X.2 Equilateral, isosceles, and scalene triangles
What is a triangle?
A triangle is a two-dimensional shape with exactly three straight sides, three vertices, and three angles. A vertex is a point where two sides meet. Because a triangle has three sides and three angles, its parts can be used to describe and classify it.
- A triangle has 3 sides.
- A triangle has 3 vertices.
- A triangle has 3 angles.
- The three interior angles of a triangle always have a total measure of 180°.
When you classify a triangle, look carefully at its sides or angles. The three side lengths are especially important when deciding whether a triangle is equilateral, isosceles, or scalene.
What does it mean to classify a triangle?
To classify a triangle means to place it into a group based on its properties. One way to classify triangles is by comparing the lengths of their sides. A triangle can have three equal sides, exactly two equal sides, or three different side lengths.
- A triangle with side lengths 5 inches, 5 inches, and 5 inches is equilateral.
- A triangle with side lengths 6 inches, 6 inches, and 4 inches is isosceles.
- A triangle with side lengths 4 inches, 5 inches, and 7 inches is scalene.
Do not classify a triangle by how it looks on a page. A drawing can be stretched, rotated, or not drawn exactly to scale. Use the given measurements or properties instead.
What is an equilateral triangle?
An equilateral triangle is a triangle with three sides that have the same length. Because all three sides are equal, all three angles are also equal. Each angle in an equilateral triangle measures 60°.
- A triangle with side lengths 4 centimeters, 4 centimeters, and 4 centimeters is equilateral.
- A triangle with side lengths 9 feet, 9 feet, and 9 feet is equilateral.
- An equilateral triangle has three equal sides and three equal angles.
The word equilateral means “equal sides.” If all three side lengths are the same, you can identify the triangle as equilateral.
What is an isosceles triangle?
An isosceles triangle is a triangle with at least two sides that have the same length. In many elementary math problems, an isosceles triangle is described as having exactly two equal sides. The angles opposite equal sides are also equal.
- A triangle with side lengths 5 inches, 5 inches, and 8 inches is isosceles.
- A triangle with side lengths 7 meters, 7 meters, and 3 meters is isosceles.
- If two sides of a triangle are equal, the two angles opposite those sides are equal.
Remember that an equilateral triangle also has at least two equal sides. Therefore, under the broad mathematical definition, every equilateral triangle is also isosceles. In school exercises, follow the definition and classification rule given in the problem.
What is a scalene triangle?
A scalene triangle is a triangle in which all three sides have different lengths. Because no two sides are equal, no two of its angles are equal.
- A triangle with side lengths 3 inches, 4 inches, and 5 inches is scalene.
- A triangle with side lengths 6 centimeters, 8 centimeters, and 10 centimeters is scalene.
- A triangle with side lengths 4 feet, 5 feet, and 6 feet is scalene.
The key word for scalene is different. If all three side lengths are different, the triangle is scalene.
How to classify a triangle by its side lengths
You can classify a triangle by comparing its three side lengths. First, check whether all three lengths are equal. If they are, the triangle is equilateral. If exactly two lengths are equal, the triangle is isosceles. If all three lengths are different, the triangle is scalene.
- 8, 8, 8 → all three sides are equal → equilateral
- 8, 8, 5 → two sides are equal → isosceles
- 5, 6, 8 → all three sides are different → scalene
Compare all three side measurements before choosing an answer. Do not stop after comparing only two sides. You need to know whether one, two, or all three sides have the same length.
Equal sides and equal angles
The sides and angles of a triangle are connected. In a triangle, equal sides are opposite equal angles. This property can help you find information about a triangle even when every measurement is not labeled.
- If two sides of a triangle are equal, the angles opposite those sides are equal.
- If a triangle has three equal sides, it has three equal angles.
- In an equilateral triangle, each angle measures 60° because the three equal angles must total 180°.
When a diagram uses matching marks on two sides, those marks usually show that the sides have equal lengths. Matching angle marks can show that angles have equal measures.
Equilateral, isosceles, and scalene triangles compared
The three classifications describe triangles by the number of equal side lengths. Equilateral triangles have three equal sides, isosceles triangles have at least two equal sides, and scalene triangles have three different side lengths.
- Equilateral: 10 cm, 10 cm, 10 cm — three equal sides.
- Isosceles: 10 cm, 10 cm, 6 cm — two equal sides.
- Scalene: 6 cm, 8 cm, 10 cm — three different sides.
A triangle can also be described by its angles. For example, a triangle may be classified as a right triangle because it has a 90° angle while also being classified by its side lengths. Classifications describe different properties, so more than one description can apply to the same triangle.
How to solve triangle classification problems
When a problem asks you to classify a triangle, use a careful process. Read the measurements or clues, compare the side lengths, determine which sides are equal, and then choose the classification that matches the evidence.
- A triangle has sides measuring 12 inches, 7 inches, and 12 inches.
- Step 1: Compare the three side lengths.
- Step 2: Notice that two sides measure 12 inches.
- Step 3: Because two sides are equal, the triangle is isosceles.
Use the information in the problem rather than making a guess from the picture. A triangle that looks like an isosceles triangle may have three different side lengths.
Important facts to remember
Triangle classification is based on properties that can be checked. For side-length classification, remember the three main groups: equilateral, isosceles, and scalene.
- Equilateral: 3 equal sides.
- Isosceles: at least 2 equal sides.
- Scalene: 3 different side lengths.
- Triangle: a two-dimensional figure with 3 straight sides, 3 vertices, and 3 angles.
- Interior angles: the three angles inside a triangle have a total measure of 180°.
A useful memory clue is to connect the names with their side properties: equilateral means equal sides, isosceles has at least two equal sides, and scalene has no equal sides.
Common Core alignment: CCSS.MATH.CONTENT.4.G.A.2 – Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size.
Notes for teachers
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