How to Find Area of Similar Shapes: Key Formulas and Quick Solutions
The concept of Area of Similar Shapes plays an essential role in geometry and mensuration, especially in topics like triangles, polygons, and real-life scale models. This concept helps students quickly calculate and compare areas when shapes are enlarged or reduced, making it highly important for school and competitive exams as well as for practical applications such as map reading and model making.
Understanding the Area of Similar Shapes
Similar shapes are geometric figures with the same shape but different sizes. They have equal corresponding angles and proportional corresponding sides. The area of similar shapes changes in a predictable way when the figure is enlarged or reduced. This is because the area depends on the square of the scale factor (the ratio of any pair of corresponding sides).
For example, when two rectangles are similar, the ratio of their corresponding sides is constant, and their areas can also be compared using this scale factor.
Formula for Area of Similar Shapes
The key formula to compare or find the area of two similar shapes is:
If the ratio of corresponding sides (scale factor) of two similar shapes is k, then:
| Quantity | Ratio Formula |
|---|---|
| Area Ratio | Area1 / Area2 = (Side1 / Side2)2 |
This means if two triangles, rectangles, or any polygons are similar, the ratio of their areas is the square of the ratio of their corresponding sides.
Worked Example
Let’s apply the area of similar shapes formula in a real problem:
Suppose you have two similar rectangles. The length of the first is 6 cm, and the length of the second is 9 cm. If the area of the first rectangle is 24 cm², what is the area of the second rectangle?
- Calculate the side ratio: Side1 / Side2 = 6 / 9 = 2 / 3.
- Find the area ratio: (2 / 3)2 = 4 / 9.
- Set up an equation: 24 / Area2 = 4 / 9.
- Cross multiply: 4 × Area2 = 24 × 9.
- Area2 = (24 × 9) / 4 = 216 / 4 = 54 cm².
Therefore, the area of the second rectangle is 54 cm².
Practice Problems
- The sides of two similar triangles are in the ratio 2:5. If the smaller triangle has an area of 18 cm², what is the area of the larger triangle?
- Two similar squares have areas 49 cm² and 121 cm². What is the ratio of their sides?
- The perimeters of two similar rectangles are in the ratio 3:4. If the area of the smaller rectangle is 27 m², what is the area of the larger?
- Two similar pentagons have corresponding sides of 7 cm and 14 cm. If the area of the smaller pentagon is 63 cm², find the area of the larger.
- The sides of two similar hexagons are in ratio 5:8. If the area of the larger is 128 cm², what is the area of the smaller?
Common Mistakes to Avoid
- Applying the side ratio directly to find area instead of squaring the ratio.
- Mixing up area and perimeter formulas for similar shapes.
- Not confirming that the shapes are actually similar before applying the area formula.
- Confusing which shape is the “first” or “second”—always clarify your reference.
Real-World Applications
Understanding the area of similar shapes is essential in many practical fields. Architects use these rules when scaling blueprints for buildings. Map makers (cartographers) apply the formula to convert real-world distances and areas into smaller, manageable drawings. Even in photography and design, enlarging or reducing images while keeping proportions uses the same geometry principle.
At Vedantu, we help students master such concepts with interactive lessons and practice resources for easy learning and high exam scores. Check out our guides on Similar Figures or Scale Factor for deeper understanding.
Page Summary
In this topic, we explored the Area of Similar Shapes and why their areas are proportional to the square of the scale factor of their corresponding sides. Mastering this concept helps students solve geometry problems faster and supports learning in related topics. Keep practicing with Vedantu for complete confidence in geometry!
FAQs on Area of Similar Shapes Formulas Explained with Step-by-Step Examples
1. What is the area formula for similar shapes?
The ratio of the areas of two similar shapes is equal to the square of the ratio of their corresponding side lengths (the scale factor). This means if you know the area of one shape and the scale factor between it and a similar shape, you can easily calculate the area of the second shape.
2. How do you calculate the area of two similar triangles?
To find the area of similar triangles, use the formula: Area₁/Area₂ = (Side₁/Side₂)², where Area₁ and Area₂ are the areas of the two similar triangles, and Side₁ and Side₂ are corresponding sides. If you know the area of one triangle and the ratio of corresponding sides, you can calculate the other triangle's area.
3. Why does area scale with the square of the scale factor?
Area scales with the square of the scale factor because area is a two-dimensional measurement. When you enlarge a shape, you're scaling both its length and width. Therefore, the increase in area is proportional to the square of the scaling factor. Consider a square: doubling the side length (scale factor of 2) quadruples (2²) its area.
4. Can I use the same formula for rectangles and triangles?
Yes, the formula Area₁/Area₂ = (Side₁/Side₂)² applies to all similar shapes, including rectangles and triangles. As long as the shapes are similar (meaning their corresponding angles are equal and corresponding sides are proportional), this formula will accurately compare their areas.
5. Where can I get a formula sheet for area of shapes?
Vedantu provides helpful formula sheets and downloadable resources for various geometrical shapes. These resources are designed to assist students in mastering area formulas for quick revision and exam preparation. Look for downloadable PDFs on the Vedantu website.
6. What are the formulas for area of shapes?
Area formulas vary depending on the shape. Common formulas include: Square: side²; Rectangle: length x width; Triangle: ½ base x height; Circle: πr². For similar shapes, remember the key relationship: Area₁/Area₂ = (Side₁/Side₂)².
7. How to find area of common shapes?
Finding the area of common shapes involves using specific formulas based on the shape's dimensions. For example, the area of a rectangle is length times width; a triangle's area is half its base times its height; a circle's area is π times the square of its radius. Remember to use the appropriate units (e.g., cm², m²).
8. What is the formula for similar figures?
There isn't one single formula for similar figures, but a key principle: the ratio of corresponding side lengths is constant (the scale factor). For areas of similar figures, the ratio of their areas is the square of the scale factor: Area₁/Area₂ = (Side₁/Side₂)². This applies to triangles, rectangles, and other polygons.
9. How do you compare the areas of similar shapes?
Compare the areas of similar shapes using the ratio of their corresponding sides. The ratio of the areas of two similar shapes is equal to the square of the ratio of their corresponding sides (scale factor squared). This allows for quick comparisons and calculations even without knowing the actual areas.
10. How does scaling affect perimeter vs area in similar shapes?
Scaling affects perimeter and area differently in similar shapes. Perimeter scales linearly with the scale factor: if you double the sides, the perimeter doubles. However, area scales quadratically—it's proportional to the square of the scale factor. Doubling the sides quadruples the area.
