Triangles: Types, Properties, Formulas and Examples

A triangle is a closed flat shape made from three straight line segments. It has three sides, three vertices and three interior angles. Triangles are among the most useful shapes in geometry because they are simple, strong and easy to measure.

Parts of a triangle

Sides and vertices

The three line segments are the sides of the triangle. The points where the sides meet are its vertices. A triangle named ABC has vertices A, B and C and sides AB, BC and CA.

Base and height

Any side may be chosen as the base. The corresponding height, also called the altitude, is the perpendicular distance from the opposite vertex to the line containing the base. The height may lie inside or outside the triangle.

Median

A median joins a vertex to the midpoint of the opposite side. Every triangle has three medians, and they meet at one point called the centroid.

Angle sum property

The three interior angles of every triangle add up to 180°. If two angles are known, subtract their sum from 180° to find the third angle.

Example

If two angles are 50° and 65°, the third angle is 180° − 50° − 65° = 65°.

Exterior angle property

An exterior angle is formed by extending one side of a triangle. Its measure equals the sum of the two non-adjacent interior angles. It is also supplementary to the adjacent interior angle, so those two angles add to 180°.

Triangle inequality

The sum of the lengths of any two sides of a triangle must be greater than the third side. Lengths 3 cm, 4 cm and 5 cm can form a triangle. Lengths 2 cm, 3 cm and 6 cm cannot, because 2 + 3 is not greater than 6.

Types of triangles by sides

Equilateral triangle

An equilateral triangle has three equal sides. Its three angles are also equal, and each angle measures 60°.

Isosceles triangle

An isosceles triangle has at least two equal sides. The angles opposite those equal sides are equal.

Scalene triangle

A scalene triangle has three sides of different lengths. Its three angles are also different.

Types of triangles by angles

Acute triangle

An acute triangle has three acute angles. Each interior angle is less than 90°.

Right triangle

A right triangle has one right angle of 90°. The side opposite the right angle is called the hypotenuse and is the longest side.

Obtuse triangle

An obtuse triangle has one angle greater than 90° but less than 180°. A triangle cannot have more than one right or obtuse angle because its total angle sum is 180°.

Perimeter of a triangle

Perimeter means the total distance around the triangle. If the side lengths are a, b and c, then perimeter = a + b + c.

Example

A triangle has sides 5 cm, 7 cm and 8 cm. Its perimeter is 5 + 7 + 8 = 20 cm.

Area of a triangle

The area of a triangle is one-half of the product of a chosen base and its perpendicular height. Area = 1/2 × base × height. The answer is written in square units.

Example

A triangle has a base of 10 cm and a perpendicular height of 6 cm. Its area is 1/2 × 10 × 6 = 30 cm².

Area of an equilateral triangle

If each side has length a, the area is (√3/4) × a². For a side of 8 cm, the area is (√3/4) × 64 = 16√3 cm², which is approximately 27.7 cm².

Pythagorean theorem

In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. If the shorter sides are a and b and the hypotenuse is c, then a² + b² = c².

Example

For a right triangle with shorter sides 6 cm and 8 cm, c² = 6² + 8² = 36 + 64 = 100. Therefore, c = 10 cm.

Congruent and similar triangles

Congruent triangles have the same shape and the same size. Their corresponding sides and angles are equal. Similar triangles have the same shape but may have different sizes. Their corresponding angles are equal, and corresponding sides are in the same ratio.

Worked examples

Example 1: Identify a triangle

A triangle has side lengths 7 cm, 7 cm and 10 cm. Two sides are equal, so it is an isosceles triangle.

Example 2: Find a missing angle

A right triangle has one angle of 90° and another of 35°. The missing angle is 180° − 90° − 35° = 55°.

Example 3: Check possible side lengths

Can 4 cm, 6 cm and 9 cm form a triangle? Yes. The smallest pair gives 4 + 6 = 10, which is greater than 9; the other two inequalities are also satisfied.

Common mistakes

Use the perpendicular height when finding area, not a sloping side unless that side is perpendicular to the base. Remember to use square units for area. The hypotenuse exists only in a right triangle. Finally, test the triangle inequality before assuming that three lengths can form a triangle.

Practice questions

1. Find the third angle when two angles are 40° and 75°. 2. Classify a triangle with sides 9 cm, 9 cm and 9 cm. 3. Find the area when base = 12 cm and height = 5 cm. 4. Can 2 cm, 5 cm and 8 cm form a triangle? 5. Find the hypotenuse when the shorter sides are 5 cm and 12 cm.

Answers

1. 65°. 2. Equilateral. 3. 30 cm². 4. No, because 2 + 5 is not greater than 8. 5. 13 cm.

Continue learning

Study related lessons on Mathematical Shapes, Geometry, Quadrilaterals and Symmetry.