给定底和高的三角形面积

AREA OF TRIANGLES :- There are three easy ways of calculating the area of triangles. Every method is most suitable with a particular type of triangle. These methods are

a) Area of a triangle in the terms of base and height

b) Area with herons formula

c) Area of an equilateral triangle

d) Area by using trigonometry

A) AREA IN THE TERMS OF BASE AND HEIGHT- This method is most useful with right angled triangle or when the base and height are given.

In this triangle, the base is taken as c. Height corresponding to the base is a perpendicular from the base touching a vertex of the triangle. To find the area of the triangle we have a simple formula : Area = 1/2 X base X corresponding height

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B) AREA WITH HERONS FORMULA - By this formula we can calculate the area of any triangle if we just know its sides.

Herons formula states that the area of a triangle can be calculated if we know its sides. It says that -

AREA = √s(s−a)(s−b)(s−c)

In this formula s represents the semi-perimeter of the triangle. Semi-perimeter means half the original perimeter or perimeter/2. And a, b, c represent the lengths of 3 sides of the triangle.

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C) AREA OF AN EQUILATERAL TRIANGLE - By this formula we can calculate the area of equilateral triangles by knowing its sides.

This theory states that the area of an equilateral triangle can be calculated if we know the side. It says that -

AREA OF AN EQUILATERAL TRIANGLE = (√3/4) a^2

In this formula a represents the measure of side of the triangle.

PROOF : Take the side of an equilateral triangle to be a. Now we draw a perpendicular bisector to one of the side in the equilateral triangle. As we can see a side is bisected. The two parts measure a/2. Now if we go by Pythagorean theoram we can calculate the perpendicular. It will measure :

√[a^2 - (a/2)^2]

= √[a^2 - (a^2/4)]

=√[4(a^2)/4 - (a^2)/4]

=√[3(a^2)/4] = √3a/2

Now when we have got the perpendicular we can multiply it with *1/2 X base *. We get that √3a/4 X a = [√3/4]a^2

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D) Area by using Trigonometry

Area of any triangle can be calculated if we know two sides and the included angle

AREA = (1/2) X ab X sinθ

where a and b are the side lengths and θ is the included angle

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The area of a triangle is $\frac{b \cdot h}{2}$ where $b$ is the base and $h$ is the height.

The area of a triangle is one half of the product of the base and the height drawn to that base.

More concisely, $A=\frac{1}{2}bh$, where $A$ is the area, $b$ is the base, and $h$ is the height.

To see this, attach two copies of the triangle together along one of the sides. This results in a parallelogram of base $b$ and height $h$. Cut the parallelogram along a line perpendicular to the base, swap the positions of the pieces, and reattach the two halves. This gives us a rectangle with base $b$ and height $h$. Clearly, the area of this rectangle is $bh$. Since the rectangle was made from two triangles, the area of the triangle is $A=\frac{1}{2}bh$.

The area of $\triangle ABC$ is $\frac{ab\sin C}{2}$, where $a$ and $b$ are side lengths and $C$ is the angle of vertex $C$.

Here we are going to look at something a little different.

Great job! You noted that something strange is going on here! In fact the triangle does not exist.

Suppose $D$ is the foot of perpendicular from $A$ to $BC$. (I.e., $D$ is a point on $BC$ and $AD\perp BC$.) We are given that $AD=8.$

We know that $BC=10,$ so let's say that $DC = x$, and $BD = 10 - x$. This works because $BD+DC = (10-x)+x = 10 = BC.$

The two smaller right triangles, $ABD$ and $ACD,$ make up the bigger triangle $ABC!$ So, by the Pythagorean Theorem:

$$\begin{align} AB^2+AC^2 = BC^2\\ (AD^2+BD^2)+(AD^2+DC^2) = BC^2\\ (8^2+(10-x)^2)+(8^2+x^2)=10^2\\ 64+(10-x)^2+64+x^2 = 100\\ 128+(10-x)^2+x^2=100\\ 128+(100-20x+x^2)+x^2=100\\ 128+100-20x+2x^2=100\\ 2x^2-20x+128=0\\ x^2-10x+64=0\\ (x - 5)^2 + 39 = 0 \end{align}$$

This equation does not have any real solutions. Thus, the triangle doesn't exist!

Here is a worksheet (http://www.mathworksheets4kids.com/triangles/area-challenging1.pdf) to practice your new skills. Beware of units, they may change on you when you aren't looking!

Suppose we had a triangle, and we only knew the base and height of said triangle. If we wanted to find the area, how would we do that?

The formula is 1/2(b*h)

Basically, you have to multiply the base and the height of the triangle, and then divide it by two.