How to Use the Area of a Triangle Calculator
- Choose the method that matches what you know.
- Pick the length unit.
- Enter the base.
- Enter the perpendicular height.
- Read the area, plus perimeter, angles and type when known.
Pick the method that matches what you know. Base and height is the classic school formula. Three sides uses Heron’s formula, so you do not need the height at all. Two sides + angle works when you know two sides and the angle between them. Coordinates finds the area from the x and y values of the three corners, which is common in coordinate geometry and surveying.
Choose a length unit and type the measurements. The area appears immediately in square units, and whenever the triangle is fully determined you also get the perimeter, the three angles and the type of triangle. The note below the result shows the formula with your numbers substituted, and the sketch is drawn to scale.
Triangle Area Formulas
Heron: s = (a + b + c) ÷ 2, A = √(s(s − a)(s − b)(s − c))
Two sides and included angle: A = ½ab sin C
Coordinates: A = ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|
All four formulas give the same area for the same triangle. The height in the first formula must be perpendicular to the base, measured from the base (or its extension) to the opposite corner. In the angle formula, C must be the angle between sides a and b.
Worked Examples
- Base and height: b = 10 cm and h = 6 cm give A = ½ × 10 × 6 = 30 cm².
- Three sides: for sides 3, 4 and 5, s = 6, so A = √(6 × 3 × 2 × 1) = √36 = 6. This is a right triangle, which you can confirm with ½ × 3 × 4 = 6.
- Two sides and an angle: sides 7 and 9 with a 40° angle between them give A = ½ × 7 × 9 × sin 40° = 20.2478. The third side is 5.78605.
- Coordinates: A(1, 2), B(5, 7) and C(8, 3) give ½|1(7 − 3) + 5(3 − 2) + 8(2 − 7)| = ½|4 + 5 − 40| = 15.5 square units.
You can check a result by using a second method. For the 3, 4, 5 triangle, Heron’s formula gives 6, and since it is a right triangle the legs 3 and 4 are also a base and height: ½ × 3 × 4 = 6.
Which Formula Should You Use?
| You know | Use | Example |
|---|---|---|
| A side and the perpendicular height | ½bh | Most textbook problems |
| All three sides | Heron’s formula | A triangular garden bed |
| Two sides and the angle between them | ½ab sin C | Trigonometry problems |
| Corner coordinates | Shoelace formula | Graph paper, maps, CAD |
An equilateral triangle with side s has the short formula A = (√3 ÷ 4)s², which is what Heron’s formula gives when all three sides are equal.
Tips and Common Mistakes
- Do not use a slanted side as the height. The height must meet the base at a right angle.
- For an obtuse triangle, the height from the top corner may fall outside the base. The formula still works.
- Three lengths only make a triangle if the two shorter ones add up to more than the longest. The calculator warns you when they do not.
- Area is in square units. If the sides are in centimeters, the area is in square centimeters.
- In the angle formula, make sure your angle unit matches: 40 degrees is very different from 40 radians.
Converting Area Units
Because area is two-dimensional, unit conversions are squared. One square meter is 100 × 100 = 10,000 square centimeters, and one square foot is 12 × 12 = 144 square inches. If your sides are in different units, convert them to one unit before calculating; mixing feet and inches is a common source of wrong answers.
Limits
The calculator works with flat (Euclidean) triangles. Very large triangles on the Earth’s surface, such as those between cities, need spherical geometry. Coordinates are treated as points on a flat grid, so the area is in square units of whatever the coordinates measure.
Frequently asked questions
What is the formula for the area of a triangle?
The basic formula is half the base times the height, A = ½bh, where the height is perpendicular to the base. Other formulas use three sides, two sides and an angle, or coordinates.
How do you find the area of a triangle with three sides?
Use Heron's formula. Find s, half the perimeter, then take the square root of s(s − a)(s − b)(s − c). For sides 3, 4 and 5 this gives an area of 6.
How do you find the area with two sides and an angle?
Multiply the two sides, multiply by the sine of the angle between them, and halve the result: A = ½ab sin C. The angle must be the one formed by those two sides.
How do I find the area from coordinates?
Use the shoelace formula: half the absolute value of x1(y2 − y3) + x2(y3 − y1) + x3(y1 − y2). It works for any three points that are not on one straight line.
Does the height have to be inside the triangle?
No. In an obtuse triangle the height from the top corner can fall outside the base, on its extension. The area formula ½bh still gives the correct answer.
What units is triangle area measured in?
Area uses square units of the length you measured. Sides in centimeters give square centimeters, and sides in feet give square feet. Convert all sides to one unit first.