Triangles

A triangle is a closed, flat figure made of three straight sides. Its three corners must not all lie on one straight line. Turning a triangle or making a scaled copy does not change its angle sizes or its shape groups.

Two Ways to Sort

We can group triangles by side lengths:

  • An equilateral triangle has three equal sides.
  • An isosceles triangle has at least two equal sides. With this inclusive definition, an equilateral triangle is also isosceles.
  • A scalene triangle has three different side lengths.

We can also group them by angle sizes:

  • An acute triangle has three angles smaller than 90°.
  • A right triangle has one angle equal to 90°. Its two sides meeting at that corner are perpendicular.
  • An obtuse triangle has one angle greater than 90° and less than 180°.

A triangle can have a label from each list. For example, a triangle with angles of 45°, 45° and 90° is both right and isosceles. Equal sides do not mean parallel sides: every pair of a triangle's sides meets at a corner.

Follow the Angles

The three inside angles of a triangle on a flat plane add to 180°. Two right angles would already use that whole total, leaving no positive third angle. That is why a triangle cannot have two right angles.

The right angle need not be at the bottom. Turn a drawing and check again with a square paper corner. Use the angles and sides, not the way the triangle points.

Stone pyramids at Giza with triangular faces.
A pyramid is a three-dimensional solid; its side faces are triangles. (Ricardo Liberato / Wikimedia Commons)

Why Frames Use Triangles

In an ideal flat frame, three rigid bars joined at their ends cannot change the triangle's shape while keeping all three lengths fixed. A four-bar frame can change its angles unless it is braced. This helps explain triangular bracing in structures. Actual strength also depends on the materials, joints and loads; “triangle” does not mean unbreakable or always strongest.

Try a Shape Investigation

Draw a right triangle with its right corner above the other two corners. Then draw an acute triangle. Explain which angles decide the groups. A roughly triangular slice with a curved edge is not an exact mathematical triangle.

Open the Grade 4 Shape Detective lesson to sort figures, draw counterexamples and explain your choices.