An equilateral triangle has a side 2 cm. Calculate the height of the triangle.

Analyze the question: An equilateral triangle has all sides equal (2 cm) and all angles 60°. The height (altitude) drops from one vertex to the midpoint of the opposite side. This splits the triangle into two congruent right-angled triangles.

Concept: The altitude of an equilateral triangle bisects the base. We then use Pythagoras' theorem in one of the right-angled triangles.

Solution:

• Hypotenuse = the slanted side = 2 cm
• Base of the right triangle = half of 2 cm = 1 cm
• Height = h

Using Pythagoras:

h² + 1² = 2²

h² + 1 = 4

h² = 3

h = √3 cm

Check with the formula: Height of an equilateral triangle = (√3 / 2) × side = (√3 / 2) × 2 = √3 cm ✓

Tip: For any equilateral triangle, remember h = (√3/2) × side. Also, the 30°-60°-90° triangle has sides in the ratio 1 : √3 : 2. Here, the side of 2 is the hypotenuse, 1 is the short side, and √3 is the height. Spotting this ratio will save you time in the exam.