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# ZingPath: Volume and Surface Area of Cones, Cylinders, and Spheres

## Observing Changes in Surface Area of Cylinders                   Searching for

## Volume and Surface Area of Cones, Cylinders, and Spheres

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### Lesson Focus

#### Observing Changes in Surface Area of Cylinders

Algebra Foundations

Observe the relationship between the height, radius, and surface area of a cylinder.

### Now You Know

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

• Explain that the surface area of a cylinder is the sum of the area of the bases and the lateral area.
• Explain that the area of the base of a cylinder is proportional to its radius squared.
• Explain that the lateral area of a cylinder is proportional to its radius and height.

### Everything You'll Have Covered

A right cylinder is a three-dimensional geometric figure.

A right cylinder has two congruent and parallel bases in which the centers of the bases are aligned directly one above the other. The bases do not have to be circles. If the bases are circles, then it is called a right circular cylinder. In the Activity Object, right circular cylinder are used, however, they may also simply be called cylinders.

The surface area of a cylinder can be found by using the formula. The surface area of a cylinder is equal to the sum of the area of the bases and the lateral area. To find the surface area of a cylinder, first we need to calculate the area of the bases. The area of the bases is equal to Next, we find the lateral area, which is the sum of all its faces excluding the bases. The lateral area is equal to the product of the length and the width (h) of the rectangle. So the surface area of the cylinder is equal to  This Activity Object will focus on the changes in a cylinder's surface area when other variables are altered. For instance, students will be able to change the height and radius of the cylinder, and then observe the results from these changes.

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

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

The lateral area of a cylinder is proportional to both its radius and its height.

The lateral area of a cylinder is equal to • When only the height is changed - the lateral area of a cylinder is proportional to its height. What this means is that the lateral area increases when the height increases, and decreases when the height decreases. When only the height of the cylinder changes, the area of the bases stays the same.
• when only the radius is changed - the lateral area of a cylinder is proportional to its radius. What this means is that if the lateral area increases when the radius increases, and decreases when the radius decreases. The circumference of the cylinder's base is always equal to the width of the rectangle, which is its lateral face.
• When the height AND radius are both changed - because the lateral area of a cylinder is proportional to its radius and height, the lateral area will change when the radius and height of the cylinder change.
• The following key vocabulary terms will be used throughout this Activity Object:
• area of the base -the area of either of the two congruent parallel faces of a prism; for a right cylinder, the area of the base = • circumference - the complete distance around a circle; equal to the length of the rectangle of the lateral face of a cylinder
• cylinder - a three-dimensional shape with two parallel circular bases that are congruent and a lateral surface that is curved.
• 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 cylinder is equal to , where r is the radius and h is the height of the cylinder
• 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 circle to any point on the circle
• surface area of a cylinder - the sum of the area of the bases and lateral area of a cylinder; • (SA: surface area, r: radius and h: height)

### Tutorial Details

 Approximate Time 20 Minutes Pre-requisite Concepts area of a circle, circles, cylinders, lateral area, pi ? 3.14, surface area Course Algebra Foundations Type of Tutorial Dynamic Modeling Key Vocabulary area of the base, base, circumference