三角形面积公式.doc
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三角形面积公式
Triangle area formula
The 1. known triangle at the end of a, h, S = ah/2
2.
known triangles, three sides a, B, C, then
(Helen formula) (p= (a+b+c) /2)
S=sqrt[p (P-A) (P~B) (P-C)]
=sqrt[ (1/2) (a+b+c) (a+b~c) (a+c-b) (b+c~a)]
V = 2/2sqrt[ (a+b+c) (a+b~c) (a+c-b) (b+c~a)]
3.
, a and B on both sides of the triangle are known. The angles of
both sides are C, and S = 1/2 * absinC, that is, the sine value of
the angle between the two sides of the product
・
4.
triangle, three sides are a, B, C, and the radius of inscribed
circle is R
Then the triangle area = (s+b+c) r/2
5.
of the three sides of a triangle are respectively a, B, C, a
radius of R
Then the triangle area is =abc/4R
6.
S =1/2 *
C D
1
E F
1
a B
1
C D1 | three order determinant, the triangle ABC in plane
Cartesian coordinate system A (a, b), B (C, d), C (E, f), where
ABC
E F 1
Take the best selection in counterclockwise order starting from
the right corner, because it made the results are generally
positive, if not according to the rules, may be negative, but it
does not matter, as long as you can get the absolute value, the
size of the triangle area will not be affected!
7.
Helen Qin Jiushao triangle line area formula:
V [S
二
(Ma+Mb+Mc) * (Mb+Mc-Ma) * (Mc+Ma- Mb) * (Ma+Mb-Mc)]/3
Ma, Mb, and Me are the middle of the triangle
8., according to trigonometric function area:
S
二
?
ab sinC=2R2 sinAsinBsinC= a2 sinBsinC/2sinA
Note: R is the radius of excircle.
9.
calculate the area by vectors:
S delta)
二？
2 root (|AB|*|AC|) - (AB*AC) ².
10.
in the Cartesian coordinate system, the area of the triangle
ABC is
S=|AB * AC|/2
That is, the area S is equal to half the norm of the vector AB and
the product of the AC vector
Extended reading: 1. according to the sine theorem:
2.5
triangle ABC=absinC/2
3.5
triangle ABC=acsinB/2
4. S triangle ABC
二
bcsinA/2
The Helen fonnula and translated hilen formula, formula, formula,
Helen dragon hero Qin Jiushao formula, the legend is the ancient
Sura of ancient Wang Xilun (Heron, also known as Hailong) the
formula found by B
・
using the three sides of the triangle to
obtain the area of the triangle
・
But according to Morris Kline
published in 1908 research, this formula is discovered by
Archimedes, published in the name of supporting hieron II (not
verified)
・
Qin Jiushao, a mathematician in the Song Dynasty, also
put forward the three oblique quadrature method, which is
basically the same as Helen
s formula.
Suppose there is a triangle whose length is a, B, C, and the area
，
S of the triangle can be obtained by the following formula:
S= root [p (P-A) (P-B) (P-C)]
And the P in the formula is half perimeter:
P
二
(a+b+c) /2
注
1
：
“测量”(《度量论》
)
手抄本中用的作为半周长，所以
S
二丁
［
P (P-A) (PB) (P-C)
］
和二丿［
(S-A) (S-B) (SC)
］两种写法
都是可以的，但多用
P
作为半周长。
由于任何
N
边的多边形都可以分割成
N-2
个三角形，所以海伦公式 可
以用作求多边形面积的公式。比如说测量土地的面积的时候，不用测
三角形的高，只需测两点间的距离，就可以方便地导出答案。
(1)
证明
与海伦在他的著作“测量”(《度量论》
)
中的原始证明不同，在此我
们用三角公式和公式变形来证明。设三角形的三边一、
B
、
C
的对角
分
别为一、
B
、
C,
则余弦定理为
(
一’
COSC
二
2 + B
八
2-C
八
2) / 2ab
S=1/2*AB * Sine