Length, area, volume, and surface area have practical applications. A landowner may be interested about the area of the space available for planting and building. If a fence is desired, a different computation is required. Similarly, the amount of water needed to fill a pool and the quantity of waterproofing needed to protect the floor and walls of the pool require different geometrical formulas.
What does it mean to measure the interior of a geometric object? It depends on the object and the context. A line segment is a finite one-dimensional object bounded by two endpoints. Length can be thought of as the measure of the interior space of a line segment, namely the distance between the endpoints. In a one-dimensional space, all objects lie on a single line. we move only forward and backward along that line. There is no height or width.
Two-dimensional geometry is the study of objects lying in a plane. Two-dimensional objects include polygons, circles, and other figures that can be drawn on a flat surface. A closed two-dimensional figure will be bounded by linear and/or curved segments that separate the interior of the figure from the rest of the plane, which we refer to as the exterior of the figure. The measure of the interior of a two-dimensional figure is called its area and involves both length and width. Of course, the boundary of the planar figure can also be measured, but the boundary consists of one-dimensional objects; namely, line segments and curves.
In three-dimensional geometry, we have length, width, and a third direction called depth (or height). The geometrical solids studied in Chapter 3, including prisms, cylinders, cones, pyramids, and polyhedra, are examples of three-dimensional solids. The measurement of the interior space of a geometrical solid is called volume. Each three-dimensional solid is bounded by surfaces which may include polygons, circles, and other two-dimensional objects. Since these surfaces are two-dimensional, we use area to describe the size of the boundary. The total of the area of these bounding surfaces is given the appropriate name, surface area.
Suppose you know that the area of a trapezoid is 100 and then you multiply the height of the trapezoid by 1.2. What is the area of the resulting trapezoid?
Next we will explore what happens when we multiply both length and width (or base and height) by the same number. For each rectangle in Task 3.5.2.a, multiply both the length and width by 4. Then compute the area of the enlarged rectangle. Then compare the resulting areas to those computed in Task 3.5.2.a..
Area is the measure of the interior of a two-dimensional figure. Refer to your work in this exploration as you explain how area is effected by stretching two-dimensional figures both vertically and horizontally.
We can also measure the perimeter of two-dimensional figures. In this task, we will explore whether multiplying the length and/or height of a figure by a fixed number has a predictable effect on the perimeter of the figure.
Suppose a triangle has perimeter 20. If we multiply the height and base of a triangle by 3, must the perimeter equal 60? Support your answer using examples. The following GeoGebra applet will allow you to explore different shaped triangles.
Perimeter is the measure of the boundary of a two-dimensional figure. Refer to your work in this section of the exploration as you explain how perimeter is affected by stretching two-dimensional figures both vertically and horizontally by the same amount. Can we make the same claim if we stretch vertically by one factor and horizontally by another factor?
Explain how the volume of a prism or cylinder is related to the volume of a solid with the same base but depth 1 unit. Describe how this agrees with the volume formulas for prisms and cylinders.
