DEFINITION

Similar shapes are shapes that have the same shape but different sizes.

This means that one of them can be bigger or smaller, but they look exactly alike. All the corners (angles) are the same, and the sides are in the same ratio.

For example:
If you draw a small triangle and then draw a big triangle that looks exactly the same, they are called similar triangles.

In short:
Similar shapes have the same shape but different sizes.

EXPLANATION

When two shapes are similar, it means:

1. The shapes look exactly the same.


2. Their angles (corners) are equal.


3. The sides of one shape are longer or shorter than the sides of the other shape in the same ratio (proportion).

If we make a shape bigger or smaller, it still remains similar as long as the sides and angles remain in the same order.

Example 1: Two Similar Triangles

Small triangle:
    /\
   /  \
  /____\

Big triangle:
        /\
       /  \
      /____\

Both triangles look the same, but the big one is larger.
They are similar triangles.

Example 2: Two Similar Rectangles

Small rectangle:
+------+
|      |
|      |
+------+

Big rectangle:
+------------+
|            |
|            |
+------------+

Both rectangles have the same shape, but different sizes.
They are similar rectangles.

Example 3: Similar Circles

Small circle:   ( )
Big circle:     (   )

All circles look the same no matter how big or small they are.
All circles are similar.

MORE EXAMPLES OF SIMILAR SHAPES

1. Two squares of different sizes.


2. Two rectangles with sides in the same ratio.


3. Two equilateral triangles of different sizes.


4. Two circles (because all circles have the same shape).


5. Two cubes where all sides are in the same proportion.

SHAPES THAT ARE NOT SIMILAR

Two shapes are not similar if they do not look exactly alike.

For example:

Square:     +----+
            |    |
            +----+

Rectangle:  +--------+
            |        |
            +--------+

The square and rectangle are not similar because their sides are not in the same proportion, even though they both have four sides.

They are not similar.

PROPERTIES OF SIMILAR SHAPES

1. Their shapes are exactly the same.


2. Their angles are equal.


3. Their sides are in the same ratio (proportion).


4. The sides that face each other (corresponding sides) are proportional.


5. One shape can be an enlargement or a reduction of the other.

EXAMPLE (SHOWING RATIOS)

Look at these two triangles below:

Triangle PQR (small)     Triangle ABC (big)
PQ = 3 cm                AB = 6 cm
QR = 4 cm                BC = 8 cm
PR = 5 cm                AC = 10 cm

Now, let us find the ratios of their sides:

AB : PQ = 6 : 3 = 2 : 1
BC : QR = 8 : 4 = 2 : 1
AC : PR = 10 : 5 = 2 : 1

The ratios are all the same (2 : 1).
Therefore, Triangle ABC is similar to Triangle PQR.

WHY WE LEARN SIMILAR SHAPES

1. It helps us to Understand how to make drawings that look the same but are bigger or smaller.


2. It helps us to Know how maps, models, and photos are made.


3. It helps us to Find missing sides of shapes by using ratio and proportion.


4. It helps us to Recognize shapes that look alike in real life.


5. It helps us to Prepare for more advanced topics like scale drawing and geometry.

REAL LIFE EXAMPLES OF SIMILAR SHAPES

1. A small photograph and a larger version of the same photograph.


2. A toy car and the real car it represents.


3. A map and the actual land area it shows.


4. A small drawing of a house and the real house.


5. Small and big circles on a logo or design.

SUMMARY

1. Similar shapes have the same shape but different sizes.


2. Their angles are equal.


3. Their sides are in the same ratio.


4. One shape can be made from the other by enlarging or reducing it.


5. All circles are similar.


6. All squares are similar.


7. But a square and a rectangle are not similar.

QUICK CHECK (Try It Yourself)

1. Are all circles similar?


2. Are all squares similar?


3. Is a triangle with sides 3 cm, 4 cm, 5 cm similar to one with sides 6 cm, 8 cm, 10 cm?


4. Are a square and a rectangle similar?


5. Are two equilateral triangles of different sizes similar?

---

DEFINITION

Enlargement means making a shape bigger or smaller, but keeping its shape the same.
It is like when you zoom in or zoom out on a picture — the picture becomes larger or smaller, but it still looks exactly the same.

Scale Factor is the number that tells us how much bigger or smaller the new shape is compared to the original one.

If the scale factor is greater than 1, the shape becomes bigger.
If the scale factor is less than 1, the shape becomes smaller.

EXPLANATION

When we enlarge a shape:

1. All the angles stay the same.


2. The sides change length, but in the same ratio.


3. The new shape is similar to the old one.

So enlargement means “same shape, different size”.
The scale factor shows how many times larger or smaller the new shape is.

Example 1: Enlargement of a Triangle

Original triangle (small):

   /\
  /  \
 /____\

Enlarged triangle (big):

       /\
      /  \
     /____\

The bigger triangle looks exactly the same, but it is larger.
This is called an enlargement.

Example 2: Enlargement of a Rectangle

Small rectangle:

+----+
|    |
+----+

Big rectangle:

+--------+
|        |
|        |
+--------+

Both rectangles have the same shape, but the second one is bigger.
They are similar and one is an enlargement of the other.

WHAT IS A SCALE FACTOR?

The scale factor tells how much the shape has been enlarged or reduced.

To find the scale factor, use:
Scale Factor = (Length of side in new shape) ÷ (Length of side in original shape)

EXAMPLE 1

A small square has sides 2 cm long.
A larger square has sides 6 cm long.

Solution:
Scale factor = New side ÷ Original side
= 6 ÷ 2
= 3

The scale factor is 3.
This means the big square is 3 times larger than the small one.

EXAMPLE 2

A small triangle has a side of 4 cm.
The enlarged triangle has a side of 8 cm.

Scale factor = 8 ÷ 4 = 2
The new triangle is twice as big.

EXAMPLE 3

A small rectangle has a side of 10 cm.
The enlarged rectangle has a side of 5 cm.

Scale factor = 5 ÷ 10 = 1/2
The new rectangle is half the size of the original.
It is a reduction (made smaller).

HOW TO KNOW IF IT IS ENLARGEMENT OR REDUCTION

Type	What Happens	Scale Factor
Enlargement	The shape becomes bigger	Greater than 1
Reduction	The shape becomes smaller	Between 0 and 1

EXAMPLE 4 (WITH TWO DIMENSIONS)

A rectangle 4 cm long and 3 cm wide is enlarged by a scale factor of 2.

New length: 4 × 2 = 8 cm
New width: 3 × 2 = 6 cm

The new rectangle is twice as large in every direction.

EXAMPLE 5 (REDUCTION)

A triangle has sides of 6 cm, 8 cm, and 10 cm.
It is reduced by a scale factor of ½.

New sides:
6 × ½ = 3 cm
8 × ½ = 4 cm
10 × ½ = 5 cm

The new triangle is smaller, but still looks exactly the same.

IMPORTANT POINTS ABOUT ENLARGEMENT AND SCALE FACTOR

1. The new shape must have equal angles with the old shape.


2. The sides are proportional (in the same ratio).


3. The scale factor tells how much the shape has grown or shrunk.


4. The centre of enlargement is the fixed point from which the shape grows or shrinks.


5. If the scale factor = 1, the shape does not change size.


6. If the scale factor > 1, the shape enlarges (gets bigger).


7. If the scale factor < 1, the shape reduces (gets smaller).

SCALE FACTOR EFFECT

Original shape (Scale factor = 1)
+----+
|    |
+----+

Enlarged shape (Scale factor = 2)
+--------+
|        |
|        |
+--------+

Reduced shape (Scale factor = ½)
+--+
|  |
+--+

WHY WE LEARN ENLARGEMENTS AND SCALE FACTOR

1. It helps us to Understand how to make pictures or models in different sizes.


2. It helps us to Make and read maps, plans, and blueprints.


3. It helps us to Draw shapes in different scales but same shape.


4. It helps us to Solve geometry problems easily.


5. It helps us to Prepare for scale drawings, ratios, and similar shapes in advanced mathematics.

REAL LIFE EXAMPLES

1. A small photograph and its enlarged print.


2. A map showing a large area on small paper.


3. A toy car model made smaller than a real car.


4. Architectural drawings of a building.


5. Zooming in or out on a phone screen.

QUICK CHECK (Try It Yourself)

1. A small square has a side of 4 cm. A big one has 12 cm. What is the scale factor?


2. A rectangle 6 cm by 4 cm is enlarged by a scale factor of 3. What is the new size?


3. A triangle with sides 8 cm, 6 cm, 4 cm is reduced by ½. What are the new sides?


4. What happens when the scale factor equals 1?


5. What type of change happens when the scale factor is less than 1?

---

DEFINITION

When two shapes are similar, it means:

1. They have the same shape but different sizes.


2. Their angles are equal.


3. Their sides are in the same ratio (called the scale factor).

Now, when shapes are similar, the lengths, areas, and volumes of the shapes are related in special ways.

These relationships help us to find unknown measurements when we know the scale factor.

RELATIONSHIP BETWEEN LENGTHS, AREAS, AND VOLUMES

If two shapes are similar and their scale factor = k, then:

1. The ratio of their lengths = k : 1


2. The ratio of their areas = k² : 1


3. The ratio of their volumes = k³ : 1

This means:

1. Lengths change in the same ratio as k.


2. Areas change in the square of the ratio (k²).


3. Volumes change in the cube of the ratio (k³).

EXPLANATION IN SIMPLE WORDS

Imagine you draw a small cube and a big cube that look exactly the same.

If every side of the big cube is twice (2 times) the side of the small cube, then:

1. The length of each side is 2 times longer.


2. The area of each face is 2 × 2 = 4 times bigger.


3. The volume is 2 × 2 × 2 = 8 times bigger.

So when you make a shape bigger, the lengths, areas, and volumes grow much faster!

TEXT DIAGRAM EXAMPLES

Example 1: Lengths

Small square (side = 2 cm)

+--+
|  |
+--+

Big square (side = 4 cm)

+----+
|    |
|    |
+----+

Scale factor = 4 ÷ 2 = 2
So, length ratio = 2 : 1

Example 2: Areas

Area of small square = 2 × 2 = 4 cm²

Area of big square = 4 × 4 = 16 cm²

Area ratio = 16 : 4 = 4 : 1

= (2²) : 1

The area ratio = square of the scale factor (k²).

Example 3: Volumes

Small cube (side = 2 cm)

Volume = 2 × 2 × 2 = 8 cm³

Big cube (side = 4 cm)

Volume = 4 × 4 × 4 = 64 cm³

Volume ratio = 64 : 8 = 8 : 1
= (2³) : 1

The volume ratio = cube of the scale factor (k³).

SUMMARY TABLE OF RELATIONSHIPS

Measurement	Ratio Relationship	Power of k	Example (if k = 2)
Length	k : 1	k¹	2 : 1
Area	k² : 1	k²	4 : 1
Volume	k³ : 1	k³	8 : 1

EXAMPLES

Example 1 (Length):

Two similar rectangles have a scale factor of 3.

If the smaller rectangle has a length of 5 cm, find the length of the bigger rectangle.

Solution:

Scale factor = 3

New length = 5 × 3 = 15 cm

The bigger rectangle has a length of 15 cm.

Example 2 (Area):

Two similar triangles have a scale factor of 2.

If the area of the smaller triangle is 6 cm², find the area of the larger triangle.

Solution:

Area ratio = k² = 2² = 4

New area = 6 × 4 = 24 cm²

The larger triangle has an area of 24 cm².

Example 3 (Volume):

Two similar cubes have a scale factor of 3.

If the smaller cube has a volume of 8 cm³, find the volume of the larger cube.

Solution:

Volume ratio = k³ = 3³ = 27

New volume = 8 × 27 = 216 cm³

The larger cube has a volume of 216 cm³.

Example 4 (Finding scale factor from areas):

The area of one circle is 9 times the area of another.

Find the scale factor between their radii.

Solution:

Area ratio = k² = 9

k = √9 = 3

The scale factor between their radii is 3.

Example 5 (Finding scale factor from volumes):

The volume of one sphere is 64 times the volume of another.

Find the scale factor between their diameters.

Solution:

Volume ratio = k³ = 64

k = ³√64 = 4

The scale factor is 4.

WHY WE LEARN THIS

1. It helps us to Understand how big or small similar shapes become.


2. It helps us to Find missing lengths, areas, or volumes in models and real life.


3. It helps us to Make correct scale drawings, maps, and plans.


4. It helps us to Understand ratios and powers in geometry and science.


5. It helps us to Work with real-world problems like building models, enlarging photos, and architecture.

REAL LIFE EXAMPLES

1. A toy car and a real car (different sizes but same shape).


2. A small globe and the Earth.


3. A map and the real land it represents.


4. Enlarged or reduced drawings.


5. A model of a building made smaller than the real one.

SUMMARY

1. Similar shapes have the same shape but different sizes.


2. Their sides (lengths) are in the ratio of k : 1.


3. Their areas are in the ratio of k² : 1.


4. Their volumes are in the ratio of k³ : 1.


5. The bigger the scale factor, the more the area and volume increase.

QUICK CHECK (Try It Yourself)

1. Two similar squares have a scale factor of 3. If the smaller one has a side of 4 cm, what is the side of the larger one?


2. The area of a small triangle is 5 cm². If the scale factor is 2, what is the area of the bigger triangle?


3. The volume of a cube is 27 cm³. If it is enlarged by a scale factor of 2, what is the new volume?


4. If two similar shapes have an area ratio of 16 : 1, what is their scale factor?


5. If two similar shapes have a volume ratio of 8 : 1, what is their scale factor?

---

ENLARGEMENTS AND SCALE FACTOR (How Shapes Change Size)

Question 1:

A small square has a side of 3 cm. It is enlarged by a scale factor of 4. Find the side of the larger square.

Solution:

Step 1: Write down the scale factor: k = 4

Step 2: Multiply the original side by the scale factor: 3 × 4 = 12 cm

The larger square has a side of 12 cm.

Question 2:

A rectangle has sides 5 cm by 3 cm. It is reduced by a scale factor of ½. Find the new dimensions.

Solution:

Step 1: Write down the scale factor: k = ½

Step 2: Multiply each side by the scale factor:

Length: 5 × ½ = 2.5 cm

Width: 3 × ½ = 1.5 cm

New rectangle dimensions: 2.5 cm by 1.5 cm

Question 3:

A triangle has sides 6 cm, 8 cm, and 10 cm. It is enlarged by a scale factor of 3. Find the lengths of the new sides.

Solution:

Step 1: Write down the scale factor: k = 3

Step 2: Multiply each side by 3:

6 × 3 = 18 cm

8 × 3 = 24 cm

10 × 3 = 30 cm

Enlarged triangle sides: 18 cm, 24 cm, 30 cm

Question 4:

A square has a side of 7 cm. Its enlarged version has a side of 21 cm. Find the scale factor.

Solution:

Step 1: Use the formula: Scale factor = New side ÷ Original side

k = 21 ÷ 7 = 3

Scale factor = 3

Question 5:

A rectangle 4 cm by 6 cm is reduced to a smaller rectangle with a length of 2 cm. Find the scale factor and the new width.

Solution:

Step 1: Scale factor = New length ÷ Original length = 2 ÷ 4 = ½

Step 2: Multiply original width by scale factor: 6 × ½ = 3 cm

New rectangle: 2 cm by 3 cm

LENGTHS, AREAS, AND VOLUMES OF SIMILAR SHAPES

Question 1:

Two similar squares have a scale factor of 3. If the smaller square has an area of 4 cm², find the area of the larger square.

Solution:

Step 1: Area ratio = k² = 3² = 9

Step 2: Multiply smaller area by ratio: 4 × 9 = 36 cm²

Larger square area = 36 cm²

Question 2:

Two similar cubes have a scale factor of 2. The smaller cube has a volume of 8 cm³. Find the volume of the larger cube.

Solution:

Step 1: Volume ratio = k³ = 2³ = 8

Step 2: Multiply smaller volume by ratio: 8 × 8 = 64 cm³

Larger cube volume = 64 cm³

Question 3:

A small triangle has sides 3 cm, 4 cm, 5 cm. A similar triangle has sides 6 cm, 8 cm, 10 cm. Find the scale factor, area ratio, and volume ratio.

Solution:

Step 1: Scale factor = New side ÷ Original side = 6 ÷ 3 = 2

Step 2: Area ratio = k² = 2² = 4 : 1

Step 3: Volume ratio = k³ = 2³ = 8 : 1

Scale factor = 2, Area ratio = 4:1, Volume ratio = 8:1

Question 4:

A rectangular prism has dimensions 2 cm × 3 cm × 4 cm. A similar prism has dimensions 6 cm × 9 cm × 12 cm. Find the scale factor, ratio of lengths, areas, and volumes.

Solution:

Step 1: Scale factor = 6 ÷ 2 = 3

Step 2: Length ratio = 3 : 1

Step 3: Area ratio = k² = 3² = 9 : 1

Step 4: Volume ratio = k³ = 3³ = 27 : 1

Scale factor = 3, Length ratio = 3:1, Area ratio = 9:1, Volume ratio = 27:1

Question 5:

A cylinder with radius 2 cm and height 5 cm is enlarged by a scale factor of 2. Find the new radius, new height, and the ratio of volumes.

Solution:

Step 1: Multiply radius and height by scale factor:

New radius = 2 × 2 = 4 cm

New height = 5 × 2 = 10 cm

Step 2: Volume ratio = k³ = 2³ = 8 : 1

New cylinder: radius = 4 cm, height = 10 cm, volume ratio = 8:1

CONCLUSION

Understanding similar shapes, enlargements, and scale factor is very important in mathematics because it helps us to see how shapes can change size while keeping their shape the same.

Key points to remember:

• Similar shapes have the same shape but different sizes. Their corresponding angles are equal, and their sides are proportional.


• Enlargement means making a shape bigger or smaller while keeping the shape the same. Reduction is making a shape smaller.


• The scale factor tells us how much larger or smaller the new shape is compared to the original shape. A scale factor greater than 1 enlarges the shape, while a scale factor less than 1 reduces it.


• The lengths of similar shapes change in the ratio of the scale factor (k), areas change in the square of the scale factor (k²), and volumes change in the cube of the scale factor (k³).


• These concepts are used in real life in maps, drawings, models, photographs, and architecture to maintain proportions while changing size.

By understanding and applying these rules, you can solve problems involving enlargements, scale factors, and similar shapes easily and accurately. It also prepares you for more advanced topics in geometry and measurement.