Triangle Side Calculator - Law of Cosines and Law of Sines

A triangle side calculator recovers the missing side of a triangle when you already know enough of the other sides and angles. The form handles SAS (two sides plus the included angle) via the Law of Cosines, and AAS or ASA (one side plus two angles) via the Law of Sines. Type the known values and the calculator returns the missing side, the other sides, all three angles, and the perimeter.

• • Surveying a triangular plot: Measure two sides and the angle between them with a tape and a clinometer, then read the third side directly.
• • Cutting rafters and braces: Recover the third edge of a roof or frame triangle from the two edges you can measure on site and the corner angle you can lay out.
• • Checking a homework problem: Verify a hand calculation for a 3-4-5, 5-12-13, or any other textbook triangle by entering the given sides and angle.

Three pieces of data are enough to fix a Euclidean triangle. With two sides and the angle between them, the calculator applies the Law of Cosines. With one side and two angles, it completes the third angle and applies the Law of Sines. The same form swaps between cases via a single selector.

Enter side lengths in any positive unit and angles in degrees. The result is reported in the same length unit as the input sides, with no extra conversion step on the way to a drawing or cut list.

For the right-triangle special case where the two legs meet at 90 degrees, ABC triangle calculator handles the Pythagorean and trig ratios in one form.

The triangle side calculator picks one of two closed-form identities based on the case you select. SAS uses the Law of Cosines; AAS or ASA uses the Law of Sines. Both paths are pure algebraic formulas with no iteration.

• a, b, c: The three side lengths, with c the side opposite angle C and the side the calculator reports as the missing side.
• A, B, C: The three interior angles in degrees, with A + B + C = 180. In SAS mode you supply C; in AAS mode you supply A and B and the calculator recovers C.
• k = a / sin(A): The common ratio from the Law of Sines. Once you know k from one side-angle pair, every other side is k * sin(opposite angle).

The Law of Cosines works for any triangle, so the SAS path is the most general option when you can measure two sides and the angle between them. The Law of Sines is the right pick when you have an angle at each end of a side you can already measure; the calculator fills in the third angle and uses the common ratio k to recover the other two sides.

If your three pieces of data are three sides, the SSS case applies; the SSS triangle calculator handles that inverse problem.

According to Wolfram MathWorld, Law of Cosines, Law of Cosines c^2 = a^2 + b^2 - 2ab*cos(C) is valid for any planar triangle with sides a, b, c and included angle C

According to Wikipedia, Law of cosines, c^2 = a^2 + b^2 - 2ab*cos(C), which rearranges to c = sqrt(a^2 + b^2 - 2ab*cos(C))

If you have three sides and want the angles, SSS triangle calculator covers the inverse problem and labels the triangle as acute, right, or obtuse.

Four ideas cover the two paths the triangle side calculator uses to recover a missing side. Each is a small, well-known result from Euclidean geometry.

The selector at the top of the form chooses between the Law of Cosines path and the Law of Sines path, and the rest of the form provides the values each path needs.

When you only need to focus on the AAS case without the SAS path, AAS triangle calculator walks the Law of Sines in a single-mode form.

Use the same form for both cases. The mode selector at the top tells the calculator which identity to apply, and the input rows change to match the chosen case.

1. 1 Pick the case that matches your data: Choose SAS if you know two sides and the angle between them. Choose AAS if you know one side and two angles.
2. 2 Enter side a: Type a positive side length in any unit. The result will be reported in the same unit.
3. 3 Enter the second piece of data: For SAS, type side b and the included angle C. For AAS, type angle A and angle B in degrees.
4. 4 Read the missing side: The result panel shows the recovered side c at the top. The label always says 'Missing side (c)' regardless of the case you picked.
5. 5 Read the other angles and sides: The result panel also lists the other two sides and all three angles, so you can see the full triangle.
6. 6 Use the perimeter for layout and cuts: Copy the perimeter for fence, trim, or framing totals. It is the sum of the three sides in the unit you entered.

A contractor measures two sides of a triangular garden bed as 8 m and 11 m and lays out the included corner at 50 degrees with a string line. Switching the form to SAS and entering a = 8, b = 11, C = 50 returns the missing side c = 8.5874 m, plus the two unknown angles and a perimeter of 27.5874 m for ordering edging.

If you need to switch between SSS, SAS, ASA, AAS, and AAA without re-entering values, triangle calculator handles all five cases behind one form.

The triangle side calculator combines the two main identities behind one form, so a single page covers the bulk of the textbook cases without bouncing the user between tools.

• • Handles both main cases: SAS via the Law of Cosines and AAS via the Law of Sines live in the same form, so a user does not have to know which sub-tool to open first.
• • Reports more than the missing side: The result panel also returns the other sides, all three angles, and the perimeter, so the user gets a complete description of the triangle in one submission.
• • Validates inputs up front: Angles must be strictly between 0 and 180, side lengths must be positive, and the two AAS angles must sum to less than 180. The form rejects bad input.
• • Works in any length unit: Type sides in metres, feet, inches, or centimetres. The result is in the same unit, with no extra conversion step on the way to a cut list.

These benefits matter most when the inputs come from a real measurement rather than a textbook. A tape, a clinometer, and the form cover most of the triangles a contractor, surveyor, or student will meet in practice.

For a single-mode form that focuses on the Law of Cosines path only, SAS triangle calculator covers the SAS case in isolation.

The triangle side calculator has a small set of factors and limits. The values are stable across the math-conversion family, but the assumptions below are worth knowing before you trust the output for a cut list or a design calculation.

• • The triangle side calculator covers SAS and AAS only. For three sides and an angle, the SSS case applies; use the dedicated SSS triangle calculator.
• • The formulas assume a planar Euclidean triangle. They do not apply to spherical triangles on a globe, where the Law of Cosines for sides takes a different form.

These limits are not weaknesses of the formulas, they are limits of the input cases. The two identities are exact for any planar Euclidean triangle; the calculator routes the input to the identity that fits.

According to Wikipedia, Law of sines, a / sin(A) = b / sin(B) = c / sin(C) holds for any triangle, and the common ratio lets you recover any unknown side

If the two known angles are on the ends of the known side, the ASA case applies and ASA triangle calculator solves it directly.