SAS Triangle Calculator (Side-Angle-Side)
Enter two sides and the angle between them (Side-Angle-Side) to fully solve the triangle. The calculator finds the third side using the Law of Cosines, determines all remaining angles with the Law of Sines, then computes area, perimeter, semiperimeter, three altitudes, three medians, the inradius and the circumradius. Results update as you type. Switch between degrees and radians for the angle input.
What is the SAS (Side-Angle-Side) case?
SAS describes a triangle where you know the lengths of two sides and the measure of the angle sitting between them - the included angle. This combination uniquely determines the entire triangle: there is no ambiguity, unlike the SSA case (two sides and a non-included angle) which can produce zero, one, or two valid triangles. Because the angle is locked between the two known sides, the Law of Cosines gives the third side directly, and the remaining angles follow from the Law of Sines plus the 180-degree angle-sum rule.
How the formulas work step by step
Step 1 - find side c with the Law of Cosines: c = sqrt(a squared + b squared - 2ab times cos(gamma)). This is a generalisation of the Pythagorean theorem (when gamma is 90 degrees, the cos term vanishes and you get the classic a squared + b squared = c squared). Step 2 - find angle alpha with the Law of Sines: sin(alpha) / a = sin(gamma) / c, so alpha = arcsin(a times sin(gamma) / c). Because gamma is the included angle and therefore less than 180 degrees, alpha is always the smaller of the two unknown angles when a is the shorter side, so the arcsin is unambiguous. Step 3 - find beta from the angle sum: beta = 180 - gamma - alpha. Step 4 - find area: A = (1/2) times a times b times sin(gamma). This formula uses only the original given values and avoids rounding errors that would accumulate if you first computed c then applied Heron's formula. All other properties (altitudes, medians, inradius, circumradius) follow from the three sides and the area.
Altitudes, medians, inradius and circumradius
Altitude h_a is the perpendicular height of the triangle measured down to side a: h_a = 2A / a. Similarly h_b = 2A / b and h_c = 2A / c. These are useful in calculating the area independently as a check. The median m_a goes from vertex A to the midpoint of the opposite side a. Its length is m_a = (1/2) times sqrt(2b squared + 2c squared - a squared), and analogous formulas give m_b and m_c. All three medians meet at the centroid, which is the triangle's centre of mass. The inradius r is the radius of the largest circle that fits inside the triangle: r = A / s, where s is the semiperimeter. The circumradius R is the radius of the circle that passes through all three vertices: R = c / (2 times sin(gamma)). Euler's inequality guarantees R is always at least 2r, with equality only for the equilateral triangle.
Common applications of the SAS triangle
SAS problems arise constantly in engineering, navigation, surveying, and architecture. A surveyor who can measure two distances from a known point and the angle between them can calculate the distance to a third point without walking to it. In navigation, knowing two legs of a journey and the turning angle lets you compute the direct return distance. Structural engineers use SAS geometry to determine truss member lengths from given support distances and joint angles. In physics, vector addition follows the same cosine-rule algebra when you know two vector magnitudes and the angle between them.
SAS triangle - standard properties overview
| Property | Formula | Notes |
|---|---|---|
| Third side c | c = sqrt(a^2 + b^2 - 2ab*cos(gamma)) | Law of Cosines |
| Angle alpha | sin(alpha) = a*sin(gamma)/c | Law of Sines |
| Angle beta | beta = 180 - alpha - gamma | Angle sum property |
| Area A | A = (1/2)*a*b*sin(gamma) | Direct from given values |
| Perimeter P | P = a + b + c | Sum of all sides |
| Semiperimeter s | s = P / 2 | Used in Heron's formula and inradius |
| Altitude h_a | h_a = 2A / a | Height to side a |
| Altitude h_b | h_b = 2A / b | Height to side b |
| Altitude h_c | h_c = 2A / c | Height to side c |
| Median m_a | m_a = (1/2)*sqrt(2b^2 + 2c^2 - a^2) | From vertex A |
| Median m_b | m_b = (1/2)*sqrt(2a^2 + 2c^2 - b^2) | From vertex B |
| Median m_c | m_c = (1/2)*sqrt(2a^2 + 2b^2 - c^2) | From vertex C |
| Inradius r | r = A / s | Radius of inscribed circle |
| Circumradius R | R = c / (2*sin(gamma)) | Radius of circumscribed circle |
All values computed from the two given sides (a, b) and the included angle (gamma) using the Law of Cosines and Law of Sines.
Frequently asked questions
What makes SAS different from the other triangle cases?
In the SAS case the angle is between the two known sides, which means it uniquely determines the triangle - there is exactly one triangle that can be formed, and the Law of Cosines gives the third side directly. Contrast this with SSA (two sides and a non-included angle) where the triangle might not exist, or two different triangles could satisfy the conditions, creating the so-called ambiguous case.
Can I use the SAS calculator if my angle is in radians?
Yes. Switch the Angle input selector to Radians before entering your value. The calculator converts internally to degrees for the displayed results, so all output angles are always shown in degrees regardless of the input mode.
Why does the calculator output all three angles even though I gave one?
The included angle you enter (gamma) is echoed back after unit conversion to confirm what was used. The two remaining angles (alpha and beta) are computed from the Law of Sines and the angle-sum property. Having all three lets you verify the result: they must sum to exactly 180 degrees.
What do the inradius and circumradius tell me?
The inradius is the radius of the inscribed circle - the largest circle that fits snugly inside the triangle touching all three sides. The circumradius is the radius of the circumscribed circle that passes through all three vertices. Their ratio R / r is related to triangle shape: for an equilateral triangle R = 2r exactly; for any other triangle R is larger than 2r. These values appear in many geometry proofs and physical models such as structural trusses and optical reflectors.
What happens if my two sides are equal (an isosceles triangle)?
If a equals b, the two base angles alpha and beta will be equal. The calculator handles this correctly: the Law of Sines gives alpha, and beta = 180 - gamma - alpha. For an isosceles triangle both base angles end up identical, which you can verify in the output.
How do I find the area of an SAS triangle?
Use the formula A = (1/2) times a times b times sin(gamma), where a and b are the two given sides and gamma is the included angle. This works for any SAS triangle without needing to find the third side first. It is equivalent to (1/2) times base times height because b times sin(gamma) is the height when a is the base.