**Introduction**

The lateral area of a hemisphere is an essential mathematical concept that students learn to grasp and apply in various scenarios. This comprehensive article seeks to provide an in-depth understanding of the topic, its key concepts, and real-life applications. We will explore the topic’s grade appropriateness, related math domain, Common Core Standards, and practical examples to help students and educators.

**Grade Appropriateness**

The concept of the lateral area of a hemisphere is typically introduced in middle school, around the 7th or 8th grade, and extends into high school geometry and pre-calculus courses.

**Math Domain**

The lateral area of a hemisphere belongs to the domain of geometry, a branch of mathematics that deals with shapes, sizes, and properties of figures.

**Applicable Common Core Standards**

The relevant Common Core Standards for studying the lateral area of a hemisphere are:

*7.G.B.4:* Know the formulas for the area and circumference of a circle and use them to solve problems.

*8.G.C.9:* Know the formulas for the volumes of cones, cylinders, and spheres, and use them to solve real-world problems.

*G.GMD.A.1:* Give an informal argument for the formulas for the circumference of a circle, the area of a circle, the volume of a cylinder, a pyramid, and a cone.

**Definition of the Topic**

The lateral area of a hemisphere refers to the curved surface area of a hemisphere, which is half of a sphere. It does not include the flat circular base.

**Key Concepts**

*Hemisphere*

A hemisphere is half of a sphere divided along a great circle, usually referred to as the equator.

*Lateral area*

The lateral area is the curved surface area of a solid, excluding its bases.

**Discussion with Illustrative Examples**

**Surface Area** is the total area covering the surface or outside part of a three-dimensional object. The formula 4πr^{2} is used to determine the surface area of a sphere, so half of that is the curved surface area of a hemisphere.

The word hemisphere has the prefix hemi, which came from the Greek term hēmi meaning “half.” So, the hemisphere means half of a sphere.

To get the **lateral area of a hemisphere (L),** we simply use the formula:

L = 2πr^{2}

where:

r is the radius of the hemisphere.

π (pi) is approximately equal to 3.14159.

Remember that the lateral surface area of a hemisphere does not include the area of the circular base. If we add the area of the base to the lateral surface area, we will get the total surface area.

**Examples with Solution**

**Example 1**

Calculate the lateral area of a hemisphere with a radius of 5 cm.

**Solution**

L = 2πr^{2}

L =(2)(3.14159)(5^{2})

L ≈ 157.08 cm^{2}

Therefore, the lateral area of the hemisphere is **157.08 cm**** ^{2}**.

Remember always to write the unit of measurement on your final answers! For surface areas, the answers should be in square units.

**Example 2**

Find the lateral surface area of a hemisphere with a radius of 7 cm.

**Solution**

L= 2πr^{2}

L=(2)(3.14159)(7^{2})

L≈ 307.88 cm^{2}

Therefore, the lateral surface area is **307.88 cm**^{2}**.**

**Example 3**

Determine the lateral surface area of a hemisphere whose diameter is 51 cm.

**Solution**

The radius of the hemisphere is 23÷2 = 11.5 m since the radius is equal to half of the diameter.

L = 2πr^{2}

L =(2)(3.14159)(11.5^{2})

L ≈ 850.95 m^{2}

Therefore, the lateral surface area is **850.95 m**^{2}**.**

**Real-life Application with Solution**

**Example 1**

A hemispherical dome with a radius of 10 meters needs to be painted. Work out how many square meters of paint will be required to cover the curved surface area.

**Solution**

L = 2πr^{2}

L =(2)(3.14159)(10^{2})

L ≈ 628.32 m^{2}

The painter will require **628.32 square meters** of paint to cover the curved surface area of the dome.

**Example 2**

What is the lateral surface area of the coconut cut into a hemisphere with a diameter of 6 inches used by a stall selling coconut juice?

**Solution**

Since the diameter of the coconut is 6 inches, its radius is 3 inches. Calculating the lateral surface of its hemisphere, we have,

L = 2πr^{2}

L =(2)(3.14159)(3^{2})

L ≈ 56.55 in^{2}

The lateral surface of the coconut cut into a hemisphere is **56.55 square inches**.

**Practice Test**

A. Complete the table below. Round off your answers to the nearest hundredths.

B. Answer the following problem.

1. Calculate the lateral area of a hemisphere with a radius of 7 cm.

2. Find the lateral area of a hemisphere with a diameter of 12 meters.

3. A hemispherical container has a radius of 4 meters. How much material is needed to create a cover for the curved surface area?

4. What is the lateral area of a hemisphere with a circumference of 18.85 cm?

5. A hemisphere has a lateral area of 100 cm^{2}. Find the radius of the hemisphere.

**Frequently Asked Questions (FAQs)**

**What is the formula for a hemisphere’s lateral area?**

The formula for a hemisphere’s lateral area is Lateral Area = 2πr^{2}, where r is the radius of the hemisphere.

**How does the lateral area of a hemisphere differ from the total surface area?**

The lateral area of a hemisphere refers only to the curved surface area, whereas the total surface area includes both the curved surface and the flat circular base.

**Can the lateral area of a hemisphere be equal to the total surface area?**

No, the lateral area of a hemisphere cannot be equal to the total surface area, as the total surface area includes the curved surface and the flat circular base, making it larger than the lateral area.

**How does a hemisphere’s lateral area compare to a sphere’s surface area?**

The lateral area of a hemisphere is exactly half the surface area of a sphere with the same radius since a hemisphere is half of a sphere.

**Is there a relationship between a hemisphere’s volume and lateral area?**

There is no direct relationship between a hemisphere’s volume and lateral area. However, both can be calculated using the hemisphere’s radius. The volume formula is $\frac{2}{3}$r^{3}, and the lateral area formula is 2πr^{2}.

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