In these lessons, we will learn the scale factors of similar figures, the ratio of lengths, perimeters, areas and volumes of similar figures; suitable for Grade 7 or Grade 8 Math.

**Related Pages**

Scale Factor & Shapes

Scaling and Area

Surface Area of Similar Figures

Grade 8

A scale factor is the factor by which all the components of an object are multiplied in order to create a proportional enlargement or reduction.

The following diagram shows an example of scale factor. Scroll down the page for more examples and solutions on how to use scale factors.

**How to use scale to determine the dimensions of a proportional model?**

Define scale factor.

Scale factor is similar to a unit scale except no units are given. Scale factor is a ratio
comparing the scaled measurement to the actual measurement.

**How to use scale factor to sketch a proportional scale model?**

**Example:**

- The scale factor of a model car is 1:24. If the actual car is 18 ft long and 8 ft wide, how long and wide will the model car be in inches?
- You need to sketch a ladybug using a scale factor of 12. If a ladybug is 8 mm long and 4 mm wide, what will be the length and width of the sketch be in cm?

**A brief course in scale factor for similar geometric figures**

Scale Factor is defined as the ratio of any two corresponding lengths in two similar geometric figures.

Similar Figures are figures such that:

- Corresponding angles are congruent.
- Corresponding side lengths are proportional.

This video explains how to find the ratio of areas and ratios of perimeters for similar polygons.

Ratio of perimeters = ratio of sides

Ratio of areas = (ratio of sides)^{2}

**Scale Factor/Perimeter Ratio/Area Ratio**

**Example:**

Given that the polygon in each pair are similar. Find the scale factor, perimeter ratio and area ratio.

**Areas and Perimeters of Similar Figures**

This video discusses how to find the ratio of the perimeters and the ratio of the areas of similar
figures from the scale factor. Also how to use these ratios to find missing perimeters and areas.

If the scale factor of two similar figures is a/b, then

- the ratio of their perimeters is a/b and
- the ratio of their areas is a
^{2}/b^{2}

**Examples:**

- The trapezoids at the right are similar. The ratio of the lengths of corresponding sides is 6/9 or 2/3.

a) What is the ratio (smaller to larger) of the perimeters?

b) What is the ratio (smaller to larger) of the areas? - Two similar polygons have corresponding sides in the ratio 5:7.

a) What is the ratio (larger to smaller) of their perimeters?

b) What is the ratio (larger to smaller) of their areas? - The area of the smaller regular pentagon is about 27.5 cm
2. What is the best approximation for the area of the larger regular pentagon?< - If the area of the smaller triangle is about 39 ft
^{2}, what is the area of the larger triangle to the nearest tenth? - The triangles are similar. What is the scale factor? What is the ratio of their perimeters?
- The areas of two similar rhombuses are 48m
^{2}and 128m^{2}. What is the ratio of their perimeters?

**How does scale factor impact side lengths, perimeter, area, and angles?**

If the scale factor from A to B is x then

The side lengths of B will be x times larger than A

The perimeter of B will be x times larger than A

The area of B will be x^{2} times larger than A

A and B will have the same shape and angles.

Scale Factor, Length, Area and Volume for similar shapes

Ratio of lengths = ratio of sides = scale factor

Ratio of surface areas = (ratio of sides)^{2} = (scale factor)^{2}

Ratio of volume = (ratio of sides)^{3} = (scale factor)^{3}

**Surface Areas And Volumes Of Similar Solids**

Similar solids have the same shape, and all their corresponding dimensions are proportional.

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

• the ratio of their corresponding areas is a^{2}:b^{2}

• the ratio of their volumes is a^{3}:b^{3}

**Examples:**

- Are the two rectangular prisms similar? If so, what is the scale factor of the first figure to the second figure?
- The square prisms at the right are similar. What is the scale factor of the smaller prism to the larger prism?
- What is the scale factor of two similar prisms with surface areas 144 m
^{2}and 324 m^{2}? - The volumes of two similar solids are 128 m
^{3}and 250 m^{3}. The surface area of the larger solid is 250 m^{2}. What is the surface area of the smaller solid?

**Similar Figures, Scale Factor, Area & Volume Ratios**

**Examples:**

- The scale factor between two similar figures is given. The surface area and volume of the
smaller figure are given. Find the surface area and volume of the larger figure.

Scale factor 5:6

SA: 275 cm^{2}

V: 3000 cm^{3} - Some information about the surface area and volume of two similar solids has been given. Find the missing value.

**3D Figures - Scale Models and Factors**

Learn about scaling 3D figures using scale factor

Problem: A 6 cm by 2 cm rectangular prism is built form small rectangular prisms of length 3 cm.

a) What is the scale factor from the smaller to the larger model?

b) Find the width and height of the smaller rectangular prisms.

c) Compare the surface area of the two rectangular prisms.

d) Compare the volume of the two rectangular prisms.

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