Surface Area of a Cone and Sphere

Understanding the surface area of different shapes is a fundamental concept in math, especially when solving real-world problems involving measurements. In this lesson, we are going to calculate the surface area of two three-dimensional geometric shapes: a cone and a sphere. Let's explore these shapes, how to derive their surface area formulas, and solve practical examples.

Surface area of a cone

A cone is a three-dimensional shape with a circular base and a top. It looks like a party hat. To find the surface area of a cone, we need to take into account both its circular base and its lateral (side) surface, often called the "oblique surface." Let's break this down into steps.

Components of the surface area of a cone

To determine the surface area of a cone we need to calculate the following:

• Base area: This is simply the area of the circular base of the cone.
• Lateral (curved) surface area: It is the area of the slanted surface of the cone.

Base Area

The base of the cone is a circle. To find the area of a circle, we use the following formula:

Area of the base = π × r²
        

Here, r is the radius of the circular base.

Lateral surface area

The lateral surface area of a cone is determined by its slant height, denoted by l. The formula for lateral surface area is:

Lateral surface area = π × r × l
        

Where r is the radius of the base and l is the slant height.

Below is a visualization of a cone, showing the radius, slant height, and altitude:

Total surface area of a cone

The total surface area of a cone is the sum of its base area and lateral surface area:

Total surface area = π × r² + π × r × l
                    = π × r (r + l)
        

With this formula, let's solve a practical problem:

Example

Consider a cone with radius 3 cm and slant height 5 cm. Let's find its total surface area.

given:
r = 3 cm
L. = 5 cm.

Total surface area = π × r (r + l)
                   = π × 3 (3 + 5)
                   = π × 3 × 8
                   = 24π cm²
        

Rounding π off as you get to about 3.14, you get:

Total surface area ≈ 24 × 3.14 = 75.36 sq.cm
        

Surface area of a sphere

A sphere is a perfectly round three-dimensional shape, like a ball. It has no edges or corners. The most distinctive feature of a sphere is that every point on its surface is the same distance from its center. Let's learn how we find its surface area.

Surface area formula of a sphere

The surface area of a sphere can be found using a simple formula that involves its radius r:

Surface area = 4 × π × r²
        

where r is the radius from the center of the sphere to any point on the surface.

Below is an illustration of a sphere:

Practical example

Example 1

Find the surface area of a sphere of radius 7 cm.

given:
R = 7 cm

Surface area = 4 × π × r²
             = 4 × π × 7²
             = 4 × π × 49
             = 196π cm²
        

Approximating π as 3.14:

Surface area ≈ 196 × 3.14 = 615.44 sq. cm
        

Example 2 - Real-world connection

If you are designing a new basketball, which is usually spherical, and you want its radius to be 12 cm, what material will be needed to cover its surface?

given:
R = 12 cm

Surface area = 4 × π × r²
             = 4 × π × 12²
             = 4 × π × 144
             = 576π cm²
        

Use π ≈ 3.14 to find the projected surface area:

Surface area ≈ 576 × 3.14 = 1809.44 sq. cm
        

Comparison and applications

Understanding the surface area of cones and spheres is essential for applications in a variety of fields, including architecture, engineering, and astronomy. For example, spacecraft designers consider the Earth's spherical shape and optimal re-entry angles. Similarly, in everyday life, food packaging, party hats, ice cream cones, and sports equipment such as soccer balls often incorporate these shapes.

In short:

• The surface area of a cone is made up of the base area and the lateral surface area. The total surface area formula incorporates both, calculated as π × r (r + l).
• The surface area formula for a sphere is 4 × π × r², which is simple and effective for calculating a completely enclosed three-dimensional volume.

The next time you encounter these shapes, you will have a solid understanding of their surfaces and how to calculate them through the formulas discussed. It is a delight to see these shapes not only in the pages of a textbook but also all around us, contributing to creative designs, engineering marvels, and natural phenomena.