ShapesProperties
- All sides are equal and every angle is 60°.
- It has three lines of symmetry.
- Each height is also a median and an angle bisector, and all three meet at one point.
- Six equilateral triangles fit together to make a regular hexagon.
Why the formulas work
Drop a height from the top corner. It cuts the base in half and makes two right-angled triangles with hypotenuse a and base a/2. By Pythagoras the height is the root of a² minus a²/4, which is (√3/2)a. Half of base times height then gives (√3/4)a². The value of √3 is about 1.732.
Worked example
Take a equilateral triangle with side 10 cm.
- Area. Area = (√3 ⁄ 4) a2. Here: (√3 ⁄ 4) × 10 × 10 = 43.3 cm².
- Perimeter. Perimeter = 3a. Here: 3 × 10 = 30 cm.
- Height. Height = (√3 ⁄ 2) a. Here: (√3 ⁄ 2) × 10 = 8.66 cm.
Finding a missing measurement
From the perimeter, divide by 3. From the height, the side is 2h ÷ √3. From the area, the side is √(4A ÷ √3): an area of about 43.3 cm² gives a side of 10 cm, since (√3 ⁄ 4) × 100 is about 43.3.
Mistakes to avoid
- Using ½ × a × a for the area. The height is not a, it is about 0.866 times a.
- Rounding √3 to 1.7 too early. Use 1.732 and round at the end.
Where you see it
Road warning signs, the rack of balls in pool and snooker, some trusses and the faces of a tetrahedron are equilateral triangles.
Try these
Work each one out, then tap to check.
A equilateral triangle has side 6 cm. Find its area, perimeter and height.
Area = 15.59 cm²; Perimeter = 18 cm; Height = 5.2 cm.
A equilateral triangle has side 14 cm. Find its area, perimeter and height.
Area = 84.87 cm²; Perimeter = 42 cm; Height = 12.12 cm.