# Area of a Triangle Calculator

Calculate the area of a triangle using base and height, Heron's formula, or two sides and an included angle. Free triangle area calculator with step-by-step results.

## What this calculates

Find the area of any triangle using the method that matches the information you have. This calculator supports base and height, Heron's formula (three sides), and the SAS formula (two sides and an included angle).

## Inputs

- **Calculation Method** — options: Base & Height, Three Sides (Heron's Formula), Two Sides & Included Angle (SAS) — Select the method based on which values you know.
- **Base** — The base of the triangle (base-height method).
- **Height** — The perpendicular height from base to opposite vertex.
- **Side a** — First side length (Heron's or SAS method).
- **Side b** — Second side length (Heron's or SAS method).
- **Side c** — Third side length (Heron's formula only).
- **Included Angle (degrees)** — The angle between sides a and b (SAS method).

## Outputs

- **Area** — The area of the triangle.
- **Perimeter** — The total length of all three sides.
- **Method Used** — formatted as text — Which formula was applied.
- **Details** — formatted as text — Additional information about the computation.

## Details

There are several ways to calculate the area of a triangle depending on which measurements you know.

**Method 1: Base and Height**

A = (1/2) x base x height

This is the simplest approach when you know the base and the perpendicular height. For a triangle with base 10 and height 6, the area is (1/2)(10)(6) = 30 square units.

**Method 2: Heron's Formula (Three Sides)**

When you know all three side lengths a, b, and c:

1. Compute the semi-perimeter: s = (a + b + c) / 2
2. Area = sqrt(s(s-a)(s-b)(s-c))

For a triangle with sides 3, 4, and 5: s = 6, area = sqrt(6 x 3 x 2 x 1) = sqrt(36) = 6 square units.

**Method 3: SAS (Two Sides and Included Angle)**

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

where C is the angle between sides a and b. For sides 8 and 5 with a 60-degree included angle: A = (1/2)(8)(5)(sin 60°) = 20(0.866) = 17.32 square units.

All three methods produce the same result when applied to the same triangle. Pick whichever matches the data you have.

## Frequently Asked Questions

**Q: What is Heron's formula?**

A: Heron's formula calculates a triangle's area when you know all three sides. First find the semi-perimeter s = (a+b+c)/2, then area = sqrt(s(s-a)(s-b)(s-c)). For a triangle with sides 5, 6, and 7: s = 9, area = sqrt(9 x 4 x 3 x 2) = sqrt(216) = 14.697 square units.

**Q: How do I find the height of a triangle?**

A: If you know the area and base, height = 2 x area / base. Otherwise, for a right triangle the height is one of the legs. For a general triangle, you can use trigonometry: if you know a side and an angle, height = side x sin(angle). You can also drop a perpendicular from any vertex to the opposite side.

**Q: When should I use the SAS formula?**

A: Use the SAS formula A = (1/2)ab sin(C) when you know two sides and the angle between them. This is common in surveying, navigation, and physics problems. The angle must be the included angle, meaning the angle formed where the two known sides meet.

**Q: Can any three side lengths form a triangle?**

A: No. The three sides must satisfy the triangle inequality: the sum of any two sides must be greater than the third. For example, sides 2, 3, and 10 cannot form a triangle because 2 + 3 = 5 < 10. This calculator checks for this and warns you if the sides are invalid.

**Q: What is the area of an equilateral triangle?**

A: For an equilateral triangle with side length s, the area is (sqrt(3)/4) x s². For a side of 6: area = (sqrt(3)/4)(36) = 9sqrt(3) = 15.588 square units. You can also use Heron's formula with a = b = c = s to get the same result.

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Source: https://vastcalc.com/calculators/math/area-of-triangle
Category: Math
Last updated: 2026-04-08