find the radius (𝑟) of the hemisphere.
Square the radius (𝑟2).
Multiply by 𝜋(approximately 3.14159).
Multiply by 2 to account for the curved surface.

Example : If the radius of the hemisphere is 5 cm: Curved Surface Area = 2π(5) 2=2×3.14159×25≈157.08cm So, the curved surface area of the hemisphere is approximately 157.08 cm².

Total Surface Area of a Hemisphere Formula

Surface Area of Hollow Hemisphere

Formula for Surface Area of a Hollow Hemisphere:

The total surface area of a hollow hemisphere is the sum of the curved surface areas of the outer and inner hemispheres and the areas of the two circular bases (outer and inner).

Surface Area of Hollow Hemisphere =2πR

Where :

𝑅 is the outer radius of the hemisphere.
𝑟 is the inner radius of the hemisphere.
𝜋 is a constant approximately equal to 3.14159.

This formula consists of : 2πR: The curved surface area of the outer hemisphere.

𝜋𝑟2:The curved surface area of the inner hemisphere.
𝜋𝑅2:The area of the outer circular base.
πr2: The area of the inner circular base.

[video width="1920" height="1080" mp4="https://www.pw.live/exams/wp-content/uploads/2024/12/Courios-jr-Video.mp4"][/video] Simplified Formula :

Surface Area of Hollow Hemisphere=3πR 2+3πr

Explanation:

The outer curved surface area and the outer base area contribute to the total surface area for the external part of the hemisphere.
The inner curved surface area and the inner base area contribute to the hollow hemisphere's internal surface area.

Example Calculation : If the outer radius of the hollow hemisphere is 6 cm and the inner radius is 4 cm, the surface area would be calculated as follows:

Surface Area of Hollow Hemisphere=3π(6)2+3π(4) 2=3π(36)+3π(16)=108π+48π=156πcm 2

Approximating: Surface Area of Hollow Hemisphere≈156×3.14159≈489.74cm2. Thus, the surface area of the hollow hemisphere is approximately 489.74 cm².

Related Articles
Subtraction Sums	Word Problems
What is an Abacus?	Math Puzzles
Multiplication Worksheets :	Multiplication Sums