The equilateral triangle along with a square is, perhaps, the simplest and symmetric figure in planimetry. Certainly, all ratios fair for a usual triangle are right as well for equilateral. However for the correct triangle all formulas become much simpler.
It is required to you
- calculator, ruler
Instruction
1. That searchperimeterequilateral trianglemeasure by length of one of its parties and increase result of measurement by three. In the form of a formula this rule can be written down as follows: Prt = Ds * 3, where: Prt – perimeter of an equilateral triangle, Ds – length of any of its parties. The perimeter of a triangle will turn out in the same units of measure, as length of its party.
2. Example. Length of the party of an equilateral triangle is equal to 10 mm. It is required to define its perimeter. Decision. Prt = 10 * 3 = 30 (mm)
3. As the equilateral triangle possesses high degree of symmetry, for calculation of its perimeter one of parameters suffices. For example, the areas, heights, the radius of the entered or circumscribed circle.
4. If the radius of an inscribed circle of an equilateral triangle is known, then for calculation of its perimeter use the following formula: Prt = 6 * √3 * r, where: r - radius of an inscribed circle. This rule follows from the fact that the radius of an inscribed circle of an equilateral triangle is expressed through length of its party by the following ratio: r = √3/6 * Ds.
5. To calculate perimeter of the correct triangle through the radius of a circumscribed circle, apply a formula: Prt = 3 * √3 * R, where: R - radius of a circumscribed circle. This formula is easily brought out of the fact that the radius of a circumscribed circle of the correct triangle is expressed through length of its party by the following ratio: R = √3/3 * Ds.
6. For calculation of perimeter of an equilateral triangle through the known area use the following ratio: Srt = Dst² * √3/4, where: Srt is the area of an equilateral triangle. From here it is possible to remove: Dst² = 4 * Srt/√3, therefore: Dst = 2 * √ (Srt/√3). Substituting this ratio in a perimeter formula through length of the party of an equilateral triangle, we receive: Prt = 3 * Dst = 3 * 2 * √ (Srt/√3) = 6 * sst / (√3) = 6sst/3^¼.