"SAS" means "Side, Angle, Side" Show
To solve an SAS triangle
Example 1
In this triangle we know:
To solve the triangle we need to find side a and angles B and C. Use The Law of Cosines to find side a first: a2 = b2 + c2 − 2bc cosA a2 = 52 + 72 − 2 × 5 × 7 × cos(49°) a2 = 25 + 49 − 70 × cos(49°) a2 = 74 − 70 × 0.6560... a2 = 74 − 45.924... = 28.075... a = √28.075... a = 5.298... a = 5.30 to 2 decimal places Now we use the The Law of Sines to find the smaller of the other two angles. Why the smaller angle? Because the inverse sine function gives answers less than 90° even for angles greater than 90°. By choosing the smaller angle (a triangle won't have two angles greater than 90°) we avoid that problem. Note: the smaller angle is the one facing the shorter side. Choose angle B: sin B / b = sin A / a sin B / 5 = sin(49°) / 5.298... Did you notice that we didn't use a = 5.30. That number is rounded to 2 decimal places. It's much better to use the unrounded number 5.298... which should still be on our calculator from the last calculation. sin B = (sin(49°) × 5) / 5.298... sin B = 0.7122... B = sin−1(0.7122...) B = 45.4° to one decimal place Now we find angle C, which is easy using 'angles of a triangle add to 180°': C = 180° − 49° − 45.4° C = 85.6° to one decimal place Now we have completely solved the triangle i.e. we have found all its angles and sides. Example 2
This is also an SAS triangle. First of all we will find r using The Law of Cosines: r2 = p2 + q2 − 2pq cos R r2 = 6.92 + 2.62 − 2 × 6.9 × 2.6 × cos(117°) r2 = 47.61 + 6.76 − 35.88 × cos(117°) r2 = 54.37 − 35.88 × (−0.4539...) r2 = 54.37 + 16.289... = 70.659... r = √70.659... r = 8.405... = 8.41 to 2 decimal places Now for The Law of Sines. Choose the smaller angle? We don't have to! Angle R is greater than 90°, so angles P and Q must be less than 90°. sin P / p = sin R / r sin P / 6.9 = sin(117°) / 8.405... sin P = ( sin(117°) × 6.9 ) / 8.405... sin P = 0.7313... P = sin−1(0.7313...) P = 47.0° to one decimal place Now we will find angle Q using 'angles of a triangle add to 180°': Q = 180° − 117° − 47.0° Q = 16.0° to one decimal place Mastering this skill needs lots of practice, so ... Created by Hanna Pamuła, PhD candidate Reviewed by Bogna Szyk and Jack Bowater Last updated: Oct 20, 2022 Finding out the missing side or angle couldn't be easier than with our great tool - right triangle side and angle calculator. Choose two given values, type them into the calculator and the remaining unknowns will be determined in a blink of an eye! If you are wondering how to find the missing side of a right triangle, keep scrolling and you'll find the formulas behind our calculator. How to find the sides of a right triangleThere are a few methods of obtaining right triangle side lengths. Depending on what is given, you can use different relationships or laws to find the missing side:
If you know two other sides of the right triangle, it's the easiest option; all you need to do is apply the Pythagorean theorem:
🙋 Our Pythagorean theorem calculator will help you if you have any doubts at this point.
Apply the law of sines or trigonometry to find the right triangle side lengths:
🙋 Refresh your knowledge with Omni's law of sines calculator!
Find the missing leg using trigonometric functions:
As we remember from basic triangle area formula, we can calculate the area by multiplying triangle height and base and dividing the result by two. A right triangle is a special case of a scalene triangle, in which one leg is the height when the second leg is the base, so the equation gets simplified to:
For example, if we know only the right triangle area and the length
of the leg
🙋 For this type of problems, see also our area of a right triangle calculator. How to find the angle of a right triangleIf you know one angle apart from the right angle, calculation of the third one is a piece of cake: Given Given However, if only two sides of a triangle are given, finding the angles of a right triangle requires applying some basic trigonometric functions: for
and for
How do you solve a right angle triangle with only one side?To solve a triangle with one side, you also need one of the non-right angled angles. If not, it is impossible:
How to find the missing side of a right triangle? How to find the angle? ExampleLet's show how to find the sides of a right triangle with this tool:
Now, let's check how does finding angles of a right triangle work:
FAQHow many lines of symmetry does a right triangle have?If a right triangle is isosceles (i.e., its two non-hypotenuse sides are the same length) it has one line of symmetry. Otherwise, the triangle will have no lines of symmetry. Can a right angled triangle have equal sides?No, a right triangle cannot have all 3 sides equal, as all three angles cannot also be equal, as one has to be 90° by definition. A right triangle can, however, have its two non-hypotenuse sides be equal in length. This would also mean the two other angles are equal to 45°. Are all right triangles similar?Not all right angled triangles are similar, although some can be. They are similar if all their angles are the same length, or if the ratio of 2 of their sides is the same. Hanna Pamuła, PhD candidate 30 60 90 triangle45 45 90 triangleArea of a right triangle… 15 more |