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# ZingPath: Surface Area

## Observing Changes in Surface Area of Cones              Searching for

## Surface Area

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### Lesson Focus

#### Observing Changes in Surface Area of Cones

Geometry

Students observe the changes in the surface area of a cone when the height and the radius are changed.

### Now You Know

After completing this tutorial, you will be able to complete the following:

• Explain that the surface area of a cone is the sum of the area of the base and the lateral area.
• Explain that the area of the base of a cone is directly proportional to its radius squared.
• Recognize that the lateral area of a cone can be calculated with the formula LA = ?rl.
• Explain the relationship between the lateral area of a cone and its height.
• Explain the relationship between the lateral area of a cone and its radius.

### Everything You'll Have Covered

A cone is a three-dimensional geometric figure

A cone is a shape whose base is a circle and whose sides taper up to a point. • arc - a continuous part of a circle between any two of its points; the measure of an arc is the measure of the angle formed by two radii with endpoints at the endpoints of the arc
• sector - the region bounded by two radii of the circle and the arc they intercept

In the diagram below, an arc can be identified between points B and C on the circumference of the circle. The shaded area represents a sector. The surface area of a cone can be found by using the formula .

The surface area of a cone is equal to the sum of the area of the base and the lateral area. To find the surface area of a cone, first we need to calculate the area of the base. The area of the base is equal to .

Next, we find the lateral area, which is the sum of all its faces excluding the base. The lateral area is equal to the area of the sector, which would be .

To find the length of l (slant height), use the relation of the length of the arc BC and the circumference of the base. The length of arc BC is equal to the circumference of the base ( ).

At the same time, the length of the arc is equal to . Set these two equations equal to one another: . We find that . Now we can replace one l in the area of the sector formula to get , which simplifies to .

The area of the base of the cone is . So, the surface area of the cone is equal to . This Activity Object will focus on the changes in a cone's surface area when other variables are altered. For instance, students will be able to change the height and radius of the cone, and then observe the results from these changes.

• When only the height is changed - the surface area increases when its height increases, and decreases when the height decreases. When only the height of the cone changes, the area of the base stays the same.
• When only the radius is changed - the surface area increases when the radius increases, and decreases when the radius decreases.
• When the height AND radius are both changed - the surface area will change when the height and radius of the cone change.

The area of the base of a cone is proportional to its radius squared.

The area of the base is equal to . The area of the cone's base increases as the radius increases and decreases as the radius does the same.

The lateral area of a cone changes when its radius or its height changes.

The lateral area increases when the radius or the height increases. Likewise, the lateral area decreases when the radius or height decreases.

The following key vocabulary terms will be used throughout this Activity Object:

• arc - a continuous part of a circle between any two of its points; the measure of an arc is the measure of the angle formed by two radii with endpoints at the endpoints of the arc
• circumference - the complete distance around a circle; equal to the length of the arc of the sector of the lateral face of a cone
• cone -a three-dimensional figure that has one circular base and one vertex
• height - the perpendicular distance to the base
• lateral area - the sum of the surface areas of all the faces of a solid, excluding the base of the solid. The lateral area of a cone is equal to , where r is the radius and l is the slant height of the cone
• net - a two-dimensional pattern of a three-dimensional figure that can be folded to form the figure For example, • radius - the length of a line segment that connects the center of the circle to any point on the circle
• sector - the region bounded by two radii of the circle and the arc they intercept
• slant height- the height of any lateral face of a cone
• surface area of a cone - the sum of the area of the base and lateral area of a cone; ### Tutorial Details

 Approximate Time 20 Minutes Pre-requisite Concepts Students should know the definitions of circle, cone, slant height, and net, and be able to use formulas to determine the area of a circle, circumference of a circle, lateral area, and surface area. Course Geometry Type of Tutorial Skills Application Key Vocabulary area of base, cone, height