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Observing Changes in the Surface Area of Regular Prisms

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Observing Changes in the Surface Area of Regular Prisms

Geometry

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The relationship between the surface area and side lengths of a regular prism is observed.

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Now You Know

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

  • Explain that the surface area of a prism is the sum of the area of the bases and the lateral area.
  • Explain that the area of the base of a regular prism is proportional to the square of the base side length.
  • Explain that the lateral area of a regular prism is proportional to the base side length and the height.

Everything You'll Have Covered

A prism is a three-dimensional geometric figure.

A prism is a polyhedron consisting of two parallel, congruent faces called bases. These bases are perpendicular to the lateral faces of the prism. The lateral faces are always rectangles. A regular prism a prism whose bases are regular polygons.

The surface area of a prism is equal to the sum of the area of its bases and the lateral area.

In a prism, the lateral faces are always rectangles, but the shapes of the bases are different. Therefore, the formula for finding the surface area of a prism will be different, depending on the prism.

In Section 2 all conclusions are made based on these regular prisms: regular triangular, square, regular pentagonal and regular hexagonal prisms.

The table below shows the area of the base, lateral area and surface area of regular triangular, square, regular pentagonal and regular hexagonal prisms.

When h is the height and l is the base side length:

The surface area of a regular prism depends on its base side length and height.

This Activity Object will focus on the changes in a regular prism's surface area when other variables are altered. For instance, students will be able to change the base side length and the height of different types of prisms, and then observe the results from these changes.

  • When only the height is changed -The surface area increases when the height increases, and decreases when the height decreases. When only the height of the prism changes, the area of the bases stays the same.
  • When only the base side length is changed - The surface area increases when its base side length increases, and decreases when the base side length decreases.
  • When the height AND base side length are both changed - because the surface area of a prism depends on its height and the base side length, the surface area will change when the height and base side length of the prism change.

The area of the base of a regular prism is proportional to the square of the base side length.

The area of the bases of a regular prism depends on the shape of the base. The area of the prism's base increases as the base side length increases and decreases as the base side length does the same.

The lateral area of a regular prism is proportional to its height and base side length.

In a prism, the lateral faces are always rectangles. When h is the height and l is the base side length, the lateral area can be calculated by the number of faces multiplied by hl. The lateral area increases when the height increases and also when the base side length increases.

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

  • 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; in a prism the lateral faces are always rectangles
  • length - the distance from one end to the other end of an object
  • net - a two-dimensional pattern of a three-dimensional figure that can be folded to form the figure For example,
  • surface area of a prism - the sum of the area of the bases and the lateral area
  • width - the breadth of an object

Tutorial Details

Approximate Time 20 Minutes
Pre-requisite Concepts Students should know how to calculate the area of rectangles and triangles, how to calculate the lateral area of regular prisms, and the definition and properties of regular polygons.
Course Geometry
Type of Tutorial Skills Application
Key Vocabulary lateral area, regular prisms, surface area