**Introduction**

Geometry is a fascinating field of mathematics that allows us to measure and understand the world around us in new and exciting ways. One such measurement is the surface area of a hemisphere, which is an essential concept for students to grasp. The surface area of a hemisphere will be thoroughly explained in this article.

**Grade Appropriateness**

The concept of surface area, including a hemisphere, is typically introduced in the 7th grade. However, it can be revisited and further explored in higher grades, particularly in high school Geometry and Calculus classes.

**Math Domain**

This topic falls under the domain of Geometry, which deals with the properties, measurement, and relationships of points, lines, angles, surfaces, and solids.

**Applicable Common Core Standards**

The Common Core Standards related to this topic are:

*7.G.B.4:* Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

*7.G.B.6:* Solve real-world and mathematical problems involving area, volume, and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

**Definition of the Topic**

In simple terms, a **hemisphere** is half of a sphere. It is what you get when you cut a sphere right down the middle. The **surface area of a hemisphere** includes the area of its circular base and the area of the curved part (like the surface area of a sphere).

**Key Concepts**

**Area = 3rπ**** ^{2}**, where

*r*is the hemisphere’s radius, is the

*formula for a hemisphere’s total surface area.*

This formula is derived by adding the curved surface area of the hemisphere (2πr²) and the base area (πr²).

**Discussion with Illustrative Examples**

The total area of a three-dimensional object’s surface or exterior is known as **surface area.**

The word **hemisphere** has the prefix hemi, which came from the Greek term hēmi meaning “half.” So, the hemisphere means half of a sphere. A hemisphere is composed of a curved surface that is half of a sphere and a circular flat base.

To get its total surface area, we need to add the area of its curved surface and the area of its circular base.

*Formulas for Calculating the Surface Area of a Hemisphere*

Curved Surface Area (C)

C = 2πr^{2}

Circular Base Area (B)

B = πr^{2}

Total Surface Area (T)

T = 3πr^{2}

For these equations,*r* = radius of the hemisphere

*Remember that a hemisphere’s radius is a line drawn from the base’s center to any point on the curved surface.*

Let us consider a hemisphere with a radius of 4 units.

To find its total surface area, we plug the radius into the formula:

T=3πr^{2}

T=(3)(π)(4)^{2}

T=3π(16)

T=48π square units

Thus, the total surface area of the hemisphere is **48π square units.**

**Examples with Solution**

**Example 1**

A hemisphere has a radius of 7 cm. What is its surface area?

**Solution**

T = 3πr^{2}

T =(3)(π)(7 cm)^{2}

T = 3π (49 cm^{2})

T=147π cm^{2}

The total surface area is **147π cm**^{2}**.**

**Example 2**

Find the hemisphere’s total surface area and curved surface area with a radius of 8 m.

Use 3.14 as the estimated value of π.

**Solution**

C=2πr²

C=(2)(3.14)(8 m)^{2}

C=401.92 m^{2}

T=3πr²

T=(3)(π)(8 m)^{2}

T=602.88 m^{2}

Therefore, the curved surface area is **401.92 m**** ^{2}**, and the total surface area is

**602.88 m**

**.**

^{2}**Example 3**

Determine the curved surface area and total surface area of a hemisphere whose diameter is 51 cm. Use 3.14 as the estimated value of π.

**Solution**

Since the radius is one-half of the diameter, the radius of the hemisphere is 51 ÷ 2 = 25.5 cm.

C = 2πr^{2}

C =(2)(3.14)(25.5)^{2}

C = 4 083.57 cm^{2}

T = 3πr^{2}

T =(3)(3.14)(25.5)^{2}

T = 6 125.36 cm^{2}

Therefore, the curved surface area is **4 083.57 cm**** ^{2}**, and the total surface area is

**6 125.36 cm**

**.**

^{2}**Example 4**

A hemisphere has a surface area of 675π cm². What is its radius?

**Solution**

We rearrange the formula to solve for r:

Surface Area = 3πr²

r=$\sqrt{\frac{Surface Area}{3π}}$

r=$\sqrt{\frac{675π}{3π}}$

r=$\sqrt{225}$

r=15

Thus, the radius of the hemisphere is **15 cm.**

**Real-life Application with Solution**

Mario is working on his mini globe hemisphere project has a radius of 50 centimeters. What is the surface area of the hemisphere? Use π =3.14.

Solution

T = 3πr^{2}

T =(3)(3.14)(50)^{2}

T = 23550

Therefore, the surface area of the mini globe hemisphere project is **23550 cm**** ^{2}**.

**Practice Test**

A. Complete the table below. Use 3.14 as the value of π and round off your answers to the nearest hundredths.

B. Answer the following problem.

1. A hemispherical cake has a diameter of 11 inches. Calculate its total surface area.

2. Find the surface area of a hemisphere with a radius of 10 cm.

3. A hemisphere has a surface area of 100π cm². What is its radius?

4. If the surface area of a hemisphere is 500π m², what is the radius?

5. Calculate the surface area of a hemisphere with a radius of 5 m.

*Answers:*

A.

1. Radius = 4 cm, Curved Surface Area = 100.48 cm^{2}, Total Surface Area = 150.72 cm^{2}.

2. Radius = 7 m, Curved Surface Area = 307.72 m^{2}, Total Surface Area = 461.58 m^{2}.

3. Radius = 5.5 cm, Curved Surface Area = 189.97 cm^{2}, Total Surface Area = 284.96 cm^{2}.

B.

1. Total Surface Area = 90.75π square inches

2. Total Surface Area = 300π square centimeters

3. Radius ≈ 5.77 cm.

4. Radius ≈ 12.91 m.

5. Total Surface Area = 75π m².

**Frequently Asked Questions (FAQs)**

### Do spheres and hemispheres have the same surface area?

No, the surface area of a hemisphere is half of a sphere’s surface area plus the area of the base of the hemisphere, which is a circle.

### How is the formula for the surface area of a hemisphere derived?

The formula for a hemisphere’s surface area is created by adding the hemisphere’s curved surface area, which is 2πr², and the area of the circular base, which is πr². Hence, the formula for a hemisphere’s surface area results in 3πr².

### Does the surface area of a hemisphere include the base?

Yes, the total surface area of a hemisphere includes both the curved surface area and the base area.

### How does the radius of a hemisphere affect its surface area?

The surface area of a hemisphere is directly proportional to the square of its radius. If the radius doubles, the surface area increases by a factor of four.

### What are some real-world applications of calculating the surface area of a hemisphere?

There are many real-world applications, such as calculating the amount of materials necessary to create a dome, the surface area of a planet (which is a sphere, but the concept is similar), or the area of a dome that needs to be painted, as in our example.

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