On This Page
- How to use the Triangle Calculator
- What is a triangle?
- Find missing sides and angles
- Right triangle calculations
- Area, height, and perimeter
- SSS, SAS, ASA, AAS, and SSA methods
- Worked triangle examples
- Triangle mistakes to avoid
- Frequently asked questions
How to Use the Triangle Calculator
If your geometry problem gives a mix of sides and angles, enter the known values here and let the calculator finish the triangle. It can find missing sides, missing angles, area, and perimeter as long as the inputs describe a real triangle.
The labels follow standard triangle notation: side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. If you are using it as a triangle side calculator, enter the known side lengths first, then add the known angle if the problem gives one.
What Is a Triangle?
In ordinary Euclidean geometry, the three interior angles of a triangle always add to 180 degrees, or π radians. The side lengths must also pass the triangle inequality rule: the sum of any two sides must be greater than the third side.
That rule is why not every set of three numbers can form a real triangle. For example, sides 2, 3, and 8 cannot close into a triangle because 2 + 3 is not greater than 8.
Find Missing Triangle Sides and Angles
If your homework asks you to find the missing side of a triangle or calculate an unknown angle, start by identifying what is already known. Three sides, two sides with the included angle, or two angles with one side usually give enough information.
The calculator chooses a method based on your inputs and shows the solved values in the result panel. This is useful for side-side-side triangle problems, trigonometry triangle questions, and “solve the triangle” exercises where the final answer needs every side and every angle.
Right Triangle Calculations
For a right triangle, the usual goal is to find the hypotenuse, a missing leg, or a missing angle. Enter the 90 degree angle along with the side information you know, and the calculator will use the matching triangle rules.
For right triangles, the longest side is the hypotenuse and it sits opposite the 90 degree angle. The calculator can still use the general triangle formulas, but the familiar right triangle relationship is:
Special right triangles such as 30-60-90 and 45-45-90 triangles follow fixed side ratios. This tool is most helpful when you want the full side, angle, area, and perimeter breakdown, not just the shortcut ratio.
Triangle Area, Height, and Perimeter
Use the result panel as a triangle area calculator when you know enough information to solve all three sides. Once the sides are known, the calculator uses Heron's formula to find area.
If a problem gives base and height directly, the simpler area formula is:
For square footage questions, use the same area formula but keep the units in feet. A triangle with base and height measured in feet has area in square feet. Perimeter is simpler: add the three side lengths.
Triangle Solving Methods
Different input patterns use different formulas. The calculator checks the values you entered and chooses the first valid method that can solve a real triangle.
SSS: Three Sides
Use SSS when all three side lengths are known. A side-side-side triangle calculator uses the Law of Cosines to find the angles, then Heron's formula to find the area.
SAS: Two Sides and the Included Angle
When two sides and the angle between them are known, the missing side can be found with the Law of Cosines. The remaining angles can then be found from the Law of Sines and the 180-degree angle sum.
ASA and AAS: Two Angles and One Side
When two angles are known, the third angle is found by subtraction. Then the Law of Sines can scale the known side to the missing sides.
SSA: Two Sides and a Non-Included Angle
SSA can sometimes be ambiguous because two different triangles may match the same information. This calculator returns one valid solution and notes when the SSA method was used.
Worked Examples
Example 1: SSS Triangle With Area and Perimeter
Problem: A math workbook gives a triangle with sides a = 5, b = 6, and c = 7. Find the missing angles, perimeter, and area.
- Use the Law of Cosines to solve the angles.
- Perimeter = 5 + 6 + 7 = 18.
- Semi-perimeter = 18 ÷ 2 = 9.
- Area = √(9(9 - 5)(9 - 6)(9 - 7)) = √216 = 14.7.
Answer: The area is about 14.7 square units, and the perimeter is 18 units.
Example 2: SAS Triangle
Problem: A survey sketch shows two sides of 10 and 14 units with the included angle of 38 degrees. Find the third side first, then solve the triangle.
- Enter side a = 10, side b = 14, and angle C = 38 degrees.
- The calculator uses c² = a² + b² - 2ab cos(C).
- After side c is found, the remaining angles and area are calculated.
Answer: This is an SAS problem because the known angle is between the two known sides.
Example 3: ASA or AAS Triangle
Problem: A geometry problem gives angle A = 42 degrees, angle B = 63 degrees, and side c = 12. Solve the triangle.
- Find angle C: 180 - 42 - 63 = 75 degrees.
- Use the Law of Sines to find side a and side b.
- Use the solved sides to calculate perimeter and area.
Answer: Two angles and one side are enough to solve a unique triangle.
Example 4: Right Triangle Hypotenuse Check
Problem: A ladder forms a right triangle with a wall. The horizontal distance from the wall is 6 ft and the vertical height is 8 ft. Find the ladder length.
- The ladder is the hypotenuse.
- Use c² = 6² + 8².
- c² = 36 + 64 = 100.
- c = 10 ft.
Answer: The ladder length is 10 ft. You can also enter the two legs and a 90 degree angle in the calculator to check the full triangle.
Example 5: Triangle Square Footage
Problem: A triangular garden bed has a base of 18 ft and a height of 11 ft. Find the square footage.
- Area = 1/2 × base × height.
- Area = 1/2 × 18 × 11.
- Area = 99 square feet.
Answer: The triangular area is 99 sq ft.
Triangle Mistakes to Avoid
- Entering angles that do not fit: The three interior angles must add to 180 degrees. If two angles already add to 180 or more, there is no triangle left to solve.
- Mixing up opposite sides and angles: Side a sits opposite angle A, side b sits opposite angle B, and side c sits opposite angle C. Swapping labels changes the solved triangle.
- Using three angles only: AAA gives the shape, but not the size. Add at least one side length if you need side lengths, area, or perimeter.
- Ignoring the SSA case: Two sides and a non-included angle can sometimes describe more than one triangle. Check the diagram and the method note before copying the answer.
- Using height as a side: Triangle height is usually an altitude, not one of the three sides, unless the problem clearly says it is a side of a right triangle.
Frequently Asked Questions
How many values are needed to solve a triangle?
You usually need three values, and at least one of them must be a side. Three angles alone define only the shape, not the size.
How do you calculate the area of a triangle?
If base and height are known, use area = 1/2 × base × height. If all three sides are known, use Heron's formula with the semi-perimeter.
How do you calculate the perimeter of a triangle?
Add the three side lengths: perimeter = a + b + c.
How do you calculate the height of a triangle?
If you know area and base, rearrange the area formula: height = 2 × area / base.
How do you find the missing side of a right triangle?
Use the Pythagorean theorem when two sides of a right triangle are known. For non-right triangles, use the Law of Cosines or Law of Sines depending on the given information.
Can this calculator use radians?
Yes. Change the angle unit to radians before entering angle values. Results will also be shown in radians.
Why is my triangle invalid?
The sides may fail the triangle inequality rule, the angles may not leave a positive third angle, or the SSA information may not form a real triangle.
What is the difference between SSS and SAS?
SSS means all three sides are known. SAS means two sides and the included angle between those sides are known.