What is the ratio of the total surface area of the solid hemisphere to the square of its radius

Our surface area of a hemisphere calculator is a handy tool that finds different types of hemisphere surface areas. Are you looking for an answer to the question of how to find the surface area of a hemisphere? Or maybe you just need to estimate it quickly? Whatever you plan to do, try this hemisphere calculator which comprises several various area of a hemisphere formulas.

Hemispheres are created by dividing a sphere into two equal halves, as you can see in the picture below. Unlike the full sphere, a hemisphere has two kind of surface areas: the base area (which is a circle) and the cap area. The notation, which we have used in this surface area of a hemisphere calculator, is as follows:

• r - radius of a hemisphere,
• d - diameter of a hemisphere,
• V - volume of a hemisphere,
• A - total surface area of a hemisphere,
• Ab - base surface area of a hemisphere,
• Ac - cap surface area of a hemisphere,
• A / V - surface to volume ratio of a hemisphere.

🙋 The interesting fact is that the total volume of two hemispheres equals the volume of one full sphere. However, the same is not true with surface areas. The total area of two hemispheres is greater than the area of a sphere. The reason is simple: hemispheres have an additional base area.

In case you need to analyze full spheres, our sphere calc is essentially the tool you're looking for!

When we split a sphere in half and take one of the resulting parts, we get a hemisphere. In each hemisphere, we can name two surface areas: base area and cap area (look at the picture above). From the area of a sphere calculator we know that the surface area of a sphere is as below:

A(sphere) = 4 * π * r².

You can think about it like two times the cap surface area of a hemisphere. Therefore, the hemisphere cap area equals:

Ac = A(sphere) / 2,

Ac = 2 * π * r².

The base surface area is a circle with the same radius as a hemisphere. Thus, according to the circle calc: find A, it can be expressed as:

Ab = π * r².

Finally, the total surface area is the sum of those two contributions:

A = Ac + Ab,

A = 2 * π * r² + π * r²,

A = 3 * π * r².

This area of a hemisphere calculator allows you to find all three types of surface areas of a specific hemisphere. Moreover, you can do every calculation in different units (SI and imperial). If you want to learn more about area unit conversion, check out our area converter!

Now, after we know what are the surface areas of a hemisphere and how to find them, let's try to derive different area of a hemisphere formulas. They can be useful in situations when we don't have the radius given. First of all, there are some basic hemisphere equations you should know:

1. Diameter of a hemisphere: d = 2 * r,
2. Volume of a hemisphere: V = 2/3 * π * r³,
3. Base surface area of a hemisphere: Ab = π * r²,
4. Cap surface area of a hemisphere: Ac = 2 * π * r²,
5. Total surface area of a hemisphere: A = 3 * π * r²,
6. Surface to volume ratio of a hemisphere: A / V = 9 / (2 * r).

The area of a hemisphere formula (for the total area) can be then derived from the above equations. We can obtain even six of them! The equations used by this area of a hemisphere calculator are as follows:

1. Given radius: A = 3 * π * r²,
2. Given diameter: A = 3/4 * π * d²,
3. Given volume: A = ³√[243/4 * π * V²]
4. Given base area: A = 3 * Ab,
5. Given cap area: A = 3/2 * Ac,
6. Given surface to volume ratio: A = 243 * π / (4 * (A/V)²).

As you know, the Earth is approximately a sphere with a radius of almost 6400 km, in which we can specify the northern and the southern hemispheres (specifically speaking, the Earth is a geoid). This division plays an essential role in geography and physics. For example, there is a force called the Coriolis effect, which acts on you whenever you travel by airplane, causing you to get off the course. When you're in the northern hemisphere, you will deflect to the right and in the southern hemisphere to the left. You can find more information about this topic in our Coriolis effect calculator.

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.

What is the Surface Area of Hemisphere?

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.

Surface Area of a Hemisphere Formula

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

How to Find the Surface Area of a Hemisphere?

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

1. Example 1: Find the total surface area of a hemisphere with radius = 21 units. (Use π = 22/7)

   Solution:

   Radius of the hemisphere = 21 units

   Using total surface area of a hemisphere formula,

   Total surface area of a hemisphere = 3 π r2 = 3 × (22/7) × 212 = 3 × 22 × 21 × 3 = 4158 units2

   Therefore, the total surface area of the hemisphere = 4158 units2

2. Example 2: Using the surface area of a hemisphere formula, find the radius of the hemisphere if its total surface area = 648 units2. (Use π = 3.14)

   Solution:

   The total surface area of a hemisphere = 648 units2

   3 π r2 = 648

   3 × 3.14 × r2 = 648

   9.42 × r2 = 648

   r2 = 648/9.42 = 68.78

   r = 8.29 units

   Therefore, the radius of the hemisphere = 8.29 units

3. Example 3: Find the curved surface area of a hemisphere whose radius measures 9 units. (Use π = 3.14)

   Solution:

   The radius (r) of the hemisphere = 9 units. So, let us substitute the value of r = 9 in the formula.

   Curved Surface area of a hemisphere = 2 π r2 = 2 × 3.14 × 92 = 2 × 3.14 × 81 = 508.68 units2

   Therefore, the curved surface area of the hemisphere = 508.68 units2

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FAQs on 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.