Triangles

Triangles are one of the most commonly studied shapes in geometry. Their names and attributes are derived from their angles and sides. The three sides of a triangle are typically named after the letters of the points that they touch: the side opposite point A is called side a, the side opposite point B is called side b, and the side opposite point C is called side c. The sum of the lengths of the two shorter sides of a triangle is always greater than the length of the longest side. This is known as the Triangle Inequality.

There are many different types of triangles, and each one has its own special properties. The most basic triangle is the equilateral triangle, which has all three sides the same length and all three angles the same size.

The next most basic triangle is the isosceles triangle, which has two sides the same length and two angles the same size.

The last basic triangle is the scalene triangle, which has all three sides different lengths and all three angles different sizes.

Each type of triangle has its own special formula for finding its area. To find the area of an equilateral triangle, you multiply the length of one side by the length of the other side and then divide by 2.

To find the area of an isosceles triangle, you multiply the length of one of the sides by the length of the other side and then divide by 2.

To find the area of a scalene triangle, you multiply the length of one side by the length of the other side, and then you subtract the length of the third side. After that, you divide by 2.

Triangles are a great gateway into geometry because so much follows from a few simple facts. Start concrete for young learners — identifying triangles, counting sides and corners, sorting them by shape — then build toward the properties: the three angles always sum to 180°, and sides and angles relate to each other.

As students advance, introduce the types (equilateral, isosceles, scalene; right, acute, obtuse) and then the powerful tools: the Pythagorean theorem for right triangles, and eventually similarity, congruence, and basic trigonometry. Draw constantly — geometry is visual, and sketching a triangle to reason about it is a skill in itself. Hands-on work with rulers, protractors, and cut-out shapes makes the abstract properties tangible.