Triangle Area Calculator
Calculates the area of any triangle from two sides and the included angle (the SAS formula): A = ½·a·b·sin(C). Perfect for surveying, navigation, and engineering tasks where the perpendicular height isn't directly available but two sides and the angle between them are known.
Last updated: May 2026
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About this calculator
When the perpendicular height of a triangle is unknown but two sides and the included angle are known, the area follows the SAS (Side-Angle-Side) formula: A = ½ × a × b × sin(C), where a and b are the two known sides and C is the angle formed between them. The formula works because the height of the triangle relative to side a can be expressed as b × sin(C), where the angle must be converted from degrees to radians (radians = degrees × π / 180) before computing sine. Substituting this into the standard A = ½ × base × height gives the SAS formula directly. Variables: side1 and side2 (two known sides in any consistent linear unit), angle (the included angle between them, in degrees). The formula is valid for any triangle as long as the angle is the one formed between the two known sides — not an exterior angle or an angle opposite one of the sides. Edge cases: at C = 90° (a right triangle), sin(90°) = 1 and the formula reduces to A = ½ × a × b, the familiar right-triangle area where the two legs serve as base and height. At C = 180° (collinear points), sin(180°) = 0 and the area is zero, which correctly represents a degenerate triangle. For angles greater than 180° or negative angles, the formula still works mathematically because sine is periodic, but such inputs typically indicate a geometric error. The result is in square units matching the input dimensions. For triangles defined by three sides only, use Heron's formula instead; for two angles and a side, use the Law of Sines first to find the missing side.
How to use
Example 1: Triangle with side1 = 8 units, side2 = 5 units, included angle = 30°. Step 1: compute sin(30°) = 0.5. Step 2: apply formula — A = 0.5 × 8 × 5 × 0.5 = 10 square units. Verify: the perpendicular height from side2 to the opposite vertex is 5 × sin(30°) = 2.5 units, so A = ½ × base × height = ½ × 8 × 2.5 = 10 — matches. Example 2: A surveyor measures two sides of a triangular plot as 120 m and 95 m, with an angle of 67° between them. Step 1: sin(67°) ≈ 0.9205. Step 2: A = 0.5 × 120 × 95 × 0.9205 ≈ 5246.9 m². Verify: at 67° the sine is close to 1 (max area occurs at 90°), so the area should be close to ½ × 120 × 95 = 5700 m² — the result 5247 m² is appropriately slightly smaller, confirming the calculation.
Frequently asked questions
How do I find the area of a triangle when I only know two sides and the included angle?
Use the SAS area formula: A = ½ × side1 × side2 × sin(angle), where the angle is the one formed between the two known sides (the included angle). This formula works for any triangle — acute, right, or obtuse. It derives from the fact that the triangle's height relative to one side can be computed as the other side multiplied by the sine of the included angle. Make sure to use the included angle — the one directly between the two sides — not an angle opposite a side. The result inherits the square of whichever linear unit you use; entering sides in meters gives area in square meters.
What is the difference between the SAS formula and the standard base-times-height formula?
The standard formula A = ½ × base × height requires the perpendicular height, which is often hard to measure directly in the field. The SAS formula A = ½ × side1 × side2 × sin(angle) calculates the same area by deriving the height trigonometrically from the included angle. Both formulas give identical results — the SAS version is more practical when two sides and an angle are known but no altitude has been measured. Surveyors and navigators favor SAS because they can measure side lengths with tape or rangefinder and angles with a theodolite or compass without needing to drop perpendiculars. For three sides only (no angle), use Heron's formula instead.
Why does the triangle area formula use the sine of the included angle?
The sine of the included angle extracts the component of the second side that is perpendicular to the first side — exactly the height needed for the area formula. If you consider side1 as the base, then the height of the triangle from the opposite vertex is side2 × sin(angle). Substituting into the base-height formula gives A = ½ × side1 × (side2 × sin(angle)) = ½ × side1 × side2 × sin(angle). This is also why area is maximized when the included angle is 90° (sin(90°) = 1) and minimized as the angle approaches 0° or 180° (where sin approaches 0). At those extremes the triangle degenerates to a line segment with zero area.
What are common mistakes when using the SAS triangle area formula?
Using an angle that is not the included angle (the angle between the two known sides) gives wrong results — make sure C is sandwiched between sides a and b. Forgetting the ½ factor doubles the area. Forgetting to set the trig function to degree mode and treating the angle as radians gives wildly wrong sine values. Confusing side lengths — using a side adjacent to a different vertex than the angle's vertex — produces incorrect output. For obtuse triangles where the included angle is between 90° and 180°, the formula still gives the correct positive area because sine remains positive in that range. Inputting an angle outside (0°, 180°) is geometrically meaningless for a valid triangle, even though the formula produces a number.
When should I NOT use this SAS triangle area formula?
If you know only the three sides and no angles (SSS configuration), use Heron's formula A = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2 — this is purely algebraic and doesn't need trig. If you know one side and two angles (ASA or AAS), first apply the Law of Sines to find a missing side, then use this SAS formula or the base-height formula. For triangles defined by three vertex coordinates (x, y), the shoelace formula is more direct. For triangles on a curved surface (spherical triangles on Earth, hyperbolic triangles in non-Euclidean geometry), use specialized spherical or hyperbolic area formulas that account for curvature. Degenerate cases — where the included angle is 0° or 180° (three collinear points) — produce area 0 correctly, but verify your inputs make geometric sense before relying on the answer.