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.
Subsection3.5.1Measuring Interiors
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.
Checkpoint3.5.1.
Match the correct type of measurement to each scenario.
Reread the above paragraphs.
The amount of ice that can fit in a cooler.
volume
The amount of vinyl to cover a floor.
area
The amount of wood to build a box.
surface area
The amount of wire to form a polygon’s boundary.
perSimeter
Subsection3.5.2Exploring Growth in Two-Dimensional Figures
How does the change in the length in one or more directions affect the area of a figure’s interior? How does it affect the size of its boundary?
Exploration3.5.1.Growth in Area.
(a)Growth and Area.
For each of the following rectangles compute the area:
(i)
A rectangle with length 3 inches and width 5 inches.
(ii)
A rectangle with length 4 cm. and width 7 cm.
(iii)
A rectangle with length 2 ft and width 10 ft.
(iv)
A rectangle with length \(\ell\) units and width \(w\) units.
(b)
For each rectangle in Task 3.5.1.a, multiply the length by 3 and keep the width unchanged. Then compute the area of the enlarged rectangle.
(i)
A rectangle with length 3 inches and width 5 inches.
Hint.
The length is 9 and the width is still 5.
(ii)
A rectangle with length 4 cm and width 7 cm.
Hint.
Multiply the length 4 by 3. Keep width 7.
(iii)
A rectangle with length 2 ft and width 10 ft.
(iv)
A rectangle with length \(\ell\) units and width \(w\) units.
Hint.
Once again we multiple \(\ell\) by 3 and keep \(w\text{.}\)
(v)
Describe how multiplying the length of a rectangle by a number \(k\) affects the area of the rectangle.
(c)
Suppose you know the area of a triangle is 52 and then you multiply the base by 5. What will the area of the resulting triangle be?
(d)
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?
Hint.
The height is unchanged. Only the area is multiplied by 1.2.
(e)
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 3.5.1.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 3.5.1.a..
(i)
A rectangle with length 3 inches and width 5 inches.
Hint.
The new length will be \(4\cdot 3=12\) inches and the new height is \(4\cdot 5=20\) inches.
(ii)
A rectangle with length 4 cm and width 7 cm.
Hint.
Again multiply both the length and the width by 4.
(iii)
A rectangle with length 2 ft and width 10 ft.
(iv)
A rectangle with length \(\ell\) units and width \(w\) units.
(v)
Suppose a triangle has area 52 and you multiply both the height and the base by 5. What is the area of the resulting triangle? Why?
(vi)
Suppose a trapezoid has area 100 square meters and you then multiply the height and both bases by 1.2. What is the area of the resulting trapezoid?
(f)
Suppose a circle has area \(64\pi\) square units. If you multiply the radius by 5, what is the area of the resulting circle? Explain.
(g)
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.
Exploration3.5.2.Growth in Perimeter.
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.
(a)
Determine the perimeter of each rectangle in Task 3.5.1.a.
(i)
A rectangle with length 3 inches and width 5 inches.
Hint.
The perimeter is the length of the boundary of a figure. For a rectangle, add the top, right, bottom, and left side lengths.
(ii)
A rectangle with length 4 cm. and width 7 cm.
(iii)
A rectangle with length 2 ft and width 10 ft.
(iv)
A rectangle with length \(\ell\) units and width \(w\) units.
(b)
For each rectangle in Task 3.5.1.a, multiply both the length and width by 4. Then compute the perimeter of the enlarged rectangle.
(i)
A rectangle with length 3 inches and width 5 inches.
Hint.
Use the original rectangles, not the ones where length has been multiplied by 3.
(ii)
A rectangle with length 4 cm and width 7 cm.
(iii)
A rectangle with length 2 ft and width 10 ft.
(iv)
A rectangle with length \(\ell\) units and width \(w\) units.
(c)
Suppose a rectangle has perimeter 52 and you multiply both the height and the base by 5. What is the perimeter of the resulting rectangle? Why?
(d)
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.
Figure3.5.2.A GeoGebra applet to see growth in the perimeter of a triangle.
(e)
Suppose a circle has radius 7.
(i)
What is the circumference of this circle?
(ii)
Now multiply the radius by 2 to get 14. What is the circumference of the larger circle?
(f)
Suppose a circle has perimeter \(64\pi\) units. If you multiply the radius by 5, what is the circumference of the resulting circle?
(g)
Does circumference behave like perimeter or area? Explain.
(h)
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?
Before we look at three dimensional figures, we observe that area and perimeter grow in different ways. SAY MORE HERE! PERHAPS A CHECKPOINT.
Exploration3.5.3.Growth in Three-Dimensional Solids.
(a)
Suppose a rectangular prism has width 2 inches, length 5 inches, and depth 7 inches.
(i)
Compute the volume of the prism.
(ii)
What happens to the volume if just the width is multiplied by 3?
(iii)
What happens to the volume of the prism if the width and the length are both multiplied by 3 but the depth remains the same?
(iv)
What happens to the volume if all three dimensions, width, length, and depth, are multiplied by 3?
(v)
In a sentence or two, describe how multiplying one or more dimensions by a constant affects the volume of a rectangular prism.
(b)
In this task we explore a connection between volume of prisms and cylinders and the depth of the solid.
(i)
The base of a prism is a square with side length 7 inches.
(A)
What is the volume of the prism if the depth is 1 inch?
(B)
What is the volume of the prism if the depth is 3 inches?
(C)
What is the volume of the prism if the depth is \(d\) inches?
(ii)
The base of a prism is a circle with radius 5 inches.
(A)
What is the volume of the prism if the depth is 1 inch?
(B)
What is the volume of the prism if the depth is 4 inches?
(C)
What is the volume of the prism if the depth is \(d\) inches?
(iii)
The base of a prism is a triangle with base 4 inches and height 9 inches.
(A)
What is the volume of the prism if the depth is 1 inch?
(B)
What is the volume of the prism if the depth is 3 inches?
(C)
What is the volume of the prism if the depth is \(d\) inches?
(iv)
A prism has a base with area 56 square inches.
(A)
What is the volume of the prism if the depth is 1 inch?
(B)
What is the volume of the prism if the depth is 3 inches?
(C)
What is the volume of the prism if the depth is \(d\) inches?
(v)
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.
