Last updated: August 5, 2026
Perimeter of a Triangle Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Quick Answer
A triangle's perimeter is the sum of all three sides, but missing sides often need to be found first. This calculator supports SSS, SAS, and ASA-style inputs, using the law of cosines or law of sines when necessary before adding the full boundary. It returns the perimeter, the derived third side, the semiperimeter, and a triangle-type label.
Add the three triangle sides for perimeter, and use the law of cosines or law of sines first if one side is missing.
Key Takeaways
- Perimeter always means adding all three triangle sides.
- SAS problems need the law of cosines before the final sum.
- ASA problems need the third angle and the law of sines.
- The triangle inequality still matters even in direct SSS mode.
- Intermediate rounding can change the final perimeter slightly.
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
P = a + b + c, with law of cosines or law of sines used to find any missing side first
Where:
- P=triangle perimeter(length)
- a,b,c=side lengths(length)
- C=included angle(degrees)
Worked Examples
SSS triangle
A triangle has sides 7, 8, and 9 cm.
- 1Perimeter = 7 + 8 + 9.
- 2Semiperimeter = 12.
- 3All sides differ, so the triangle is scalene.
SAS triangle
Sides 7 m and 9 m include a 60° angle.
- 1Use the law of cosines to find the third side.
- 2c = √(7² + 9² − 2·7·9·cos60°) ≈ 8.185.
- 3Perimeter ≈ 7 + 9 + 8.185 = 24.185.
ASA triangle
Angles 50° and 60° include a side of 10 ft.
- 1Find the third angle: 180 − 50 − 60 = 70°.
- 2Use the law of sines to compute the remaining sides.
- 3Add all three sides to get a perimeter of about 27.368 ft.
Introduction
Triangle perimeter problems do not always arrive with three side lengths already visible. Sometimes you have all three sides, sometimes two sides and the included angle, and sometimes two angles with one known side. This calculator handles all three common pathways, automatically finds any missing side with the law of cosines or law of sines, and then adds the full boundary so you can move from partial triangle data to a trustworthy perimeter quickly.
Triangle perimeter basics
The perimeter of any triangle is still the sum of its three side lengths, but the real work lies in recovering missing sides when the given information is incomplete. In introductory geometry, this teaches students that formulas are often part of a longer chain of reasoning rather than a one-line plug-in. The calculator mirrors that structure by solving missing values first and then returning the perimeter.
When SSS is enough
If all three sides are known, the calculation is immediate: add them. Even in this simplest mode, it is still wise to check the triangle inequality. That check confirms the lengths can actually form a triangle rather than a degenerate segment. The calculator also reports triangle type so the side set tells you more than just a total boundary length.
How SAS uses the law of cosines
When two sides and the included angle are given, the third side is found with the law of cosines. This is more reliable than trying to estimate from a sketch because the included angle controls how far apart the side endpoints are. Once the missing side is computed, the perimeter becomes a standard side sum. SAS mode appears frequently in surveying, navigation, and statics diagrams.
How ASA uses the law of sines
With two known angles and the included side, first find the third angle from the 180° angle sum. Then the law of sines connects that known side to the unknown sides. This is useful in textbook problems where angle information is emphasized more than direct measurements. It also helps students see that perimeter can still be recovered even when only one side is initially known.
Worked example
Suppose you know sides 7 and 9 with an included angle of 60°. The law of cosines gives the missing side as √(7² + 9² − 2·7·9·cos60°) ≈ 8.185. Adding the three sides yields a perimeter near 24.185. This example highlights why the angle matters: if the included angle changed, the third side and the perimeter would both change, even though the two given sides stayed the same.
Choosing the right method
Pick SSS when the problem already lists three sides, SAS when the known angle lies between the two known sides, and ASA when the known side lies between the two known angles. Misidentifying the setup can lead to incorrect use of a law or to extra work. Reading the triangle diagram carefully before entering values is the simplest way to avoid that mistake.
Common errors
Students sometimes use degrees in the calculator but imagine radians in the formula, or they apply the law of sines before computing the missing angle in an ASA problem. Another common issue is rounding the derived side too early, which slightly changes the final perimeter. Retaining a few extra decimal places during intermediate steps keeps the answer stable.
Applications
Triangle perimeters appear in roof framing, cable routing, surveying parcels, truss design, and map-based distance estimates. In those settings, incomplete triangle data is normal, so the ability to move between SSS, SAS, and ASA information is more practical than memorizing only one direct formula. The calculator turns that geometry workflow into a quick, repeatable process.
Quick Reference Card
General triangle perimeter cheat sheet
Quick reference • Perimeter of a Triangle Calculator
Perimeter = a + b + cValid range: Sides > 0 and angles between 0° and 180°
Common Values
⚠ Watch Out
- •Use the included angle in SAS, not a non-included angle.
- •Do not let the two known ASA angles add to 180° or more.
- •Check the triangle inequality in SSS problems.
- •Avoid rounding the derived side too early.
Pro Tips
- →SAS uses the law of cosines for the missing side.
- →ASA needs the third angle before the law of sines can be applied.
- →Semiperimeter is a handy cross-check for later area work.
- →Sketch the triangle and label known parts before choosing a mode.
FAQs
What is the perimeter of a triangle?
It is the sum of the lengths of all three sides.
When should I use SAS mode?
Use SAS when you know two sides and the angle included between those two sides.
When should I use ASA mode?
Use ASA when you know two angles and the side between them, then use the law of sines to recover the other sides.
Why does the calculator show semiperimeter?
Semiperimeter is useful for related formulas such as Heron's formula and other triangle checks.
Can an angle be 180° in a valid triangle?
No. Individual angles must be positive and less than 180°, and the total must equal 180°.
Why might the triangle type change across methods?
The calculator classifies the finished side set. Different input values naturally lead to different triangle shapes.
Should I round the third side before adding?
No. Keep extra decimals until the end so the perimeter stays more accurate.