Triangle Area Calculator

The area of any triangle is the amount of flat space enclosed within its three sides. Unlike a rectangle where the math is almost automatic, triangles require a bit more care because their shape can vary so much. A tall skinny triangle and a flat wide one can have the same area even though they look completely different.

The good news is there are several formulas to choose from, and each one is designed for a different set of known measurements. If you know the base and height, use one formula. If you only know the three side lengths, use Heron's formula. If you have two sides and the angle between them, there's a trigonometric approach for that too. You don't need all the measurements at once; you just need the right combination for whichever formula fits your situation.

All of these methods give you the area in square units, so if your measurements are in feet, the area comes out in square feet. If you're working in meters, you get square meters. Keeping your units consistent before you start saves a lot of confusion at the end.

There isn't just one triangle area formula. There are several, each suited to different situations. The three most common are the base-height formula, Heron's formula, and the trigonometric formula using two sides and an included angle.

• Base and height: Area = ½ × base × height
• Heron's formula (three sides known): Area = √[s(s−a)(s−b)(s−c)], where s is the semi-perimeter
• Two sides and included angle: Area = ½ × a × b × sin(C)
• Coordinate method (vertices known): Area = ½ |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|

Each formula is exact, not an approximation, so the choice really comes down to what information you have available. The sections below walk through every one of them in detail.

This is the method most people learn first, and honestly it's the easiest to use when you have the right measurements. All you need is one side of the triangle (the base) and the perpendicular height measured straight up from that base to the opposite vertex.

The reason there's a ½ in the formula is straightforward: a triangle is exactly half of a parallelogram with the same base and height. Slice a rectangle diagonally corner to corner and you get two identical triangles, each with half the area. That geometric relationship is where the formula comes from.

Sometimes you don't know the height of a triangle, but you do know all three side lengths. That's exactly the situation Heron's formula was designed for. It was developed by Hero of Alexandria around the first century AD, and it's been a staple of geometry ever since.

The formula works for any triangle regardless of its shape: acute, obtuse, right, scalene, whatever. As long as the three side lengths form a valid triangle (meaning the sum of any two sides is greater than the third), you can use it.

If you know two sides of a triangle and the angle sitting between them (called the included angle), you can find the area without knowing the height at all. This approach is common in trigonometry and shows up a lot in physics, engineering, and surveying problems.

The formula is elegant because it essentially uses the sine of the angle to figure out what the height would have been, without you needing to measure it directly. It's a practical shortcut when you've got angle measurements from a protractor or a digital tool but not a clean perpendicular height.

Right triangles are the simplest case. Because one of the angles is exactly 90 degrees, the two sides that form that right angle (called the legs) act as the base and height automatically. You don't have to hunt for a perpendicular measurement because it's already built into the shape.

The formula becomes: Area = ½ × leg₁ × leg₂

So a right triangle with legs of 6 feet and 8 feet has an area of ½ × 6 × 8 = 24 square feet. The hypotenuse (the longest side, opposite the right angle) doesn't factor into the area calculation at all. This is why right triangles are so common in practical applications: they're easy to measure and easy to calculate.

An equilateral triangle has all three sides equal and all three angles equal to 60 degrees. Because of its perfect symmetry, there's a simplified formula that skips the need to find the height separately.

The formula is: Area = (√3 / 4) × a², where a is the side length.

If each side is 10 cm, the area is (√3 / 4) × 100 = 25√3 ≈ 43.3 square centimeters. The value √3 / 4 is approximately 0.433, so you can also just multiply 0.433 by the side length squared for a quick decimal result. Equilateral triangles come up a lot in design, architecture, and tiling patterns, so this formula is worth memorizing if you work with them regularly.

An isosceles triangle has two equal sides and two equal base angles. You can use the base-height formula if you know the height, or Heron's formula if you only know the side lengths. Because of its symmetry, the height of an isosceles triangle always bisects the base, which makes it easier to calculate the height using the Pythagorean theorem if needed: h = √(a² − (b/2)²), where a is the equal side and b is the base.

A scalene triangle has all three sides different lengths and no equal angles. There's nothing special about its geometry that simplifies the calculation, so your best options are Heron's formula (when all three sides are known) or the trigonometric formula (when two sides and an included angle are known). The base-height formula also works perfectly well if you can determine the height.

For both types, the formulas described in the earlier sections apply without modification. The distinction in triangle type doesn't change the math; it just determines which measurements you're likely to have available.

Here are a few worked examples covering different scenarios so you can see the formulas in action.

• Example 1 (Base and Height): A triangle with base 14 cm and height 9 cm. Area = ½ × 14 × 9 = 63 cm².
• Example 2 (Heron's Formula): Sides of 3, 4, and 5 units. Semi-perimeter s = (3+4+5)/2 = 6. Area = √[6(6−3)(6−4)(6−5)] = √[6×3×2×1] = √36 = 6 square units. (This is actually a right triangle, which you can verify: 3² + 4² = 5².)
• Example 3 (Two Sides and Angle): Sides of 7 m and 10 m with an included angle of 45°. Area = ½ × 7 × 10 × sin(45°) = 35 × 0.7071 ≈ 24.75 m².
• Example 4 (Equilateral): Side length of 6 inches. Area = (√3/4) × 36 = 9√3 ≈ 15.59 in².
• Example 5 (Right Triangle): Legs of 5 ft and 12 ft. Area = ½ × 5 × 12 = 30 ft².

Notice how the answer changes dramatically depending on the method and inputs, even for similar-looking numbers. Always double-check which formula matches your known values before starting.

If you know the exact positions of all three vertices on a coordinate plane, you can calculate the area using the coordinate (shoelace) formula. This is especially useful in mapping, computer graphics, and analytical geometry where vertex coordinates are easier to obtain than physical measurements.

The formula is: Area = ½ |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|

The vertical bars mean you take the absolute value of the result, since area can't be negative. Let's say the three vertices are A(2, 1), B(6, 1), and C(4, 5). Plug in: Area = ½ |2(1−5) + 6(5−1) + 4(1−1)| = ½ |2(−4) + 6(4) + 4(0)| = ½ |−8 + 24 + 0| = ½ × 16 = 8 square units.

This formula works regardless of where the triangle sits on the plane, and you don't need to figure out which side is the base or where the height falls. If you're working with GPS coordinates or design software, this method is often the most practical one available.

Once you have the area in one unit, you may need to convert it to another. Because area is two-dimensional, conversion factors are squared compared to linear unit conversions. Here's a quick reference:

For example, if your triangle area is 50 square feet and you need it in square meters, multiply 50 × 0.0929 = 4.645 m². Always apply the squared conversion factor, not the linear one, or your answer will be way off.

Triangle area calculations come up in a surprising range of real-world situations. Here are some of the most common ones:

• Construction and roofing: Calculating the area of triangular roof sections to estimate shingles, insulation, or material costs.
• Land surveying: Irregular plots of land are often broken into triangles to calculate total area more easily.
• Interior design: Figuring out how much paint, flooring, or wallpaper a triangular space needs.
• Engineering: Structural analysis often involves triangular trusses and load distribution across triangular elements.
• Navigation and mapping: Triangulation is a method used to determine positions and distances based on triangle geometry.
• Art and graphic design: Working with triangular shapes in layouts, patterns, and digital illustrations.
• Physics: Calculating the work done by a force using area under a force-displacement graph, which is often triangular.

In almost every field that involves measurement or spatial reasoning, triangle area pops up sooner or later. Getting comfortable with at least two or three of the formulas above means you'll be ready for it whenever it does.