Surface Area of a Hemisphere

The surface area of a hemisphere is defined as the region covered by the faces of the hemisphere. The total surface area of a hemisphere includes the curved surface area and the base area of the hemisphere. Let us learn more about the total surface area of a hemisphere and the curved surface area of a hemisphere along with solved examples.

The surface area of a hemisphere is the total area of all its faces. Hemisphere is a three-dimensional shape which is obtained when a sphere is cut along a plane passing through the center of the sphere. In other words, a hemisphere is half of a sphere. The hemisphere can either be hollow or solid. The surface area of a hemisphere is measured in square units. Observe the figure given below to see a sphere and a hemisphere.

The formula for the surface area of a hemisphere can be given for a solid and for a hollow hemisphere. The surface area of a hemisphere is the total area that its faces cover. It can be classified into two categories:

• The curved surface area of a hemisphere (CSA)
• The total surface area of a hemisphere (TSA)

Curved Surface Area of a Hemisphere Formula

The curved surface area of a hemisphere is the area covered by its curved surface. It is exactly half of the surface area of a sphere. The curved surface area of a hemisphere with radius 'r' can be calculated using the following formula.

Curved surface area of a hemisphere = 1/2 (curved surface area of a sphere) = 1/2 (4 π r²) = 2 π r²

Where, r is the radius of the hemisphere.

Total Surface Area of a Hemisphere Formula

The total surface area of the hemisphere is defined as the total space occupied by the curved surface and the base surface of the hemisphere. The total surface area of a hemisphere can be calculated by finding the sum of the areas of its curved surface and base surface. It should be noted that the base of a hemisphere is a circle. For a hemisphere, if its radius is given, then its total surface area can be calculated using the formula:

Surface area of a hemisphere = Curved Surface Area + Base Area = 2 π r² + π r² = 3 π r²

Where r is the radius of the hemisphere.

Surface Area of a Hollow Hemisphere Formula

The formula for the surface area of a hollow hemisphere is, TSA of hollow hemisphere = 3πR² + πr². Let us understand the derivation of this formula.

It should be noted that there are two diameters for the circular bases of a hollow hemisphere, one for the inner circular base (hollow region), and one for the outside circular base. If 'r' is the radius of the internal hemisphere, and 'R' is the radius of the external hemisphere, we can derive the following formulas.

• The curved surface area of the outer hemisphere = 2πR²
• The curved surface area of the inner hemisphere = 2πr²
• If we observe the figure given below, we can see that a ring is formed. The Area of the ring = π(R² - r²)
• So, 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

As we learned in the previous section, the surface area of a hemisphere can be calculated using different formulas for a hollow or a solid hemisphere, depending on whether we are finding the curved surface area or the total surface area.

Curved Surface Area of Hemisphere

The formula to calculate the Curved Surface Area (CSA) of a hemisphere with radius 'r' is given as, CSA of hemisphere = 2πr². The following steps show how to find the curved surface area of a hemisphere.

• Step 1: Identify the radius (r) of the hemisphere.
• Step 2: Substitute the value of the radius (r) and find the curved surface area of the hemisphere using the formula, CSA of hemisphere = 2πr²
• Step 3: Represent the final answer obtained with square units.

Example: Find the curved surface area of a hemisphere whose radius measures 8 units. (Use π = 3.14)

Solution:

The radius (r) of the hemisphere = 8 units. So, let us substitute the value of r = 8 in the formula.
Curved Surface area of a hemisphere = 2 π r² = 2 × 3.14 × 8² = 2 × 3.14 × 64 = 401.92 units²

Therefore, the curved surface area of the hemisphere = 401.92 units²

Total Surface Area of Hemisphere

The formula to calculate the total surface area (TSA) of a hemisphere with radius 'r' is given as, TSA of hemisphere = 3πr². Thus, we follow the steps shown below to find the total area of a hemisphere.

• Step 1: Identify the radius (r) of the hemisphere.
• Step 2: Substitute the value of the radius (r) and find the total surface area of a hemisphere using the formula, TSA of hemisphere = 3πr²
• Step 3: Represent the answer in square units.

Example: Find the total surface area of a hemisphere whose radius measures 7 units. (Use π = 22/7)

Solution:

The radius (r) of the hemisphere = 7 units. So, let us substitute the value of r = 7 in the formula.
Total Surface area of a hemisphere = 3 π r² = 3 × (22/7) × 7² = 3 × 22 × 7 = 462 units²

Therefore, the total surface area of the hemisphere = 462 units²

Surface Area of Hollow Hemisphere

The surface area of a hollow hemisphere can be calculated using the following steps:

• Step 1: Identify the inner radius (r) and the outer radius (R) of the hollow hemisphere.
• Step 2: Find the total surface area of the hemisphere using the formula, TSA of hollow hemisphere = 3πR² + πr²
• Step 3: Represent the final answer in square units.

Example: Find the surface area of a hollow hemisphere whose outer radius measures 14 units and inner radius is 10 units. (Use π = 3.14)

Solution:

The outer radius (R) of the hemisphere = 14 units, the inner radius (r) = 10 units. So, let us substitute the value of R = 14, and r = 10 in the formula.

TSA of hollow hemisphere = 3πR² + πr²

= [3π × (14)²] + [π × (10)²]

= (3 × 3.14 × 196) + (3.14 × 100)

= 1846.32 + 314

= 2160.32 units²

Therefore, the surface area of the hollow hemisphere = 2160.32 units²

☛ Related Articles

• Surface Area of Cuboid
• Surface Area of Cube
• Surface Area of Cylinder
• Surface Area of Prism
• Surface Area of Cone
• Surface Area of Sphere
• Difference Between Area and Surface Area
• Surface Area Formulas

What is the Surface Area of a Hemisphere?

The surface area of a hemisphere is the total area of all its faces. Since a hemisphere consists of a curved surface and a flat circular surface, we need to combine these areas to get the total surface area of a hemisphere.

What is the Formula for the Surface Area of a Hemisphere?

The formula that is used to find the total surface area of a hemisphere is expressed as, total surface area of a hemisphere = 3πr²; where 'r' is the radius of the hemisphere.

How to Find the Curved Surface Area of a Hemisphere?

The curved surface area (CSA) of a hemisphere can be calculated using the following steps.

• Step 1: Identify the radius (r) of the hemisphere.
• Step 2: Substitute the value of the radius (r) and find the curved surface area of the hemisphere using the formula, CSA of hemisphere = 2πr²
• Step 3: Express the area in square units.

How to Find the Surface Area of a Hemisphere?

The surface area of a hemisphere can be calculated depending upon the kind of hemisphere which can be hollow or solid, and also upon the kind of surface area that is required. The following list shows the formulas that are used to find the total surface area, the curved surface area of a hemisphere, and the total surface area of a hollow hemisphere.

• The total surface area of a hemisphere = 3πr², where 'r' is the radius of the hemisphere.
• The curved surface area of a hemisphere = 2πr², where 'r' is the radius of the hemisphere.
• The total surface area of a hollow hemisphere = 2π (R² + r²) + π(R² - r²), where R is the outer radius of the hemisphere and 'r' is the inner radius of the hemisphere. This formula can also be expressed as 3πR² + πr²

What is the Total Surface Area of a Hollow Hemisphere?

The total space occupied or covered by the curved surface and the base surface of the hemisphere is defined as the total surface area of a hemisphere. For a hollow hemisphere, if its inner radius (r) and outer radius (R) are given, then its area can be calculated by the formula: total surface area of a hollow hemisphere = outer curved surface area + inner curved surface area + base area. This is written as TSA of hollow hemisphere = 2π (R² + r²) + π(R² - r²). It can also be simplified and written as, TSA of hollow hemisphere = 3πR² + πr²

What is the Base Area of a Hemisphere?

The base area of a hemisphere is the area of its base which is a circle. So, if the radius (r) of a hemisphere is given, then its base area can be calculated using the formula: base area of a hemisphere = area of the base circle = π r²

What is the Lateral Surface Area of a Hemisphere?

The lateral surface area of a hemisphere is also known as the curved surface of the hemisphere. For a hemisphere, if its radius (r) is given, then its lateral surface area can be calculated using the formula: lateral surface area = 1/2 (lateral surface area of a sphere) = 1/2 (4 π r²) = 2πr²

What is the Total Surface Area of a Hemispherical Shell?

The total surface area of a hemispherical shell can be calculated using the formula for the total surface area of a hollow hemisphere because a hemispherical shell is a hollow hemisphere. This can be calculated using the formula, TSA of a hollow hemisphere = 3πR² + πr², where, r is the radius of the internal hemispherical shell, and R is the radius of the external hemispherical shell.

What is the Difference Between Curved Surface Area and Surface Area of Hemisphere?

The Curved Surface Area (CSA) is that part of the hemisphere which is curved. The formula used to calculate the curved surface area of a hemisphere is, CSA of hemisphere = 2πr². Whereas, the total surface area of the hemisphere includes the curved area and the rest of the area of the hemisphere. The formula used to calculate the total surface area of the hemisphere is, TSA of hemisphere = 3πr².