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Math Foundations

Observe the relationship between the surface area and dimensions of a square pyramid.

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

- After completing this Activity Object, learners will be able to:
- Explain that the surface area of a square pyramid is the sum of the area of the base and the lateral area.
- Recognize the formula for the area of the base, lateral area and surface area of a square pyramid.
- Explain that the area of the base of a square pyramid is proportional to the base side length squared.
- Explain how the slant height of a square pyramid changes when the height or base side length changes.
- Explain how the lateral area of a square pyramid changes when the height or base side length changes.

A square pyramid is a three-dimensional geometric figure.

A square pyramid has a square base with four isosceles triangular faces meeting at a point (the apex).

The surface area of a square pyramid can be found by using the formula

The surface area of a pyramid is equal to the sum of its area of the base and the lateral area. To find the surface area of a square pyramid, first we need to calculate the area of the base. The area of the base is equal to where a is the base side length. Next, we find the lateral area which is the surface area of the solid figure excluding the area of the base. The lateral area is equal to , where a is the base side length and l is the slant height of the pyramid. So, the surface area of the pyramid is equal to

The surface area of a square pyramid changes when its base side length or height changes.

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

- When the height is changed - the surface area of a pyramid depends on its height. What this means is that the surface area increases when its height increases, and decreases when the height decreases.
- When the base side length is changed - the surface area of a pyramid is on its base side length. What this means is that the surface area increases when its base side length increases, and decreases when the base side length decreases.

The area of the base of a square pyramid is proportional to the base side length squared.

The area of the base of a square pyramid is equal to The area of the pyramid's base increases as the base side length increases and decreases as the base side length does the same.

The slant height of a square pyramid changes when the height or base side length changes.

The slant height of a square pyramid is equal to The slant height increases (decreases) when the height increases or when the base side length increases (decreases).

The lateral area of a square pyramid depends on height and is directly proportional to base side length.

The lateral area of a square pyramid is equal to The lateral area increases (decreases) when the slant height (decreases) or when the base side length (decreases).

- The following key vocabulary terms will be used throughout this Activity Object:
- height - the perpendicular distance to the base from the apex
- 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 pyramid is equal to where a is the base side length and l is the slant height of the pyramid
- net - a two-dimensional pattern of a three-dimensional figure that can be folded to form the figure
- For example,
- slant height - the height of any lateral face of a pyramid
- square pyramid - a polyhedron with a square base and four triangular sides meeting at a point (the apex)
- surface area of a square pyramid - the sum of the area of the base and the lateral area;

Approximate Time | 20 Minutes |

Pre-requisite Concepts | lateral area, nets, Pythagorean theorem, slant height, square pyramids, surface area |

Course | Math Foundations |

Type of Tutorial | Dynamic Modeling |

Key Vocabulary | Area of the base, base side length, height |