# Scalene Triangle Formulas

Posted In: Mathematics

A scalene triangle is a triangle that has three unequal sides.

Scalene Triangle

Sides of triangle: a,b,c
Semiperimeter: \$p={a+b+c}/2\$
Angles of triangle: α,β,γ
Altitudes to the sides a,b,c: h_a,h_b,h_c
Medians to the sides a,b,c: m_a,m_b,m_c
Bisectors of the angles α,β,γ: t_a,t_b,t_c
Area: S

1. \$α+β+γ=180°\$
2. \$a+b>c,\$
\$b+c>a,\$
\$c+a>b\$
3. \$a-b<c,\$
\$b-c<a,\$
\$a-c<b\$
4. Midline
\$q=a/2, q‖a\$
5. Law of Cosines
\$a^2=b^2+c^2-2bc\ cos\ α\$
\$b^2=a^2+c^2-2ac\ cos\ β\$
\$c^2=a^2+b^2-2ab\ cos\ γ\$
6. Law of Sines
\$a/{sin\ α}=b/{sin\ β}=c/{sin\ γ}=2R\$
where R is the radius of the circumscribed circle.
7. \$R=a/{2sin\ α}=b/{2sin\ β}=c/{2sin\ γ}\$
\$={bc}/{2h_a}={ac}/{2h_b}={ab}/{2h_c}={acb}/{4S}\$
8. \$r^2={(p-a)(p-b)(p-c)}/p,\$
\$1/r=1/h_a+1/h_b+1/h_c\$
9. \$sin{α/2}=√{{(p-b)(p-c)}/{bc}}\$
\$cos{α/2}=√{{p(p-a)}/{bc}}\$
\$tan{α/2}=√{{(p-b)(p-c)}/{p(p-a)}}\$
10. \$h_a=2/a√{p(p-a)(p-b)(p-c)}\$
\$h_b=2/b√{p(p-a)(p-b)(p-c)}\$
\$h_c=2/c√{p(p-a)(p-b)(p-c)}\$
11. \$h_a=bsin\ γ=csin\ β\$
\$h_b=asin\ γ=csin\ α\$
\$h_c=asin\ β=bsin\ α\$
12. \$m_a^2={b^2+c^2}/2-a^2/4\$,
\$m_b^2={a^2+c^2}/2-b^2/4\$,
\$m_c^2={a^2+b^2}/2-c^2/4\$
13. \$AM=2/3m_a\$
\$BM=2/3m_b\$
\$CM=2/3m_c\$
14. \$t_a^2={4bcp(p-a)}/(b+c)^2,\$
\$t_b^2={4acp(p-b)}/(a+c)^2,\$
\$t_c^2={4abp(p-c)}/(a+b)^2\$
15. \$S={ah_a}/2={bh_b}/2={ch_c}/2,\$
\$S={ab\ sin\ γ}/2={ac\ sin\ β}/2={bc\ sin\ α}/2,\$
\$S=√{p(p-a)(p-b)(p-c)}/2,\$ Heron’s Formula,
\$S=pr,\$
\$S={abc}/{4R},\$
\$S=2R^2sin\ α\ sin\ β\ sin\ γ,\$
\$S=p^2tan{α/2}\ tan{β/2}\ tan{γ/2}\$

Post Tags: Geometry