How to Calculate the Area and Perimeter of Triangles

Triangles are the simplest polygons, yet they appear everywhere in engineering, architecture, navigation, and art. Knowing how to calculate a triangle's area and perimeter is a foundational skill in geometry. This guide covers three methods for area, perimeter calculations, worked examples for different triangle types, and the special properties of equilateral, isosceles, and right triangles.

Method 1: Base and Height

The most familiar formula for the area of a triangle uses the base length b and the perpendicular height h:

This works for any triangle, provided you can identify (or measure) the height perpendicular to the chosen base. If the triangle is right-angled, the two shorter sides serve as the base and height directly.

Method 2: Heron's Formula

When you know all three side lengths but not the height, Heron's formula is ideal. Given sides a, b, and c, first calculate the semi-perimeter:

Then the area is:

Heron's formula is particularly useful in surveying and construction, where side lengths can be measured directly with a tape but heights are harder to obtain.

Method 3: Trigonometric Formula

When you know two sides and the included angle, use the trigonometric area formula. If sides a and b enclose angle C:

This method is common in navigation, physics (calculating the area of force parallelograms), and any context where angles are measured directly.

Perimeter of a Triangle

The perimeter is simply the sum of all three sides:

If you only know two sides and an angle, use the cosine rule to find the third side first:

Worked Example 1: Right Triangle (Base and Height)

Problem: A right triangle has legs of 6 cm and 8 cm. Find its area and perimeter.

Since the legs are perpendicular, they serve as the base and height:

For the perimeter, find the hypotenuse using Pythagoras:

The area is 24 cm² and the perimeter is 24 cm. (This is actually a well-known 3-4-5 triple scaled by 2.)

Worked Example 2: Scalene Triangle (Heron's Formula)

Problem: A triangle has sides of 7 m, 9 m, and 12 m. Find the area.

Calculate the semi-perimeter:

Apply Heron's formula:

The area is approximately 31.30 m². The perimeter is simply m.

Worked Example 3: Using the Trigonometric Method

Problem: Two sides of a triangle measure 10 cm and 14 cm, and the included angle is 30°. Find the area.

The area is 35 cm². Notice how a relatively small angle (30°) produces a smaller area than you might expect for those side lengths.

Worked Example 4: Equilateral Triangle

Problem: Find the area and perimeter of an equilateral triangle with side length 6 cm.

For an equilateral triangle, there is a dedicated formula:

Special Triangle Properties

Equilateral Triangle

• All three sides are equal; all angles are 60°.
• Area formula: .
• Height: .

Isosceles Triangle

• Two sides are equal; the base angles are equal.
• The height from the apex bisects the base, creating two congruent right triangles.
• If the equal sides are a and the base is b, the height is .

Right Triangle

• One angle is 90°. The two legs are perpendicular, so area is simply .
• The hypotenuse is found via Pythagoras: .
• Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25.

Tips for Triangle Calculations

• Always check the triangle inequality: any side must be less than the sum of the other two. If , the triangle is impossible.
• Ensure your calculator is set to the correct angle mode (degrees or radians) when using trigonometric methods.
• Heron's formula avoids trigonometry entirely, making it ideal when only side lengths are known.

Try It Yourself

Ready to practise? Use our free Triangle Calculator to compute area, perimeter, angles, and side lengths for any triangle. Enter the values you know and the calculator will determine the rest.