Skip to content# Similar Shapes: Step-by-Step Examples with Worksheet

## What are similar shapes?

## Working with Similar Shapes Involving Area or Volume

## Exam Tips

## Conclusion

## Worksheet on Similar Shapes

**Similar Shapes**

- The similar shapes are the core component of geometry which reveals the proportional relationship between the measures of corresponding angles and sides.
- Fundamental to geometry is the recognition of shapes in different fields, like architecture and engineering, that shape comparison and scaling depend on.

In this article, we will discuss:

**What are similar shapes?****Working with Similar Shapes Involving Area or Volume**

Here is one more link to practice a few extra questions: Maths Genie Similar Shapes Questions

The similar shapes in mathematics are those in which the first one is a proportional enlargement of the other.

**Scale Factors:**

When two shapes are similar and connected by a scale factor, denoted as ‘k’:

- In equivalent areas, areas are connected with an area factor ‘
**k**‘.^{2} - Equivalent volumes are characterized by the volume factor
**k**.^{3}

- In equivalent areas, areas are connected with an area factor ‘

**Step #1 – Identification:**

- Identify the known quantities which are units of lengths, areas or volumes.

**Step #2 – Direction:**

Find out whether the shapes are getting larger or smaller.

**Step #3 – Scale Factor Calculation:**

- Identify the Scale Factor (‘k’) by two given lengths, areas or volumes.
Look whether the

**scale factor is greater than one for expansion or less than one for contraction.**- Express the scale factor as ‘k = s.f.’ for lengths, ‘
**k**‘ for areas, and ‘^{2}= s.f.**k**‘ for volumes.^{3}= s.f.

**Step #4 – Utilizing Scale Factor:**

- Use the known scale factor for magnifying other relevant lengths, areas, or volumes.
- Relationships:

**Area Scale Factor = (Length Scale Factor) ^{2}**

**Length Scale Factor = √(Area Scale Factor)**

**Volume Scale Factor = (Length Scale Factor) ^{3}**

**Length Scale Factor = ∛(Volume Scale Factor)**

- When conducting calculations, it is crucial to take extra care not to mix up the identities of the shapes involved. To maintain clarity, consider the following guidelines:

While doing calculations, it is important to be extra careful not to mingle and merge the identities of the figures. To maintain clarity, consider the following guidelines:

**Labelling Shapes:**- Assign distinct labels to each shape involved in the comparison.

**Equation Representation:**All the time make equations to get clarity and for a perfect understanding.

**For example:**

If shape A is deemed similar to shape B, ensure the following relationships are maintained:

- The length of shape A is represented as k times the length of shape B:

**Length A = k (length B)**

- The area of shape A is correlated with the area of shape B by a factor of k
^{2 }:

- The area of shape A is correlated with the area of shape B by a factor of k

**area A = k ^{2} (area B)**

- The volume of shape A is connected to the volume of shape B through a factor of k
^{3}:

- The volume of shape A is connected to the volume of shape B through a factor of k

**Volume A = k ^{3} (Volume B)**

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**Solved Example:**

**Question 1:**

**Solid P and solid Q are mathematically similar.**

**The volume of solid P is 32 cm ^{3}**

**The volume of solid Q is 108 cm ^{3}.**

**The height of solid P is 10 cm.**

**Find the height of solid Q.**

Solution:

Calculate k^{3}, the scale factor of enlargement for the volumes, using Q = k^{3} (volume P), Or

**k ^{3} = large volume/smaller volume**

**108 = 32k ^{3}**

**K ^{3} = 108/32 = 27/8**

For similar shapes, If the volume scale factor is k^{3} for similar shapes, then the length scale factor is k.

Find k.

Substitute into the formula for the heights of similar shapes. **Height Q = k(height P).**

h = 10k

Height of Q = 15 cm

The height of Solid Q is 15 cm.

- Understanding similar shapes is of fundamental significance for various applications in mathematics, such as geometry and real-world examples.
- The scaling factor sets in play corresponding proportional relationships between the sides, angles, area and volume of the shapes being related.
- By specifying the practical examples, we have shown the efficiency of these fundamentals in processing the actual world problems similarly emphasizing the usefulness of similarity in mathematical modeling and analysis.

**Question 1:** Below are two similar triangles. The area of triangle P is 20cm^{2} Work out the area of triangle Q.

**Question 2:** Below are two similar parallelograms. The area of parallelogram P is 28cm^{2} Work out the area of parallelogram Q

**Question 3:** Shown below are two mathematically similar parallelograms. Find x.

**Question 4:** The areas of two mathematically similar shapes are in the ratio 49:81
The length of the smaller shape is 24.5cm
Work out the length of the larger shape..