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Triangle Calculator

MathTriangles and Trigonometry
What you know

Side a is opposite angle A.

Area —
Perimeter
—
Triangle type
—
Side a
—
Side b
—
Side c
—
Angle A (°)
—
Angle B (°)
—
Angle C (°)
—
Height onto side a
—

A triangle is fully determined by three sides (SSS), two sides and the angle between them (SAS), or two angles and the side between them (ASA). Enter one of those sets and the calculator returns the missing sides and angles, the area and the perimeter. Sides 3, 4 and 5 give a right triangle with area 6 and perimeter 12.

About this tool

Students checking trigonometry homework, carpenters laying out a roof gable and surveyors closing a small plot all meet the same problem: some of a triangle's measurements are known and the rest must be worked out. This calculator solves the triangle from three standard starting sets. It reports every side and angle, the area by Heron's formula, the perimeter, the height dropped onto side a, and whether the triangle is acute, right or obtuse and scalene, isosceles or equilateral. The working lines show which law was applied at each stage. Side lengths have no fixed unit, so the area comes out in the square of whatever unit you typed. It does not handle the ambiguous SSA case, where two sides and a non-included angle can fit two different triangles.

How to use it

  1. Pick what you know

    Choose three sides, two sides with the angle between them, or two angles with the side between them.

  2. Enter the measurements

    Type the sides in any one unit and the angles in degrees. Side a is always opposite angle A, b opposite B, c opposite C.

  3. Read the solved triangle

    The area leads, followed by the perimeter, the triangle type, all three sides, all three angles and the height onto side a.

  4. Check the working

    The steps list the law of cosines or law of sines used, then the semi-perimeter and Heron's formula for the area.

Examples

Sides 3, 4 and 5

What you know
Three sides (SSS)
Side a
3
Side b
4
Side c
5

Result Area: 6
Perimeter: 12
Triangle type: Right scalene
Side a: 3
Side b: 4
Side c: 5
Angle A (°): 36.87
Angle B (°): 53.13
Angle C (°): 90
Height onto side a: 4

  1. A = arccos((b² + c² − a²) ÷ 2bc) = 36.87°
  2. B = arccos((a² + c² − b²) ÷ 2ac) = 53.13°
  3. C = 180° − A − B = 90°
  4. s = (a + b + c) ÷ 2 = 6
  5. Area = √(s(s − a)(s − b)(s − c)) = 6

The 3-4-5 triangle is right-angled, so its area is half the product of the two legs: 6 square units.

Sides 5 and 7 with a 60° angle between

What you know
Two sides + angle (SAS)
Side b
5
Side c
7
Angle A (between b and c)
60

Result Area: 15.1554
Perimeter: 18.245
Triangle type: Acute scalene
Side a: 6.245
Side b: 5
Side c: 7
Angle A (°): 60
Angle B (°): 43.9
Angle C (°): 76.1
Height onto side a: 4.8536

  1. a = √(b² + c² − 2bc·cos A) = 6.245
  2. B = arccos((a² + c² − b²) ÷ 2ac) = 43.9°
  3. C = 180° − A − B = 76.1°
  4. s = (a + b + c) ÷ 2 = 9.1225
  5. Area = √(s(s − a)(s − b)(s − c)) = 15.1554

The law of cosines gives the third side as √39 = 6.245, and the area is ½·5·7·sin 60° = 15.1554.

Side 10 between two 60° angles

What you know
Two angles + side (ASA)
Side a
10
Angle B
60
Angle C
60

Result Area: 43.3013
Perimeter: 30
Triangle type: Acute equilateral
Side a: 10
Side b: 10
Side c: 10
Angle A (°): 60
Angle B (°): 60
Angle C (°): 60
Height onto side a: 8.6603

  1. A = 180° − B − C = 60°
  2. b = a · sin B ÷ sin A = 10
  3. c = a · sin C ÷ sin A = 10
  4. s = (a + b + c) ÷ 2 = 15
  5. Area = √(s(s − a)(s − b)(s − c)) = 43.3013

Two 60° angles force the third to 60°, so this is an equilateral triangle with area 25√3 = 43.3013.

How it is calculated

cos A = (b² + c² − a²) ÷ 2bc; a ÷ sin A = b ÷ sin B = c ÷ sin C; Area = √(s(s − a)(s − b)(s − c)), s = (a + b + c) ÷ 2

a, b, c
side lengths, each opposite the angle with the same letter
A, B, C
interior angles in degrees, summing to 180°
s
semi-perimeter, half of a + b + c

With three sides, the law of cosines gives angles A and B, and C is what remains of 180°. With two sides and their included angle, the law of cosines first gives the third side. With two angles and the side between them, the third angle comes from the 180° sum and the law of sines scales the other two sides. Once all three sides are known, Heron's formula gives the area, and the height onto side a is twice the area divided by a.

When not to use it

  • If you know two sides and an angle that is not between them, the data may fit two triangles and this tool will not choose for you.
  • For triangles drawn on a sphere, such as long navigation routes, angles add to more than 180° and plane formulas do not apply.

Common mistakes

  • Mixing units, such as one side in feet and another in inches, which produces a wrong but plausible-looking triangle.
  • Entering a SAS angle that is not the one between the two given sides.
  • Typing three sides where the two shorter ones add up to no more than the longest, which cannot close into a triangle.

Frequently asked questions

Why does the calculator reject sides 1, 2 and 5?

The triangle inequality says the two shorter sides must together exceed the longest. 1 + 2 = 3 is less than 5, so the two short sides cannot reach each other's ends and no triangle exists.

What units does the area come out in?

The square of whatever length unit you used for the sides. Sides in metres give square metres, sides in inches give square inches. Angles are always in degrees.

How is the triangle type decided?

By the largest angle: below 90° it is acute, exactly 90° right, above 90° obtuse. By sides: three equal is equilateral, two equal is isosceles, none equal is scalene.

Why use Heron's formula instead of half base times height?

Heron's formula needs only the three sides, which are always known once the triangle is solved, so no separate height measurement is required. The calculator still reports the height onto side a, found as twice the area divided by a.

Can I solve a triangle from three angles alone?

No. Three angles fix the shape but not the size, since every scaled copy has the same angles. At least one side length is needed, which is why the ASA mode asks for the side between the two angles.

What if I know two angles and a side that is not between them?

Subtract the two known angles from 180° to get the third. The given side then sits between two known angles, so relabel the triangle and use the ASA mode.

Cite this tool

Using this tool in an assignment, article or report? Copy a citation, or a link for your website.

APA

Tools Directory Editorial Team. (2026, October 13). Triangle Calculator. Tools Directory. Retrieved October 26, 2026, from https://tools.milunir.com/math/triangles-and-trigonometry/triangle-calculator/

MLA

Tools Directory Editorial Team. “Triangle Calculator.” Tools Directory, 13 Oct. 2026, https://tools.milunir.com/math/triangles-and-trigonometry/triangle-calculator/. Accessed 26 Oct. 2026.

Chicago

Tools Directory Editorial Team. “Triangle Calculator.” Tools Directory. Last modified October 13, 2026. Accessed October 26, 2026. https://tools.milunir.com/math/triangles-and-trigonometry/triangle-calculator/.

Link<a href="https://tools.milunir.com/math/triangles-and-trigonometry/triangle-calculator/">Triangle Calculator – Tools Directory</a>