- Which three values do I need to solve a triangle?
- Any three of the six parts, as long as at least one is a side. That gives five cases: SSS (three sides), SAS (two sides and the angle between them), ASA (two angles and the side between them), AAS (two angles and a side that isn't between them) and SSA (two sides and an angle that isn't between them). Three angles alone won't work. Angles fix the shape, but a triangle with angles 50°, 60° and 70° can be any size, so you need one side to pin it down.
- When does the calculator use the law of sines vs the law of cosines?
- The law of cosines, c² = a² + b² − 2ab·cos C, handles the cases where you know more sides than angles: SSS and SAS. For b = 5, c = 7 and A = 60°, it gives a = 6.245. The law of sines, a / sin A = b / sin B = c / sin C, handles the cases built around two known angles or an angle with its opposite side: ASA, AAS and SSA. With a = 10, A = 45° and B = 60°, the third angle is 75°, and the law of sines gives b = 12.2474 and c = 13.6603. The small tag above the result tells you which case and which law were used.
- Why do some inputs give two triangles?
- That's the ambiguous SSA case. If you know an angle, the side opposite it, and one more side, the opposite side can sometimes swing into two different positions and still close the triangle. With a = 6, b = 8 and A = 30°, angle B can be 41.8103° or 138.1897°. The first triangle has c = 11.4003 and area 22.8007, and the second has c = 2.4561 and area 4.9121. Both are correct. Your problem's picture or wording tells you which one you want, so the calculator shows both under Triangle 1 and Triangle 2.
- Why does it say no triangle fits?
- Some combinations can't close. In the SSA case, the side opposite the known angle has to be long enough to reach the third side. With a = 2, b = 8 and A = 30°, side a would need to be at least 4, so no triangle exists. With three sides, the two shorter ones have to add up to more than the longest: 1, 2 and 3 just lie flat in a line. Two angles that already add up to 180° or more also leave no room for a third.
- How is the area calculated?
- Once all three sides are known, the area comes from Heron's formula: with s = (a + b + c) / 2, the area is √(s(s − a)(s − b)(s − c)). For sides 5, 5 and 8, s = 9 and the area is √(9 × 4 × 4 × 1) = 12. The calculator uses a rearranged version of the formula that stays accurate for long, thin triangles, where the textbook version loses digits. Area is in square units of whatever length you typed, so sides in feet give square feet.
- What do the heights, medians, inradius and circumradius mean?
- A height (altitude) is the perpendicular distance from a corner to the opposite side: hc is the height onto side c. A median runs from a corner to the midpoint of the opposite side. The inradius r is the radius of the largest circle that fits inside the triangle, and the circumradius R is the radius of the circle through all three corners. For the 3-4-5 triangle, hc = 2.4, the median to a is 4.272, r = 1 and R = 2.5. R is exactly half the hypotenuse in every right triangle.
- How does it decide if a triangle is right, acute or obtuse?
- It looks at the largest angle. Exactly 90° is right, over 90° is obtuse, and under 90° is acute. The side class comes first: Equilateral when all three sides match, Isosceles when two match, Scalene when none do. So 3-4-5 reads Scalene · right, 5-5-8 reads Isosceles · obtuse with a 106.2602° angle, and a triangle with all sides 1 reads Equilateral · acute with 60° angles and area 0.433. Tiny floating-point differences are ignored, so a computed 89.99999999° still counts as right.
- Should I use degrees or radians?
- Use whatever your problem uses. School geometry and building work almost always use degrees. Calculus, physics and programming often use radians, where a full turn is 2π and a straight angle is π ≈ 3.1416. Switching the unit converts the angles you've typed, so 30° becomes 0.5236, and every result angle follows along. The 3-4-5 triangle's angles read 0.6435 rad, 0.9273 rad and 1.5708 rad. For standalone conversions, the Degrees to Radians Converter shows the π form too.
