Triangle Area Calculator

Calculate the area of any triangle using four different methods with step-by-step solutions.

Select Method & Enter Values

Result

Area
30
square units

Step-by-Step Solution

How to Calculate the Area of a Triangle

The area of a triangle can be computed using several different methods depending on the information available. Whether you know the base and height, all three sides, two sides with an included angle, or the coordinates of the vertices, there is a formula that works for your situation.

Triangle Area Formulas

Base and Height

The simplest formula when you know the perpendicular height from the base to the opposite vertex.

A = (1/2) x b x h

Heron's Formula

Uses all three side lengths. First compute the semi-perimeter s = (a+b+c)/2.

A = sqrt(s(s-a)(s-b)(s-c))

SAS (Two Sides + Angle)

When you know two sides and the included angle between them.

A = (1/2) x a x b x sin(C)

Coordinate (Shoelace)

When you know the (x, y) coordinates of all three vertices.

A = (1/2)|x1(y2-y3)+x2(y3-y1)+x3(y1-y2)|

When to Use Each Method

  • Base and Height: Best for textbook problems where the height is given or easily measured.
  • Heron's Formula: Ideal when you know all three side lengths but not the height.
  • SAS Formula: Use when two sides and the angle between them are known (e.g., surveying).
  • Coordinate Method: Perfect for coordinate geometry problems where vertices are given as points on a plane.

Practical Applications

Triangle area calculations are used in land surveying, architecture, computer graphics, physics (calculating cross-sectional areas), and engineering. The coordinate method is especially common in computational geometry for polygons and mesh calculations.