Triangles

What you will learn

• Name triangles by their sides and by their angles.
• Use the angle sum to find a missing angle.
• Say why the angles of a triangle add to $180^\circ$.

Before you start: you should know the types of angle and angles on a straight line. See Angle Rules on Lines and Around a Point.

The idea

A triangle is a flat shape with three straight sides. It is the simplest polygon and also the most important. Every other polygon can be cut into triangles, and triangles are rigid: push on a square frame and it collapses into a rhombus, but a triangular frame keeps its shape.

Classifying triangles

By sides:

• Equilateral: all three sides equal, all angles $60^\circ$.
• Isosceles: two equal sides, and the two angles opposite them are equal.
• Scalene: no equal sides.

By angles: acute (all angles under $90^\circ$), right-angled (one angle exactly $90^\circ$) or obtuse (one angle over $90^\circ$).

Why the angles add to 180°

Draw a line through one corner parallel to the opposite side. The two angles at the base reappear at the top as alternate angles, sitting alongside the third angle on a straight line. Angles on a line make $180^\circ$, so the triangle’s angles do too.

Worked examples

Example 1: Missing angle

A triangle has angles $48^\circ$ and $75^\circ$. Find the third angle and say what type it is.

1. $48 + 75 = 123$.
2. $180 - 123 = 57$.
3. All three angles are under $90^\circ$.

Answer: $57^\circ$, an acute triangle.

Example 2: Isosceles

An isosceles triangle has one angle of $100^\circ$. Find the other two angles.

1. The $100^\circ$ angle cannot be a base angle, because two of them would already make $200^\circ$.
2. The other two are equal and share $180 - 100 = 80$.
3. Each is $80 \div 2 = 40$.

Answer: $40^\circ$ and $40^\circ$

Example 3: Area

A triangle has a base of $12$ cm and a perpendicular height of $7$ cm. Find its area.

1. A triangle is half of a rectangle with the same base and height.
2. $A = \tfrac{1}{2} \times 12 \times 7 = 42$.

Answer: $42 \text{ cm}^2$

Watch out: the height used for area is the perpendicular height, not the length of a slanted side.

Why it matters

Triangles hold up the world. Roof trusses, cranes, electricity pylons and bicycle frames are all built from triangles because of their rigidity. Builders check that corners are square with a 3-4-5 rope, and GPS receivers locate you using a three-dimensional cousin of triangulation.