Scalene triangle

A scalene triangle is a triangle whose three sides all have different lengths. Because the sides are all different, its three angles are all different too. A scalene triangle can be acute, right or obtuse, and like every triangle, its angles add up to 180°.

What is a scalene triangle?

Triangles are sorted by their sides into three groups. A scalene triangle is the one with no equal sides.

Triangle ABC has sides of 9, 7 and 5, and its angles are 50.7°, 33.6° and 95.7°. No two sides are equal and no two angles are equal, so it is scalene. The figure is drawn to scale, so you can see that the largest angle, at C, sits across from the longest side, AB.

Scalene vs isosceles vs equilateral

So a triangle is scalene if you can show that no two sides match, or that no two angles match. Either test works, because equal sides always sit opposite equal angles.

Properties of a scalene triangle

• All three sides are different lengths.
• All three angles are different.
• The angles add up to 180°, as in every triangle.
• It has no lines of symmetry, because no fold can match one side to another.
• The longest side faces the largest angle, and the shortest side faces the smallest angle.
• Any two sides add up to more than the third side. This is the triangle inequality, and it is true of every triangle.

Types of scalene triangles

Triangles are also sorted by their largest angle. A scalene triangle can be any of the three types:

The check in the last column compares the square of the longest side c with the sum of the squares of the other two, a² + b². If c² is smaller, the triangle is acute. If they are equal, it is right, by the converse of the Pythagorean theorem. If c² is larger, it is obtuse.

Most right triangles are scalene. The 3-4-5 triangle and every triangle similar to it, such as 6-8-10, have three different sides. The only right triangles that are not scalene are the isosceles ones, with angles of 45°, 45° and 90°.

Scalene triangle angles

Since the angles add up to 180°, you can find the third angle from the other two.

Example 1: the missing angle

Two angles of a triangle are 40° and 75°. Find the third angle, and say what type of triangle it is.

The angles are 40°, 75° and 65°. All three are different, so it is a scalene triangle, and all are less than 90°, so it is an acute scalene triangle.

Example 2: angles from the sides

When you know all three sides, the law of cosines (called the cosine rule in the UK) gives each angle. For triangle ABC above, with sides 5, 7 and 9, the largest angle C is opposite the longest side, 9:

The same method gives the other angles as 33.6° and 50.7°, and 95.7° + 33.6° + 50.7° = 180°. The largest angle is over 90°, which confirms the triangle is obtuse. The Trigonometry calculator solves any triangle from three measurements and shows each step.

Try a law of cosines question from the Trigonometry course. Its triangles have three different sides, just like these.

Perimeter of a scalene triangle

The perimeter is the sum of the three sides: P = a + b + c. There is no shortcut, because no two sides are the same. The 5, 7, 9 triangle has a perimeter of 5 + 7 + 9 = 21 units.

Area of a scalene triangle

Which formula you use depends on what you know.

When you know the base and height

The height must be perpendicular to the base. In a scalene triangle the height is not one of the sides (unless it is a right triangle), and for an obtuse triangle it can fall outside the triangle. A scalene triangle with a base of 12 cm and a height of 5 cm has an area of ½ × 12 × 5 = 30 cm².

When you know all three sides: Heron's formula

First find s, half the perimeter. Then:

Example 3: area of a triangle with sides 7, 8 and 9

The area is about 26.8 square units.

When you know two sides and the angle between them

For the 5, 7, 9 triangle, the angle between the sides 5 and 7 is C = 95.7°, so A = ½ × 5 × 7 × sin 95.7° = 17.4 square units. Heron's formula gives the same answer.

Can these side lengths make a triangle?

Before you classify three lengths as a scalene triangle, check they can make a triangle at all. Each pair of sides must add up to more than the third. The quick way is to check the two shortest sides against the longest.

• 4, 6, 9: 4 + 6 = 10, which is more than 9, so this is a triangle, and a scalene one.
• 3, 4, 8: 3 + 4 = 7, which is less than 8, so the two short sides cannot meet. No triangle.

Test what you have learned so far.

Scalene triangles in real life

Scalene triangles are everywhere once you look: the sail of a sailboat, a slice of pizza cut unevenly, the triangle made by a ladder, a wall and the ground, and the triangles surveyors use to measure land. Engineers use them in roof trusses and bridges. Because every side and angle is different, solving them usually takes the law of sines or the law of cosines rather than a single shortcut.

Common mistakes

• Calling a right triangle "not scalene". A right triangle can be scalene, and most are.
• Using a side as the height. In ½bh, the height is perpendicular to the base. It is only a side in a right triangle.
• Judging by the picture. Diagrams are often not drawn to scale. Use the measurements and the tick marks: matching tick marks mean equal sides.
• Forgetting to halve the perimeter in Heron's formula. s is half of a + b + c.
• Putting the wrong side opposite the angle in the law of cosines. The side you subtract, c², must be the one across from the angle you want.
• Skipping the triangle inequality. Three different numbers do not always make a scalene triangle. Check that the two shorter sides add up to more than the longest.

Practice with a short timed paper

Triangle classification, area and the laws of sines and cosines come up across high school geometry and the SAT. The daily Geometry, Algebra 2 & SAT Math paper is a short timed practice test of 8 questions, with a new one every day.

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Maths · GeometryHow to find the area of a triangle: every methodHow to find the area of a triangle: ½ × base × height, right-angled and obtuse triangles, ½ab sin C, Heron's formula and coordinates, with examples.