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Scalene Triangle Calculator

MathTriangles and Trigonometry

Any length unit, used the same for all three sides.

Area (square units) —
Triangle type
—
Perimeter
—
Angle A (opposite a), °
—
Angle B (opposite b), °
—
Angle C (opposite c), °
—
Height to side a
—
Height to side b
—
Height to side c
—
Circumradius R
—
Inradius r
—

A scalene triangle has three different side lengths, so its area comes from Heron's formula: A = √(s(s − a)(s − b)(s − c)), where s is half the perimeter. For sides 5, 6 and 7, s = 9 and the area is √216 ≈ 14.70. The angles follow from the law of cosines; the largest, opposite side 7, is 78.46°.

About this tool

Students checking geometry homework, carpenters laying out an irregular gable, and surveyors working with a plot that has no right angle all meet the same problem: three known sides and nothing symmetrical to lean on. This calculator takes the three lengths and returns the area, perimeter, each interior angle, the height dropped onto every side, and the radii of the inscribed and circumscribed circles. It also names the triangle as acute, right or obtuse, and flags it if two sides turn out to be equal. The working lists the semi-perimeter, the Heron product and each cosine step. One limit: it needs all three sides. If you know two sides and an angle instead, find the third side with the law of cosines first.

How to use it

  1. Enter the three sides

    Type lengths a, b and c in one unit. The area is reported in that unit squared.

  2. Read area and angles

    The headline is the area; angles A, B and C sit opposite sides a, b and c.

  3. Check the type and heights

    The type line says whether the triangle is acute, right or obtuse, and the three heights give the altitude onto each side.

Examples

Sides 3, 4 and 5

Side a
3
Side b
4
Side c
5

Result Area (square units): 6
Triangle type: Scalene, right
Perimeter: 12
Angle A (opposite a), °: 36.87
Angle B (opposite b), °: 53.13
Angle C (opposite c), °: 90
Height to side a: 4
Height to side b: 3
Height to side c: 2.4
Circumradius R: 2.5
Inradius r: 1

  1. s = (3 + 4 + 5) ÷ 2 = 6
  2. Area = √(s(s − a)(s − b)(s − c)) = √(36) = 6
  3. Angle A = arccos((b² + c² − a²) ÷ 2bc) = 36.87°
  4. Angle B = arccos((a² + c² − b²) ÷ 2ac) = 53.13°
  5. Angle C = 180° − A − B = 90°
  6. Height to a = 2 × Area ÷ a = 4

The classic Pythagorean triple: all sides differ, the angle opposite 5 is exactly 90°, and the area is half of 3 × 4.

Sides 5, 6 and 7

Side a
5
Side b
6
Side c
7

Result Area (square units): 14.6969
Triangle type: Scalene, acute
Perimeter: 18
Angle A (opposite a), °: 44.42
Angle B (opposite b), °: 57.12
Angle C (opposite c), °: 78.46
Height to side a: 5.8788
Height to side b: 4.899
Height to side c: 4.1991
Circumradius R: 3.5722
Inradius r: 1.633

  1. s = (5 + 6 + 7) ÷ 2 = 9
  2. Area = √(s(s − a)(s − b)(s − c)) = √(216) = 14.6969
  3. Angle A = arccos((b² + c² − a²) ÷ 2bc) = 44.42°
  4. Angle B = arccos((a² + c² − b²) ÷ 2ac) = 57.12°
  5. Angle C = 180° − A − B = 78.46°
  6. Height to a = 2 × Area ÷ a = 5.8788

Half the perimeter is 9, so Heron's formula gives √216, about 14.70 square units, and every angle stays under 90°.

Sides 7, 8 and 13

Side a
7
Side b
8
Side c
13

Result Area (square units): 24.2487
Triangle type: Scalene, obtuse
Perimeter: 28
Angle A (opposite a), °: 27.8
Angle B (opposite b), °: 32.2
Angle C (opposite c), °: 120
Height to side a: 6.9282
Height to side b: 6.0622
Height to side c: 3.7306
Circumradius R: 7.5056
Inradius r: 1.7321

  1. s = (7 + 8 + 13) ÷ 2 = 14
  2. Area = √(s(s − a)(s − b)(s − c)) = √(588) = 24.2487
  3. Angle A = arccos((b² + c² − a²) ÷ 2bc) = 27.8°
  4. Angle B = arccos((a² + c² − b²) ÷ 2ac) = 32.2°
  5. Angle C = 180° − A − B = 120°
  6. Height to a = 2 × Area ÷ a = 6.9282

cos C = (49 + 64 − 169) ÷ 112 = −0.5, so the angle opposite the 13 side is 120° and the triangle is obtuse.

How it is calculated

s = (a + b + c) ÷ 2; Area = √(s(s − a)(s − b)(s − c)); cos A = (b² + c² − a²) ÷ 2bc; hₐ = 2·Area ÷ a; R = abc ÷ 4·Area; r = Area ÷ s

a, b, c
the three side lengths
s
semi-perimeter, half of a + b + c
A, B, C
interior angles opposite a, b and c
hₐ
altitude dropped perpendicular onto side a
R, r
circumradius and inradius

Heron's formula needs no angle or height, which suits a scalene triangle where no side is a natural base. Angles A and B come from the law of cosines and C is 180° minus the other two. Each altitude is twice the area divided by the side it meets. Before any of this, the triangle inequality is checked: the longest side must be strictly shorter than the sum of the other two.

When not to use it

  • You know only two sides and the angle between them; use a SAS or law of cosines calculator.
  • The shape is drawn on a sphere or a map over long distances, where plane geometry no longer holds.
  • You need the side lengths from coordinates; compute the distances between the points first.

Common mistakes

  • Mixing units, such as two sides in metres and one in centimetres.
  • Entering sides that break the triangle inequality, like 1, 2 and 5.
  • Reading angle A as the angle at side a rather than the angle opposite it.
  • Using half base times height with a slanted side instead of the perpendicular height.

Frequently asked questions

What makes a triangle scalene?

All three sides have different lengths, and as a result all three angles differ too. A scalene triangle can be acute, right or obtuse; the 3-4-5 triangle is both scalene and right-angled.

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

Half base times height needs a perpendicular height, which you rarely measure directly on an irregular triangle. Heron's formula works from the three sides alone, so it fits the data most people actually have.

How do I know if my three lengths can form a triangle?

The longest side must be shorter than the other two added together. Sides 7, 8 and 13 work because 13 is less than 15; sides 1, 2 and 5 do not, and the tool reports an error on the offending side.

How is the triangle classed as acute, right or obtuse?

By its largest angle, which always sits opposite the longest side. Under 90° is acute, exactly 90° is right, and over 90° is obtuse. For 7, 8 and 13 the angle opposite 13 is 120°, so that triangle is obtuse.

What are the inradius and circumradius?

The inradius is the radius of the largest circle that fits inside the triangle touching all three sides, equal to area divided by semi-perimeter. The circumradius is the radius of the circle through all three corners, abc divided by four times the area. For 3-4-5 they are 1 and 2.5.

Why does an obtuse triangle have heights outside it?

The altitude onto either short side of an obtuse triangle falls on the extension of that side, beyond the triangle. Its length is still twice the area divided by the side, so the value shown is correct even though the foot lies outside.

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 24). Scalene Triangle Calculator. Tools Directory. Retrieved October 26, 2026, from https://tools.milunir.com/math/triangles-and-trigonometry/scalene-triangle-calculator/

MLA

Tools Directory Editorial Team. “Scalene Triangle Calculator.” Tools Directory, 24 Oct. 2026, https://tools.milunir.com/math/triangles-and-trigonometry/scalene-triangle-calculator/. Accessed 26 Oct. 2026.

Chicago

Tools Directory Editorial Team. “Scalene Triangle Calculator.” Tools Directory. Last modified October 24, 2026. Accessed October 26, 2026. https://tools.milunir.com/math/triangles-and-trigonometry/scalene-triangle-calculator/.

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