Why is the area formula of triangle divided by two and multiplied by the height? Everyone has studied mathematics in primary school. There are all kinds of knowledge in mathematics, some of which are about graphic area. Let me see why the area formula of a triangle is to multiply the base by the height and divide it by two.
Why is the area formula of triangle divided by the base and multiplied by two? 1 Because two identical triangles can be spliced into a parallelogram, the area of the triangle is half that of the spliced parallelogram.
If S is used to represent the area of the triangle, and α and H are used to represent the base and height of the triangle respectively, then the formula of the area of the triangle is S=ah÷2.
Extended data:
Area formula of common geometric figures:
1, rectangular area = length× width S=ab
2. Area of a square = side length × side length S = a× a.
3. Area of triangle = base × height ÷2 S=ah÷2.
4. parallelogram area = bottom x height S=ah.
5. trapezoidal area = (upper bottom+lower bottom) × height ÷ 2s = (a+b) h ÷ 2.
Classification of triangles:
Ordinary triangles are divided into ordinary triangles (three sides are unequal) and isosceles triangles (isosceles triangles with unequal waist and equal waist and bottom).
According to the angle, there are right triangle, acute triangle and obtuse triangle, among which acute triangle and obtuse triangle are collectively called oblique triangle.
1, acute triangle: the largest of the three internal angles of a triangle is less than 90 degrees.
2. Right triangle: The largest of the three internal angles of a triangle is equal to 90 degrees.
3. obtuse triangle: the largest of the three internal angles of a triangle is greater than 90 degrees and less than 180 degrees.
Why the area formula of triangle is a mathematical formula to find the area of triangle by dividing the bottom by the height by two and then dividing by two;
Base × height ÷2
The mathematical formula for finding the triangle area is like this, right? So why can we get the area of triangle by this formula?
"I haven't thought about this problem ..."
"When I was in primary school, that's what the teacher taught me ..."
This is the beginning of the wrong mathematics learning method.
Of course, someone will answer:
"That's because the area of a triangle is half that of a corresponding quadrilateral."
So I'm going to ask again,
Why is the area calculation formula of quadrilateral "base × height"?
To answer this "why", then you must have a deep understanding and understanding of the mathematical definition of the calculation area.
Let's first calculate how many benchmark small squares are included in the figure below (for example,1cm 2 square).
In the figure below, the length and width of each grid are 1cm.
The rectangle in the picture is 8 cm long (bottom) and 5 cm wide (height).
In a rectangle, there are 8 squares in a horizontal row and 5 squares in a vertical column. So, how many small squares are there in this rectangle?
8×5=40
Because the area of 1 small square is 1cm 2, the area of a rectangle is 40cm 2.
Then, when calculating the rectangular area,
We can use the method of "base × height" to calculate.
So, how does the parallelogram calculate the area?
In this parallelogram, many small squares are incomplete, so it is difficult for us to count how many small squares it contains. Then, we will make the following deformation.
In this way, it becomes the same as the previous rectangle:
Base × height
Then the number of small squares it contains can be calculated soon.
Next, let's return to the triangle.
Like parallelogram, it is difficult to count how many small squares it contains because there are many incomplete ones. As shown in the picture below, we turn this triangle upside down.
In this way, it is the same as the parallelogram in front.
Then, we can calculate the area of the parallelogram as follows:
Base × height
The parallelogram in the figure is composed of the first two triangles, so we can draw the parallelogram with an area twice that of the original triangle. Then the area of the triangle should be:
Base × height ÷2
How's it going? Do you feel anything?
This is the principle behind the triangle area formula. If you can understand the whole principle, there is no need to memorize either the triangular area formula or the quadrilateral area formula. Similarly, regarding the area formula of trapezoid:
(Upper sole+lower sole) × Height ÷2
We can also draw inferences from others, so as to easily find out the "principle" behind it. Not only that, this way of thinking (through the superposition calculation of small cells, the overall area is obtained) will definitely help you understand the "integral" course you are about to learn.
If you don't want to memorize mathematical definitions and formulas, you must find out the "principle" behind it from the beginning. Besides, you can't just
It is to understand such a mathematical definition and its relationship with other mathematical definitions, which requires you to have a comprehensive grasp of these principles.
Furthermore, when you master the principles behind mathematical formulas, your curiosity is greatly satisfied. You will naturally feel:
"Oh, so that's it!"
"How interesting!"
Then make you feel that learning mathematics is actually very interesting, which is also the interest that can be brought by the "not memorizing" learning method. When you understand the principle behind a certain mathematical formula, you should think about how to use it flexibly instead of remembering it deliberately. This is the key to learning mathematics.
How to find the area of three sides of a triangle is known. The method of finding various triangle areas is as follows:
1, given the triangle base a and height h, then S=ah/2.
2. Given three sides A, B and C of a triangle, then
(Helen formula) (p=(a+b+c)/2)
S=sqrt[p(p-a)(p-b)(p-c)]
= sqrt[( 1/ 16)(a+b+c)(a+b-c)(a+c-b)(b+ c-a)]
= 1/4 sqrt[(a+b+c)(a+b-c)(a+c-b)(b+c-a)]
3. Given the two sides A and B of a triangle and the included angle C and S between the two sides, S= 1/2.
AbsinC, that is, the product of two sides multiplied by the sine of the included angle.
4. Let the three sides of a triangle be A, B and C respectively, and the radius of the inscribed circle be R..
Then the triangle area =(a+b+c)r/2.
5. Let the three sides of a triangle be A, B and C respectively, and the radius of the circumscribed circle be R..
Triangle area =abc/4R.
6. Determinant form
It's a third-order determinant, this triangle
In a plane rectangular coordinate system
, here
It is best to start from the upper right corner and select them in counterclockwise order, because the results obtained in this way are generally positive. If you don't follow this rule, you may get a negative value, but it doesn't matter, just take the absolute value and it won't affect the size of the triangle area.
This formula can be proved by the area formula of "the product of two sides multiplied by the sine value of the included angle"
7, Helen-Qin triangle midline area formula:
s =√[(Ma+m b+Mc)*(m b+Mc-Ma)*(Mc+Ma-Mb)*(Ma+m b-Mc)]/3
Where ma, MB and MC are the length of the midline of the triangle,
8, according to the trigonometric function area:
s = ab sinC = 2R Sina sinb sinC = asinBsinC/2 Sina
Note: where r is the radius of the circumscribed circle.
9, according to the vector area:
Where (x 1, y 1, z 1) and (x2, y2, z2) are the coordinate expressions of vectors AB and AC in the spatial rectangular coordinate system, namely:
The triangle area formed by adjacent sides of the vector is equal to half of the parallelogram area formed by adjacent sides of the vector.
Extended data
The triangle area formula refers to the area of the triangle calculated by the formula. A closed figure composed of three line segments on the same plane but not on the same straight line is called a triangle with the symbol △.
Common triangles are divided into isosceles triangles (isosceles triangles with unequal waist and bottom and isosceles triangles with equal waist and bottom) and isosceles triangles; According to the angle, there are right triangle, acute triangle and obtuse triangle, among which acute triangle and obtuse triangle are collectively called oblique triangle.