Triangle Area Calculator

Calculate triangle properties including area, perimeter, and angles

Area Calculation

Length Unit

Base (cm)

Height (cm)

Triangle Properties

Area Formula: ½ × base × height

Heron's Formula: √[s(s-a)(s-b)(s-c)] where s = (a+b+c)/2

Law of Cosines: c² = a² + b² - 2ab cos(C)

Law of Sines: a/sin(A) = b/sin(B) = c/sin(C)

Triangle Inequality: Sum of any two sides > third side

Angle Sum: All angles sum to 180°

What Is a Triangle?

A triangle is a basic shape formed by three straight lines that meet at three points called vertices. Each vertex has an angle. The triangle's simple structure makes it unique. It is the simplest closed shape with three sides. Despite its simplicity, a triangle plays a vital role in fields such as mathematics, construction, design, and science. Here are some essential features of a triangle.

Key Characteristics of a Triangle

A triangle has several fundamental properties. Before exploring its types, let's review these essential features:

Three sides: Each side is a straight line segment connecting two vertices.
Three angles: An angle is formed at each vertex where two sides intersect.
A closed shape: Its sides enclose a specific area.
Flat (2D): A triangle lies on a plane, having length and width but no thickness.

A vital rule is that the sum of a triangle's interior angles is always 180 degrees. This remains true regardless of the triangle's size or shape. This basic property is fundamental in geometry and key to solving many angle-related problems.

Types of Triangles (Based on Sides)

Triangles are categorized by their side lengths:

Equilateral triangle: All three sides are equal, and each interior angle measures 60°.
Isosceles triangle: Two sides are equal, resulting in two equal angles.
Scalene triangle: All three sides and angles are distinct.

Types of Triangles (Based on Angles)

Triangles can also be classified by their angles:

Acute triangle: All three angles are less than 90°.
Right triangle: One angle is exactly 90°, and the side opposite this angle, called the hypotenuse, is always the longest side.
Obtuse triangle: One angle exceeds 90°, while the other two are acute.

Area of a Triangle

The area of a triangle is the space enclosed by its three sides. Whether the triangle is narrow, wide, tilted, or balanced, its area is always expressed in square units, such as cm², m², or in². A triangle's area is exactly half that of a parallelogram with the same base and height. This explains why the standard area formula for a triangle includes a ½.

1) Calculating Area with Base and Height

Formula

The area (A) of a triangle is calculated as:

A = 1/2 bh

'A' denotes the area.
'b' is the length of any side designated as the base.
'h' signifies the height, the perpendicular distance from the chosen base to the opposite vertex.

Understanding "height"

Height is not limited to a side; in a right triangle, the legs serve as the base and height. The height is the perpendicular line from the base to the opposite vertex, forming a right angle with the base. Any side can be used as the base, but the height must drop straight from it. In triangles with larger angles, the height might extend outside the triangle, yet it still intersects the base at a right angle.

2) Area Calculation Using All Three Sides (Heron's Formula)

This approach is effective when you know the side lengths a, b, and c but not the height.

Step 1: Calculate the semiperimeter:

The semiperimeter is half of the triangle's perimeter. To determine it, sum up all three sides and divide by two: s = (a + b + c) / 2.

Step 2: Apply Heron's formula:

A = √(s(s - a)(s - b)(s - c))

Heron's formula allows you to find the area of a triangle using only its three sides. It does not require knowing the angles or height, making it especially useful in practical situations where measuring height can be challenging.

3) Calculating Area Using Two Sides and Included Angle (SAS Area Formula)

Knowing two sides and the included angle allows you to directly compute the area with the sine function. For instance, if sides a and b form the included angle C, the area is A = (1/2)ab sin(C). This formula can be adapted based on the known values: if sides b and c with included angle A are known, then A = (1/2)bc sin(A); if sides a and c with included angle B are known, then A = (1/2)ac sin(B). The sine function is used because it represents the "vertical component" of the triangle, which helps determine the height from a slanted side. For example, when side b is slanted, its height is b sin(C), making the area A = (1/2)a(b sin(C)).

4) Using Two Angles and the Included Side (ASA / AAS)

Here, you are given two angles (A and B) and the side between them (usually labeled c, opposite angle C).

Step 1: Find the third angle.

Since the sum of angles in a triangle is 180°:

C = 180° - (A + B)

Step 2: Use the area formula.

When side c is between angles A and B, the area is:

A = (c² sin A sin B) / (2 sin C)

Why this formula works.

Knowing the angles allows the Law of Sines to relate the unknown sides to the known side. This formula combines the Law of Sines with the area formula, providing a clear expression.

Practical Tips:

Ensure the angle in the sine formula is the included angle between the two known sides.
For a base-height calculation, confirm the height is perpendicular to the base.
When applying Heron's formula, verify the side lengths satisfy the triangle inequality: each side must be shorter than the sum of the other two.
Label your triangle clearly to avoid mixing up which angle pairs with which side.