How to Find the Ratio of Areas of Two Triangles

Finding the ratio of the areas of two triangles can be a useful skill in various mathematical and real-world applications. The approach to determine this ratio is fundamentally based on understanding how the dimensions of triangles affect their areas. In this article, we will explore several methods to find the ratio of the areas of two triangles, dissecting the principles behind each method and providing practical examples and exercises to reinforce these concepts.

To begin with, consider that the area of a triangle can be expressed using the formula:

Area = (1/2) × base × height

Thus, to find the ratio of the areas of two triangles, you need to compare their base lengths and heights, or any other factors that influence the area. Let’s explore some common scenarios:

Method 1: Using Base and Height

If two triangles have the same height, the ratio of their areas will be directly proportional to the ratio of their base lengths. For example, if Triangle A has a base of 4 cm and Triangle B has a base of 6 cm, and both triangles have the same height, the ratio of their areas is:

$\text{Ratio of Areas}=\frac{\text{Base of Triangle A}}{\text{Base of Triangle B}}=\frac{4}{6}=\frac{2}{3}$Ratio of Areas=Base of Triangle BBase of Triangle A​=64​=32​

Similarly, if the base lengths are the same, the ratio of their areas will be proportional to the ratio of their heights. For example, if Triangle A has a height of 5 cm and Triangle B has a height of 8 cm, and both triangles have the same base length, the ratio of their areas is:

$\text{Ratio of Areas}=\frac{\text{Height of Triangle A}}{\text{Height of Triangle B}}=\frac{5}{8}$Ratio of Areas=Height of Triangle BHeight of Triangle A​=85​

Method 2: Using Heron’s Formula

When the base and height are not known, but the side lengths of the triangles are known, Heron’s formula can be used. Heron’s formula allows for the calculation of the area of a triangle when the lengths of all three sides are known.

Heron’s formula is given by:

$\text{Area}=\sqrt{s(s-a)(s-b)(s-c)}$Area=s(s−a)(s−b)(s−c)​

where $s=\frac{a+b+c}{2}$s=2a+b+c​ is the semi-perimeter of the triangle, and $a$a, $b$b, and $c$c are the side lengths.

To find the ratio of the areas of two triangles using Heron’s formula, calculate the area of each triangle using the side lengths and then find the ratio of these areas.

For instance, let’s say Triangle A has side lengths 6 cm, 8 cm, and 10 cm, and Triangle B has side lengths 5 cm, 12 cm, and 13 cm. First, compute the semi-perimeter for each triangle:

For Triangle A:

$s_A=\frac{6+8+10}{2}=12$sA​=26+8+10​=12

Then, the area is:

${\text{Area}}_A=\sqrt{12\times (12-6)\times (12-8)\times (12-10)}=\sqrt{12\times 6\times 4\times 2}=24\text{ square cm}$AreaA​=12×(12−6)×(12−8)×(12−10)​=12×6×4×2​=24 square cm

For Triangle B:

$s_B=\frac{5+12+13}{2}=15$sB​=25+12+13​=15

Then, the area is:

${\text{Area}}_B=\sqrt{15\times (15-5)\times (15-12)\times (15-13)}=\sqrt{15\times 10\times 3\times 2}=30\text{ square cm}$AreaB​=15×(15−5)×(15−12)×(15−13)​=15×10×3×2​=30 square cm

Thus, the ratio of their areas is:

$\text{Ratio of Areas}=\frac{{\text{Area}}_A}{{\text{Area}}_B}=\frac{24}{30}=\frac{4}{5}$Ratio of Areas=AreaB​AreaA​​=3024​=54​

Method 3: Using Trigonometry

When two sides and the included angle of each triangle are known, you can use the trigonometric formula for the area of a triangle:

$\text{Area}=\frac{1}{2}\times a\times b\times \sin \left(C\right)$Area=21​×a×b×sin(C)

where $a$a and $b$b are the lengths of the two sides, and $C$C is the included angle.

To find the ratio of the areas of two triangles using this method, calculate the area of each triangle using their respective sides and angles, then find the ratio of these areas.

For example, let’s say Triangle A has sides of lengths 7 cm and 5 cm with an included angle of 30 degrees, and Triangle B has sides of lengths 10 cm and 8 cm with an included angle of 45 degrees.

For Triangle A:

${\text{Area}}_A=\frac{1}{2}\times 7\times 5\times \sin \left(30^{\circ }\right)=\frac{1}{2}\times 7\times 5\times 0.5=17.5\text{ square cm}$AreaA​=21​×7×5×sin(30∘)=21​×7×5×0.5=17.5 square cm

For Triangle B:

${\text{Area}}_B=\frac{1}{2}\times 10\times 8\times \sin \left(45^{\circ }\right)=\frac{1}{2}\times 10\times 8\times \frac{\sqrt{2}}{2}=40\text{ square cm}$AreaB​=21​×10×8×sin(45∘)=21​×10×8×22​​=40 square cm

Thus, the ratio of their areas is:

$\text{Ratio of Areas}=\frac{{\text{Area}}_A}{{\text{Area}}_B}=\frac{17.5}{40}=\frac{7}{16}$Ratio of Areas=AreaB​AreaA​​=4017.5​=167​

Practical Applications and Examples

Understanding how to find the ratio of the areas of triangles can be highly practical in various scenarios such as architecture, engineering, and even everyday problem-solving. For example, if you are designing a garden and need to compare the areas of different triangular sections, knowing how to find their area ratios will help you make more informed decisions.

Example 1: Garden Design

Imagine you are landscaping a triangular garden bed. The main garden is a triangle with sides 10 m, 12 m, and 14 m. You want to add a triangular flower bed with sides 5 m, 6 m, and 7 m. Using Heron's formula, you can find the area of both triangles and determine the ratio to understand the proportion of space each bed will occupy.

Example 2: Engineering

In engineering, when designing triangular components or supports, engineers need to ensure that the strength and material requirements are proportional to the area of the components. By calculating area ratios, they can determine material usage and cost more effectively.

Conclusion

In summary, finding the ratio of the areas of two triangles involves using various methods depending on the available information. Whether you use base and height, Heron's formula, or trigonometric formulas, understanding these methods allows for precise calculations and practical applications. Mastering these techniques equips you with the ability to handle diverse geometric problems efficiently.