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equilateral triangle

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Equilateral Triangle

An equilateral triangle is a triangle in which all three sides have the same length. As a consequence, all interior angles are the same (60 ∘60^\circ).

Area and Height of an Equilateral Triangle

EquilateralTriangleLabelled.png

Consider an equilateral triangle with side length 11 and let hh represent its height. The area is then 12h\frac{1}{2} h. More generally, if an equilateral triangle has base length xx then its height is hxh x and so its area is 12hx 2\frac{1}{2} h x^2.

Cutting the triangle in half from a vertex to the midpoint of the opposite side results in two triangles that can be put back together to form an isosceles triangle with angles 120 ∘120^\circ, 30 ∘30^\circ, and 30 ∘30^\circ. The base of this triangle is twice the height of the original, so is 2h2 h.

EquilateralTriangleDissected.png

Three of these triangles fit together to form a new equilateral triangle. As this new triangle has base 2h2 h its area is 12h(2h) 2=2h 3\frac{1}{2} h (2 h)^2 = 2 h^3. It also is three copies of the original triangle so its area is 32h\frac{3}{2} h. Putting these together, h 2=34h^2 = \frac{3}{4} and so h=32h = \frac{\sqrt{3}}{2}.

The height and area of an equilateral triangle of base length 11 are therefore respectively

32,34 \frac{\sqrt{3}}{2}, \qquad \frac{\sqrt{3}}{4}

and of base length xx

32x,34x 2 \frac{\sqrt{3}}{2} x, \qquad \frac{\sqrt{3}}{4} x^2

category: triangles

Revised on August 31, 2021 18:45:45 by Andrew Stacey
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