Triangle Area Calculator — base-height, Heron and SAS

The triangle area calculator above solves the same problem three different ways, because the measurements available in practice vary. Sometimes you know a base and a perpendicular height. Sometimes you have three side lengths from a survey or a drawing. Sometimes you have two sides and the angle between them. Each route has its own formula, and the tool shows the one it used with your numbers in place.

Arb Digital publishes this alongside a full set of free geometry tools. Beyond the area, it reports the perimeter, the semi-perimeter, the perpendicular height on side a, and the classification of the triangle — right, acute or obtuse — which is often what the wider problem actually needed.

What This Triangle Area Calculator Does

Choose a method, enter the matching measurements, and the tool returns the area in squared units. The base-and-height method is the classic half-base-times-height. Heron's formula works from three side lengths without needing any angle at all. The SAS method uses two sides and the angle between them, which is the form that appears most often in surveying and trigonometry.

Whichever method you pick, the calculator derives the perpendicular height on side a from the area, so you can see the equivalent height even when you never measured one. It also validates the input: three lengths that cannot close into a triangle are rejected with an explanation, and an SAS angle of 0 or 180 degrees produces a flat, zero-area shape, which the tool states plainly rather than returning a meaningless number.

How to Use It

1. Select your method. Base and height, three sides, or two sides plus the included angle.
2. Fill the relevant fields. For base-height use the first two boxes; for Heron use all three side boxes; for SAS use sides a and b plus the angle.
3. Choose your unit. All lengths must be in the same unit — the area comes out in its square.
4. Check the validity message. If three sides cannot form a triangle, the tool says so and explains which inequality fails.
5. Read the working line for the substituted formula, including the semi-perimeter step in Heron's method.

The Formula / How It's Calculated

Base and height: A = ½ × b × h. A triangle is exactly half the rectangle that encloses it on the same base, which is why the half appears. A base of 13 cm with a perpendicular height of 14 cm gives ½ × 13 × 14 = 91 cm².

Heron's formula: A = √(s(s−a)(s−b)(s−c)), where s is the semi-perimeter (a + b + c) ÷ 2. For the loaded 13, 14, 15 triangle: s = 42 ÷ 2 = 21, and A = √(21 × 8 × 7 × 6) = √7056 = 84 cm². That triangle is a well-known example precisely because the answer is a whole number despite none of the sides being a right-angle pair. MathWorld's page on Heron's formula gives the derivation and its history.

SAS: A = ½ × a × b × sin(C), where C is the angle between sides a and b. With a = 13, b = 14 and C = 60°, sin 60° = 0.8660, so A = ½ × 13 × 14 × 0.8660 = 78.81 cm². The reason this works is that b × sin(C) is the perpendicular height on side a — the SAS formula is the base-height formula with trigonometry doing the measuring for you. The tool also recovers the missing third side using the law of cosines, c² = a² + b² − 2ab·cos(C), which is set out in MathWorld's entry on the law of cosines and reduces to Pythagoras when the angle is 90 degrees.

The Triangle Inequality: When Three Lengths Are Not a Triangle

Any two sides of a triangle must add up to more than the third. Sides of 2, 3 and 10 cannot close — the short pair cannot reach across the long one no matter how they are angled. Heron's formula detects this automatically: one of the bracketed terms goes negative, the product under the square root turns negative, and there is no real answer.

That is a genuinely useful diagnostic. If Heron's formula returns an error rather than an area, the measurements are wrong, not the formula. In practice this catches transcription errors, mixed units, and cases where a diagonal has been recorded as a side. The boundary case, where two sides add to exactly the third, gives an area of zero: a degenerate triangle, which is really just a straight line.

Why the Height Must Be Perpendicular

The height in A = ½bh is the perpendicular distance from the base to the opposite vertex, not the length of a sloping side. In an obtuse triangle, that perpendicular can fall outside the triangle entirely, meeting the extension of the base rather than the base itself. The formula still works — the height is measured to the line containing the base, not to the segment.

Using a slant length instead of the true height always overstates the area, because the slant is the hypotenuse of a right triangle whose vertical leg is the real height. That is a systematic error, never a random one, and it is why roof areas, ramp surfaces and hillside plots are so often calculated too large. If you have the slant length and the base angle, recover the true height with the Pythagorean theorem calculator before applying the area formula.

Every Triangle With the Same Base and Height Has the Same Area

Slide the top vertex of a triangle sideways, keeping its height the same, and the shape changes completely while the area does not move at all. A tall, symmetric triangle and a long, leaning one with the same base and height are equal in area. This is the reason area alone can never tell you the shape of a triangle.

The practical use is comparison. A plot of land recorded only as an area could be almost any shape, and its perimeter — the fencing cost — could vary enormously. Of all triangles with a given perimeter, the equilateral one encloses the most area, which is a specific case of the general result that regular shapes are the most efficient enclosers. That is exactly why the tool reports the perimeter alongside the area: two triangles with equal area can have very different boundary lengths and very different costs.

Classifying a Triangle From Its Sides

Compare the square of the longest side with the sum of the squares of the other two. If a² + b² = c², the triangle has a right angle. If the sum is greater, every angle is acute. If it is smaller, one angle is obtuse. That single comparison, an extension of Pythagoras, classifies any triangle from three lengths alone.

The 13-14-15 triangle loaded in the calculator is acute: 13² + 14² = 365, comfortably more than 15² = 225. A 3-4-5 triangle is exactly right-angled, since 9 + 16 = 25. And 2-3-4 is obtuse, because 4 + 9 = 13 is less than 16. The tool applies this test automatically and reports the result in the fourth box, which is a quick way to check a drawing against its dimensions.

Choosing Between Heron and SAS in Real Work

Heron's formula only needs a tape measure, which makes it the practical choice on site: measure three sides, no angles required, no perpendicular to establish. Its weakness is numerical. For a very thin sliver of a triangle, where one side is nearly the sum of the other two, the bracketed terms nearly cancel and the result loses accuracy in floating-point arithmetic. Surveyors working with long, narrow parcels use a rearranged, numerically stable version of the same formula for exactly this reason.

SAS is more accurate for thin triangles but requires an angle measurement, which is harder to take precisely than a length. As a rule: use three sides when the triangle is reasonably well-proportioned and lengths are what you have, and use SAS when you have reliable angle data or when the triangle is very thin. The scientific calculator covers the trigonometric evaluation if you want to check the sine step by hand, and the area converter moves the result between systems.

Common Mistakes to Avoid

• Using a sloping side as the height — only the perpendicular distance to the base belongs in ½bh, and a slant always overstates the area.
• Forgetting the half — base times height gives the enclosing rectangle, which is exactly twice the triangle.
• Using a non-included angle in the SAS formula — the angle must sit between the two sides you entered, not opposite one of them.
• Mixing units — a base in metres with a height in centimetres produces an answer that is wrong by a factor of a hundred.
• Trusting Heron's formula on a very thin triangle — near-cancellation in the brackets erodes precision, so prefer SAS for slivers.

Related Free Tools From Arb Digital

Right triangles are handled directly by the Pythagorean theorem calculator, and curved shapes by the circle calculator. Heron's formula ends in a square root, which the square root calculator can simplify exactly, and the scientific calculator handles the sine step in the SAS method. Convert your result with the area converter, or browse the full free online tools hub.

Frequently Asked Questions

This tool is provided for general educational and planning use. Verify critical dimensions on site before cutting, ordering or building from any calculated figure.