Similar Solids: Definition, Properties, Area & Volume

Definitions

Making a scaled version of a solid, creates a similar solid. The scaling happens by multiplying each defining dimension by the same number. This number is the scale factor α and for the sphere, the radius r is the defining dimension. For other solids, the defining dimension(s) are lengths of sides, radii and heights. But it is still the same procedure:

• determine the defining dimension(s) for the solid
• multiply each of those dimensions by a scale factor α

Creating a similar solid has predictable effects on the properties of surface area and volume.

Surface Area of a Sphere

Let's continue with the global map. The ''orange slice'' has a technical name. It's called a gore and resembles a pointed ellipse. The shape reminds Henri of his trusty canoe.

A gore

Imagine the bottom tip at the origin where both x and y are zero. We are at the south pole of the globe. Traveling to the north pole is half the circumference of the globe. Half the circumference is half of 2π r or just π r. Thus, the values for y are from 0 to π r.

What about x?

The sine of 0 is zero and the sine of π is zero. The gore's x value is zero at two places: y = 0 and y = π r. Consider sin(y/r). When y is zero, sin(0/r) is zero. When y is π r, sin(π r/r) = sin(π) = 0. The maximum width of the gore is at the ''equator'' where the y-axis midpoint is π r/2. Meaning sin(y/r) is sin(π r/(2r)) = sin(π/2) = 1. Great! sin(y/r) works well in the equation for x.

The larger the r, the larger the sphere and the larger the gore's width. The gore subtends an angle φ that relates the width of the gore to the radius of the circle through the tangent. The tangent of φ/2 is the half the width of the gore divided by r. Thus, at the midpoint of y, the half-width is r tan(φ/2). Combining this with sin(y/r):

Why the plus/minus? To get the full width, this equation is evaluated twice; once for the plus and once for the minus.

To find the area of one gore, integrate this function over y. This gives 4 r2 tan(φ/2). In terms of n gores, φ is 2π/n meaning φ/2 is π/n. Thus, the total surface area of the sphere with n gores is A = 4 nr2 tan(π/n).

As n gets larger, π/n gets smaller, and for these small values, π/n approximates tan(π/n). Thus, as n get larger, A = 4 nr2 tan(π/n) becomes A = 4 nr2 (π/n). The n's cancel leaving A = 4π r2 (the equation for the surface area of a sphere).

Henri enjoyed the journey from the south pole to the north pole but is glad to have finally arrived at the sphere surface area equation, A = 4π r2.

Scaling

A sphere with radius 2 has been mapped with six gores. Below it is a similar sphere with radius 4:

Sphere surface areas

A rectangle with length 2π r and width π r would fit over a gore map. The smaller sphere with r = 2 has length 2π r = 2π(2) = 4π and width π r = π2 = 2π. The area of the rectangle is length times width = (4π)(2π) = 8 π^2. For the larger sphere of radius r = 4, the area is (8π)(4π) = 32π; four times greater surface area.

Thus, scaling by α increases the surface area by a factor of α^2.

This is important to Henri because scaling his canoe by α means α^2 more surface area and α^2 more paint to cover his canoe.

Volume of a Sphere

A sphere of radius r has a volume V = 4π r3/3. Here's a sphere with radius r = 2:

Small sphere

If a scale factor α of 2 is used to create a similar sphere, the new radius is α(2) = 4 but the new volume is α^3 = 8 times larger. Clearly, the smaller sphere easily fits inside the larger sphere whose volume is 8 times larger.

Small sphere inside large sphere

To see where α^3 comes from, substitude αr for r in V= 4π r3/3. We get Vnew = 4π (αr)3/3 = (α)^3 4π r3/3 = (α)^3 V.

Other Solids

A similar solid is produced by multiplying the defining dimension(s) by a scale factor α. The new surface area will be the α^2 times the old surface area. The new volume will be α^3 times the old volume. This holds true for all solids.

A partial list of solids and defining dimensions:

• sphere: radius r
• cylinder: radius r, length l
• cube: side s
• cone: radius r, height h
• rectangular solid: length l, width w, height h

Lesson Summary

A similar solid is created by making a scaled version of the original solid. The scaling is accomplished by multiplying each defining dimension by a scale factor. The defining dimension is specific for each type of solid. For a sphere it is the radius r; for a cube it is the side s. Gores are pointed elliptical slices used to transform a sphere into a map and useful for visualizing the surface area of a sphere. The surface area and the volume are propertiesof a solid. The surface area of a scaled solid is the square of the scale factor times the surface area of the original solid. For volumes, the original volume is multiplied by the scale factor cubed.