Area of Triangle Calculator

A tool to calculate area of rectangle and also support various units.

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Understanding Triangle

A comprehensive guide to understand about Triangle, area of Triangle and methods to calculate area of triangle.

What is a Triangle?

A triangle is a closed two-dimensional geometric shape with three straight sides and three vertices (corners). It's also known as a three-sided polygon. The sum of the interior angles of a triangle always equals 180 degrees.

What is the Area of a Triangle?

The area of a triangle is a measure of the space enclosed within its three sides. Triangles are fundamental shapes in geometry, and calculating their area is essential in various fields such as architecture, engineering, and physics. There are several methods to calculate the area of a triangle, depending on the given information about the triangle's sides and angles.

In this blog, we’ll explore four common methods to calculate the area of a triangle: using the base and height, using three sides (SSS), using two sides and the angle between them (SAS), and using two angles and the side between them (ASA). Each method will be explained with a formula and an example.

Methods to Calculate the Area of a Triangle

Here, we have defined total four ways to calculate area of triangle, You can go with any one from above area of triangle calculator based on your given values.

1. Base and Height Method

This is the most straightforward method to calculate the area of a triangle when the base and the corresponding height (perpendicular distance from the base to the opposite vertex) are known. The formula is::

\( \text{Area} = \frac{\text{Base × Height}}{\text{2}}\)

The unit of area is in square units (e.g., square meters, square inches).

Steps to calculate area of triangle:
1
Determine the height (h) of triangle.
2
Determine the base (b) of triangle.
3
Multiply the height (h) by the base (b).
4
This result gives you the area of the triangle.
Example: Calculate the area of a triangle with a base of 4 meters and a height of 3 meters.
- \( \text{Area} = \frac{\text{3 × 4}}{\text{2}}\)
- Area = 6 square meters
So, the area of the triangle is 6 square meters.

2. Three Sides (SSS) Method - Heron's Formula

When all three sides of a triangle are known, Heron's formula can be used to calculate the area. First, compute the semi-perimeter \( s \), then use it to find the area. The formulas are:

\( s = \frac{a + b + c}{2} \)

Area = \( \sqrt{s(s-a)(s-b)(s-c)} \)

where \( a \), \( b \), and \( c \) are the lengths of the sides.

Example: Calculate the area of a triangle with sides 3 meters, 4 meters, and 5 meters:
1
Calculate the semi-perimeter \( s \): \( s = \frac{3 + 4 + 5}{2} = 6 \)
2

Apply Heron's formula:

Area = \( \sqrt{6(6-3)(6-4)(6-5)} = \sqrt{6 \times 3 \times 2 \times 1} \)

Area = \( \sqrt{36} = 6 \) square meters

3
So, the area of the triangle is 6 square meters.

3. Two Sides + Angle Between (SAS) Method

If two sides and the angle between them are known, the area can be calculated using the sine of the angle. The formula is:

Area = \( \frac{1}{2} \times a \times b \times \sin(\theta) \)

where \( a \) and \( b \) are the two sides, and \( \theta \) is the angle between them (in radians). If the angle is given in degrees, convert it to radians using \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \).

Example: Calculate the area of a triangle with sides 4 meters and 5 meters, and the angle between them as 30 degrees.:
1
Convert the angle to radians: \( 30^\circ \times \frac{\pi}{180} = \frac{\pi}{6} \)
2
Calculate \( \sin(30^\circ) = 0.5 \)
3

Apply the formula:

Area = \( \frac{1}{2} \times 4 \times 5 \times \sin(30^\circ) \)

Area = \( \frac{1}{2} \times 4 \times 5 \times 0.5 = 5 \) square meters

So, the area of the triangle is 5 square meters.

4. Two Angles + Side Between (ASA) Method

When two angles and the side between them are known, we can find the area by first calculating the third angle, then using the Law of Sines to find the other sides, and finally applying the area formula with two sides and the included angle. The steps are:

Third angle: \( \gamma = 180^\circ - \alpha - \beta \)

Use Law of Sines: \( \frac{c}{\sin(\gamma)} = \frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} \)

Area: \( \text{Area} = \frac{1}{2} \times a \times b \times \sin(\gamma) \)

Example: Calculate the area of a triangle with angles 30 degrees and 60 degrees, and the side between them as 4 meters:
1
Calculate the third angle: \( \gamma = 180^\circ - 30^\circ - 60^\circ = 90^\circ \)
2

Use the Law of Sines to find the other sides:

Let the side between the angles be \( c \). Then:

\( \frac{c}{\sin(\gamma)} = \frac{a}{\sin(\alpha)} \)

\( a = \frac{c \times \sin(\alpha)}{\sin(\gamma)} = \frac{4 \times \sin(30^\circ)}{\sin(90^\circ)} = \frac{4 \times 0.5}{1} = 2 \) meters

\( b = \frac{c \times \sin(\beta)}{\sin(\gamma)} = \frac{4 \times \sin(60^\circ)}{\sin(90^\circ)} = \frac{4 \times \frac{\sqrt{3}}{2}}{1} \approx 3.46 \) meters

3

Calculate the area::

Area = \( \frac{1}{2} \times a \times b \times \sin(\gamma) \).

Area = \( \frac{1}{2} \times 2 \times 3.46 \times \sin(90^\circ)\)

Area \( \approx \frac{1}{2} \times 2 \times 3.46 \times 1 \approx 3.46 \) square meters So, the area of the triangle is approximately 3.46 square meters.