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xGeometry

Equilateral Triangle

Side
$$S$$
Height
$$h$$
$$2p = S \times 3$$
Perimeter
$$A = \frac{S \times h}{2}$$
Area
$$S = \frac{A \times 2}{h}$$
$$h = \frac{A \times 2}{S}$$
$$S = \frac{2p}{3}$$
Side
Knowing side or height only
$$2p = 2 h \sqrt{3}$$
Perimeter (with height)
$$h = \frac{2p}{2\sqrt{3}}$$
Height (from perimeter)
$$A = \frac{\sqrt{3}}{4} {S}^2$$
Area (with side)
$$A = \frac{{h}^2}{\sqrt{3}}$$
Area (with height)
$$S = \frac{2h}{\sqrt{3}}$$
Side (with height)
$$h = \frac{S\sqrt{3}}{2}$$
Height (with side)
Inscribed circumference
$$r = \frac{1}{2 \sqrt{3}} S$$
Inscribed radius (with side)
$$r = \frac{1}{3} h$$
Inscribed radius (with height)
$$S = 2r\sqrt{3}$$
Side
$$h = 3r$$
Height
$$2p = 6r\sqrt{3}$$
Perimeter
$$A = 3{r}^2 \sqrt{3}$$
Area
Circumscribed circumference
$$R = \frac{\sqrt{3}}{3} S$$
Circumscribed radius (with side)
$$R = \frac{2}{3} h$$
Circumscribed radius (with side)
$$S = R\sqrt{3}$$
Side
$$h = \frac{3}{2}R$$
Height
$$2p = 3\sqrt{3}R$$
Perimeter
$$A = \frac{3\sqrt{3}}{4}{R}^2$$
Area

Definition

An equilateral triangle is a triangle with all sides congruent

Properties

  1. All sides are congruent
  2. All angles are congruent
  3. All the  Generic Triangle formulas are valid
  4. The height relative to the base divides the shape in two congruent right triangles. For these are valid the  Right Triangle formulas
Equilateral Triangle
Equilateral Triangle Formulas
Data Formula
Perimeter 2p = S × 3
Area A = (S × h) / 2
Side S = 2p / 3
Side (with height) S = (2 × h) / √3
Height (with side) h = (S√3) / 2
Height (from perimeter) h = 2p / (2√3)
Perimeter (with height) A = 2 × h × √3
Area (with side) A = (√3 / 4) × S2
Area (with height) A = h2 / √3
Side S = (A × 2) / h
Height h = (A × 2) / S
Equilateral Triangle - Inscribed circumference
Data Formula
Inscribed radius (with side) r = [1 / (2√3)] × S
Inscribed radius (with height) r = 1/3 × h
Side S = 2r√3
Height h = 3r
Perimeter 2p = 6r√3
Area A = 3r2√3
Equilateral Triangle - Circumscribed circumference
Data Formula
Circumscribed radius (with side) r = (√3 / 3) × S
Circumscribed radius (with height) r = 2/3 × h
Side S = R√3
Height h = 3/2 × R
Perimeter 2p = 3√3R
Area A = [(3√3) / 4] × R2