How to find the area of a triangle might be difficult for many people, especially if you’re unsure of which formula to apply. Whether you’re working with an equilateral triangle, a right triangle, or another variety, knowing how to calculate the area is critical for geometry students and professionals. This article will bring you through the many formulae for calculating the area of a triangle, along with examples and explanations to ensure you have all the tools you need to solve any triangle problem.

Part 1. What is the Area of a Triangle?

The area of a triangle is the two-dimensional space encompassed by its three sides. It is usually measured in square units, like square centimeters (cm²), square meters (m²), or square inches. The area of a triangle is proportional to the lengths of its sides and the angles between them, as demonstrated in the different formulas and methods presented in this page.

Part 2. Area of a Triangle Formula

Depending on the information given, a triangle’s area can be calculated using one of numerous formulas. The most widely used formulas include:

This table summarizes the several formulas for estimating the area of a triangle:

Formula	Equation
Base and Height Formula	Area = 1/2 × base × height
Coordinate Formula	Area = 1/2 ×
Heron’s Formula	Area = √(s(s-a)(s-b)(s-c)) where s = (a + b + c)/2 and a, b, c are the side lengths
Known Sides (Right-Angled)	Area = 1/2 × base × height
Known Sides (General)	Area = 1/2 × a × h where a is the base and h is the perpendicular height
Equilateral Triangle	Area = √3/4 × side length^2

Part 3. How to Find the Area of a Triangle?

To calculate the area of a triangle, you must know at least one of the following: the base and height, the lengths of all three sides, or the lengths of two sides and their angle. Let us look at how to use these various ways to determine the area of different sorts of triangles.

Area of Equilateral Triangle

An equilateral triangle is a triangle with all three sides of equal length. To find the area of an equilateral triangle with a side length of ‘s’, you can use the formula:

Area = (√3/4) × s²

Here’s a step-by-step explanation with an example:

Steps to find the area of an equilateral triangle:

1. Identify the side length of the equilateral triangle.
2. Plug the side length into the formula: Area = (√3 / 4) × side length^2.
3. Calculate the result to find the area of the equilateral triangle.

Example: Find the area of an equilateral triangle with a side length of 8 cm.

Step 1: The side length of the equilateral triangle is 8 cm.

Step 2: Plug the side length into the formula:

Area = (√3 / 4) × 8^2

= (√3 / 4) × 64

= (1.732 / 4) × 64

= 0.433 × 64

= 27.73 square cm

Therefore, the area of the equilateral triangle with a side length of 8 cm is 27.73 square cm.

Explanation:

The formula for the area of an equilateral triangle is derived from the general triangle area formula:

Area = (1/2) × base × height

For an equilateral triangle, the base and height are related by the fact that the height is the perpendicular distance from the base to the opposite vertex. This height can be calculated using trigonometry as:

height = (side length × √3) / 2

Substituting this into the general triangle area formula, we get:

Area = (1/2) × side length × (side length × √3 / 2)

= (√3 / 4) × side length^2

This is the simplified formula used to find the area of an equilateral triangle given the side length.

The key steps are to identify the side length, plug it into the formula, and calculate the final area. This provides a straightforward way to determine the area of any equilateral triangle.

Area of Right Triangle

A right triangle is a triangle with one 90-degree angle. If you know the lengths of the base (b) and height (h) of a right triangle, you can use the formula:

Area = 1/2 × b × h

Steps to find the area of a right triangle:

1. Identify the base and height of the right triangle.
2. Plug the base and height values into the formula: Area = 1/2 × base × height.
3. Calculate the result to find the area of the right triangle.

Example: Find the area of a right triangle with a base of 6 cm and a height of 8 cm.

Step 1: The base of the right triangle is 6 cm, and the height is 8 cm.

Step 2: Plug the base and height values into the formula:

Area = 1/2 × 6 cm × 8 cm

= 1/2 × 48 cm²

= 24 cm²

Therefore, the area of the right triangle with a base of 6 cm and a height of 8 cm is 24 square cm.

Explanation:

The formula for the area of a right triangle is derived from the general triangle area formula:

Area = 1/2 × base × height

For a right triangle, the base and height are perpendicular to each other, forming a right angle. This allows us to use the simpler formula of 1/2 × base × height to calculate the area.

The key steps are to identify the base and height of the right triangle, plug them into the formula, and calculate the final area. This provides a straightforward way to determine the area of any right triangle.

It’s important to note that the base and height must be perpendicular to each other for this formula to be applicable. If the triangle is not right-angled, you would need to use a different formula, such as Heron’s formula or the formula using the side lengths and the perpendicular height.

Area of the Isosceles Triangle

An isosceles triangle is a triangle with at least two sides of equal length. If you know the length of the base (b) and the height (h), you can use the formula:

Area = 1/2 × b × h

Steps to find the area of an isosceles triangle:

1. Identify the base and height of the isosceles triangle.
2. Plug the base and height values into the formula: Area = 1/2 × base × height.
3. Calculate the result to find the area of the isosceles triangle.

Example: Find the area of an isosceles triangle with a base of 10 cm and a height of 8 cm.

Step 1: The base of the isosceles triangle is 10 cm, and the height is 8 cm.

Step 2: Plug the base and height values into the formula:

Area = 1/2 × 10 cm × 8 cm

= 1/2 × 80 cm²

= 40 cm²

Therefore, the area of the isosceles triangle with a base of 10 cm and a height of 8 cm is 40 square cm.

Explanation:

The formula for the area of an isosceles triangle is the same as the formula for the area of a right triangle, where the base and height are perpendicular to each other.

In an isosceles triangle, the base and the two equal sides form a right angle. This allows us to use the simpler formula of 1/2 × base × height to calculate the area.

The key steps are to identify the base and height of the isosceles triangle, plug them into the formula, and calculate the final area. This provides a straightforward way to determine the area of any isosceles triangle.

It’s important to note that this formula is applicable only for isosceles triangles, where two sides are equal in length. If the triangle is not isosceles, you would need to use a different formula, such as Heron’s formula or the formula using the side lengths and the perpendicular height.

Area of Triangle with 3 Sides (With Formula)

If you know the lengths of all three sides of a triangle (a, b, and c), you can use Heron’s formula to find the area:

Area = √[s(s-a)(s-b)(s-c)], where s = (a+b+c)/2

Where:

• a, b, and c are the lengths of the three sides of the triangle
• s is the semi-perimeter, which is calculated as: s = (a + b + c) / 2

Here’s a step-by-step explanation with an example:

Steps to find the area of a triangle with three known side lengths:

1. Identify the lengths of the three sides of the triangle (a, b, and c).
2. Calculate the semi-perimeter (s) using the formula: s = (a + b + c) / 2.
3. Plug the side lengths and the semi-perimeter into Heron’s formula: Area = √(s(s-a)(s-b)(s-c))
4. Calculate the result to find the area of the triangle.

Example: Find the area of a triangle with side lengths of 6 cm, 8 cm, and 10 cm.

Step 1: The lengths of the three sides are: a = 6 cm, b = 8 cm, and c = 10 cm.

Step 2: Calculate the semi-perimeter (s):

s = (a + b + c) / 2

s = (6 cm + 8 cm + 10 cm) / 2

s = 24 cm / 2

s = 12 cm

Step 3: Plug the side lengths and the semi-perimeter into Heron’s formula:

Area = √(s(s-a)(s-b)(s-c))

= √(12 cm (12-6)(12-8)(12-10))

= √(12 cm × 6 × 4 × 2)

= √(576 cm²)

= 24 cm²

Therefore, the area of the triangle with side lengths of 6 cm, 8 cm, and 10 cm is 24 square cm.

Explanation:

Heron’s formula is a general formula for calculating the area of a triangle given the lengths of its three sides. It is useful when the triangle is not a right triangle or an isosceles triangle, and the base and height are not easily identifiable.

The formula uses the semi-perimeter (s) as an intermediate value, which is the half of the sum of the three side lengths. This formula works for any triangle, regardless of its shape or the relationship between the sides.

The key steps are to identify the three side lengths, calculate the semi-perimeter, and then plug all of these values into Heron’s formula to find the final area of the triangle.

This method provides a reliable way to determine the area of a triangle when only the side lengths are known, without the need for additional information about the triangle’s shape or dimensions.

Part 4. Area of a Triangle Calculator

Area of a Triangle Calculator

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Part 5. Area of a Triangle Solved Examples

Example 1: The base of a triangle is 10 cm and the height is 6 cm. Find the area of the triangle.

Solution:

The formula for the area of a triangle is: Area = 1/2 × base × height

Substituting the given values:

Area = 1/2 × 10 cm × 6 cm = 30 square cm

Example 2: A right-angled triangle has perpendicular sides of 8 cm and 6 cm. Find the area of the triangle.

Solution:

For a right-angled triangle, the area formula is: Area = 1/2 × base × height

Here, the base is 8 cm and the height is 6 cm. Substituting these values:

Area = 1/2 × 8 cm × 6 cm = 24 square cm

Example 3: The sides of a triangle are 8 cm, 6 cm, and 10 cm. Find the area of the triangle.

Solution:

For a general triangle, we can use Heron’s formula to calculate the area:

s = (a + b + c) / 2 (where a, b, c are the side lengths)

Area = √(s(s-a)(s-b)(s-c))

Substituting the given values:

s = (8 + 6 + 10) / 2 = 12

Area = √(12 × (12-8) × (12-6) × (12-10)) = 24 square cm

Example 4: An equilateral triangle has a side length of 12 cm. Find the area of the triangle.

Solution:

For an equilateral triangle, the area formula is: Area = √3/4 × side length^2

Substituting the given side length:

Area = √3/4 × 12^2 = 62.43 square cm

Example 5: A triangular-shaped farmland has vertices at the coordinates (5,3), (10,7), and (2,10). Find the area of the farmland.

Solution: We can use the coordinate formula to calculate the area of the triangle: Area = 1/2 × |(x1y2 – x2y1) + (x2y3 – x3y2) + (x3y1 – x1y3)|

Substituting the given coordinates:

x1=5, y1=3

x2=10, y2=7

x3=2, y3=10

• Area = 1/2 × |(57 – 10(1010 – 2(23 – 510)| = 1/2 × |35 – 30 + 100 – 14 – 15 – 50| = 1/2 × 26 = 13 square units

In summary, these 5 examples cover various methods for calculating the area of a triangle, including:

1. Known base and height
2. Known perpendicular sides of a right-angled triangle
3. Known side lengths
4. Known side length of an equilateral triangle
5. Known coordinates of the vertices

Frequently Asked Questions on Area of a Triangle

Q1. What is the formula for the surface area of a triangular prism?

A2. The surface area of a triangular prism is calculated by adding the areas of the three rectangular faces and the two triangular bases. The formula is:

Surface Area = 2 × (1/2 × base × height) + length × (base + height)

Q2. How do you find the area of a triangle with three sides?

A3. To find the area of a triangle with three known sides, you can use Heron’s formula:

Area = √[s(s-a)(s-b)(s-c)], where s = (a+b+c)/2 and a, b, c are the lengths of the three sides.

Summary:

In this comprehensive article, we’ve looked at the numerous methods and formulas for calculating the area of a triangle. From the basic formula Area = 1/2 × base × height to the more difficult Heron’s formula, we covered the essential tools needed to find the area of numerous sorts of triangles, including equilateral, right, isosceles, and those with three known sides. Understanding these concepts and performing the solved cases will prepare you to handle a variety of triangular area challenges in your academic and professional activities.