Triangle Theorem Inputs
Example Data Table
| Case | Known Values | Theorem Used | Expected Output |
|---|---|---|---|
| Right triangle | a = 3, b = 4, c = 5 | Pythagorean Theorem and Law of Cosines | Area, angles, perimeter, right triangle check |
| SAS triangle | b = 8, c = 10, A = 45° | Law of Cosines | Missing side, missing angles, area, medians |
| AAS triangle | A = 50°, B = 60°, a = 12 | Angle Sum Theorem and Law of Sines | Third angle, missing sides, radii |
| SSA triangle | A = 35°, a = 9, b = 12 | Law of Sines | Primary result and possible alternate result |
Formula Used
Angle Sum Theorem
A + B + C = 180°. This checks whether all triangle angles form a valid triangle.
Pythagorean Theorem
For a right triangle, the longest side satisfies c² = a² + b² after sorting sides.
Law of Sines
a / sin(A) = b / sin(B) = c / sin(C). This solves ASA, AAS, and SSA cases.
Law of Cosines
a² = b² + c² - 2bc cos(A). Related forms solve other sides and angles.
Heron Area Formula
s = (a + b + c) / 2. Area = √[s(s - a)(s - b)(s - c)].
Medians
ma = 1/2 √(2b² + 2c² - a²). Similar forms apply to other medians.
Angle Bisectors
la = √[bc(1 - a² / (b + c)²)]. Similar forms apply to other angle bisectors.
Radii
Inradius = area / s. Circumradius = abc / 4 area.
How to Use This Calculator
- Enter any valid SSS, SAS, ASA, AAS, or SSA triangle values.
- Use side labels a, b, and c. Their opposite angles are A, B, and C.
- Select the theorem focus if you want to label the report.
- Add a second triangle if you need congruence or similarity checking.
- Press the calculate button to show results below the header.
- Use the CSV or PDF button to save the result summary.
Triangle Theorems Calculator Guide
Why Triangle Theorems Matter
Triangle theorems connect sides, angles, area, and special segments. They help students verify geometric work without guessing. They also support engineering sketches, layout checks, navigation problems, and classroom demonstrations. A triangle looks simple, yet it can describe slopes, braces, roof forms, survey paths, and force diagrams. This calculator brings those ideas into one place.
Solving Different Input Patterns
You can start with three sides, two sides with an included angle, two angles with one side, or selected side-angle-side cases. The tool identifies the best theorem route automatically. For SSS data, it uses the Law of Cosines. For AAS and ASA data, it applies the angle sum rule first. Then it uses the Law of Sines to scale each side.
Advanced Triangle Measurements
The result does more than solve missing sides. It reports perimeter, semiperimeter, area, heights, medians, angle bisectors, inradius, and circumradius. These values are useful when one theorem is not enough. For example, a designer may need area. A teacher may need medians. A student may need the exact theorem path for homework notes.
Checking Triangle Type
The calculator classifies the triangle by side and angle behavior. It can detect equilateral, isosceles, and scalene forms. It also checks acute, right, and obtuse angle types. The Pythagorean check compares the square of the longest side with the sum of the other two side squares. This gives a clear right triangle test.
Congruence and Similarity Review
A second triangle can be entered for comparison. When both triangles are solved, the tool checks direct side congruence and proportional similarity. This supports SSS congruence and AAA style similarity review. It is helpful for proofs, worksheets, and quick geometry verification. Always keep labels consistent when comparing two triangles, because corresponding sides and angles depend on matching names.
FAQs
1. What values can I enter?
You can enter SSS, SAS, ASA, AAS, or valid SSA information. The calculator needs enough values to define at least one triangle.
2. Why does SSA sometimes show another solution?
SSA can be ambiguous. One side and a non-included angle may form two valid triangles. The calculator shows the primary solution and notes an alternate when it exists.
3. Which side is opposite angle A?
Side a is opposite angle A. Side b is opposite angle B. Side c is opposite angle C. Keep this labeling consistent for accurate results.
4. Can this check right triangles?
Yes. It compares the longest side squared with the sum of the other two side squares. A close match confirms the right triangle theorem.
5. Does it calculate triangle area?
Yes. It calculates area using Heron's formula and a trigonometric area formula when the triangle is solved.
6. What does the second triangle section do?
It lets you compare another triangle. The tool checks direct SSS congruence and proportional side similarity when enough data is available.
7. Can I export my results?
Yes. After calculation, use the CSV button for spreadsheet data or the PDF button for a printable summary.
8. Why are angles entered in degrees?
Degrees are common in classroom geometry. The calculator converts degrees internally when using sine, cosine, and trigonometric area formulas.