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Similar Solids

Geometry is a very vast subject in mathematics. Basically, geometry is the study of lines, shapes, angles and various problems based on those. There are, generally, two types of geometrical figures studied by us in middle school - two dimensional figures and three dimensional figures. In geometry, we learn about various concepts related to geometrical shapes.

Two most common and very frequently used concepts are:

1) Congruency

2) Similarity


Here, we are going to learn similarity of shapes. Let us first know what does congruence of shapes mean. Two figures are said to be congruent, if they are of the same size as well as shape. i.e. two congruent figures when put one onto another, cover each other fully. On the other hand, two geometrical figures having the same shape are called similar shapes or solids.

When we talk about two dimensions, two shapes are said to be similar if the ratios of their corresponding sides (height or radii) are same. In the same way, in reference to three dimensions, the two solids are known as similar solids if their corresponding sides are in the same ratio. So, in a similar solid, the corresponding angles are of same ratio and corresponding sides are in proportion. In this page, go ahead with us and focus more on similar shapes and solids and properties shared by them.

 

Definition

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Two entities in three-dimensional space are known as similar solid when they have same shape and proportional dimension. The shape of similar solids are same but their size may differ though are proportional to each other. The factor by which the size of solids differ is known as scale factor.
How to find if two given solids are similar? Take each dimension of one solid and find the its ratio with corresponding dimension of the other solid. If the ratios of all the dimensions come same, the solids are similar.

Definition of Similar Solids
Are these triangles similar? Let us check. The ratio between perpendiculars is 10:5 ,that is, 2:1 but the ratio between bases is 6:4 , that is, 3:2. Hence, the proportion of difference in the measure of all dimensions is not equal. It can be concluded that these triangles are not similar.

Scale Factor of Similar Solids

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Scale factor of two similar solids is the ratio of their sizes. Suppose, there is a rectangular solid with length l, height h, and width, w. There is another solid with same shape and respective sizes as $l_1$, $h_1$ and $w_1$.

There scale factor will be:

$s$ = $\frac{l}{l1}$ = $\frac{h}{h_1}$$\frac{w}{w_1}$
Let us take an example and understand. We have figures of two solids given here.
Scale Factor of Similar Solids
There are two dimensions given for these solids of cylindrical shapes. The ratio of the heights of these two solids is 12:9 = 4:3. Again, the ratio of the radius is 8:6 = 4:3. Hence, the size of these solids is varying by a constant ratio which is 4:3. Hence, the scale factor is 4:3. The surface area and volumes of these solids will also vary accordingly.

Properties of Similar Solids

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In geometry, when two solids are similar, the ratio of the lengths of a pair of comparable dimensions is the same as the ratio of any other pair of related sides of the two solids.

Then,

1) Lengths of associated sides are equal.

2) The ratio of the related perimeters and medians equals the ratio lengths of any pair of related sides.

In geometry, when two solids are similar, the ratios of the lengths of a pair of related side of two similar solids are the same.

One of way of presenting this fact is to say that the angle of a solid does not change, if change the size of the solid by altering the lengths with a frequent value. This is not true is non-Euclidean trigonometry. If we boost the size of a (large) solid on the exterior of the Earth, then the angles will increase. We cause as face to the reader the problem of prove for Euclid’s axioms that similar solids indeed have relative sides in geometry.

Areas and volumes of similar solids

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If the scale factor of two similar solids is a:b, then:

1) The surface areas of the solids will be of the ratio $a^{2}:b^{2}$.

2) The volumes of the solids will be in the ratio $a^{3}:b^{3}$.
Suppose, for a given solid the scale factor is 2:3. Then, their surface areas will be of the ratio 4:9 and their volumes will be of the ratio 8:27. Let us check the figure given here.
Area of Similar Solids
The ratio of their dimensions are 8:4 and 12:6. Both equal to 2:1. Hence, the scale factor is 2:1. The ratio of their surface areas will be 4:1.
The ratio of their volumes will be 8:1.


Similar Solids Solved Examples

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Given below are some examples on similar solids:

Example 1:

In the similar solids shown below, the measurements of the sides of one triangle are 5", 6", and 12". Find the sides of a similar solid whose shortest side is 24".
Similar Solids Example

Solution:

Step 1: Sketch the figure and label it.

Step 2: Set up equations that can be used to find the missing sides. Use the fact that the rations of the sides must be the same

$\frac{12}{24}$ = $\frac{5}{p}$ $\frac{12}{24}$ =$\frac{6}{q}$

$\frac{1}{2}$=$\frac{5}{p}$ $\frac{1}{2}$=$\frac{6}{q}$

p=5 $\times$ 2q=6 $\times$ 2

p=10 q=12

One use of similar solids is for indirect measurement. The meaning of indirect extent will be obvious with a picture.

Scale Factor: The ratio of any two corresponding sides in the two similar figures or similar geometric solid figures. So, in the given example the scale factor of the bigger solid to the smaller solid is 24:12 = 2:1.

Example 2: Two similar square prism are given below. Find the scale factor, the ratio of their surface areas, and the ratio of their volumes.

Properties of Similar Solids

Solution: The scale factor is

$\frac{a}{b}$ $\frac{20}{4}$ = 5

The ratio of the surface areas is

$\frac{a^2}{b^2}$ = $\frac{4^2}{1^2}$ = 16

The ratio of the volumes is

$\frac{a^3}{b^3}$ = $\frac{4^3}{1^3}$ = 64

Similar Solids Practice Problems

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Two Triangular Prism A and B are similar. The scale factor of Prism A to Prism B is 3/4, on the basis of given information give the answer of following question (Problem 1 to 3)

Problem 1: If the length of a side of Prism A is 9 feet, what is the length of the
corresponding side of other Prism B?

Problem 2: If Prism A has a surface area of 88.8 feet2, what is the surface area
of Prism B?

Problem 3: Find the surface area of the prism B, If the volume of Prism A is given as 35.1 feet3?

Problem 4: Find the height of the triangular base of Prism A if the height of the triangular base of Prism B, is 3.5 feet.

Answer:

1) 12 feet

2) 157.9 $feet^2$

3) 83.2 $feet^3$

4) 2.6 feet

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