Area of Triangles Formulas

Friends, in this topic, we will talk about “Area of Triangles Formulas Pdf”.

The altitude of a triangle is the perpendicular drawn from the vertex of the triangle to the opposite side. Also, known as the height of the triangle, the altitude makes a right angle triangle with the base.

General Area Formula for a triangle with known height and base;

The area of a triangle is equal to half of multiplied the base by the height. This is the most commonly used formula to calculate the area of ​​the triangle.

! Regardless of the used edge and height that belongs to this edge, the area is always the same.

A, B, C: corner

a (|BC|), b(|AC|), c(|AB|): edge

h: height

NOTE: 

In an ABC triangle, the height may not always be inside of the triangle. In other words, if the triangle is a wide-angle triangle, the height belongs to the triangle is taken from outside of the triangle as below.

In some special triangles (isosceles triangles, etc.), height information may not be provided. In such cases, we must first find the height. “Pythagoras relation” is used when finding the height.

According to the Pythagoras relation;
In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two side lengths.

c: hypotenuse

       a² + b² = c²

 

Area of an equilateral triangle;

In special triangles such as equilateral triangles, we use some special formulas instead of the general area formula in triangle given above. This is because the equilateral triangle has three sides of equal length, so the lengths of the heights of these three sides are also equal. There is also a constant ratio between edge length and height. Therefore, the area of an equilateral triangle with only one edge can be easily found.

If ABC is an equilateral triangle;

a = b = c

\( ​h_a = h_b = h_c \)

m(A) = m(B) = m(C) = 60 º

           Area = ​\( \displaystyle\frac{a^2.\sqrt{3}}{4} \)

 

NOTE:
The area of the triangle “30º – 30º – 120º”, which is one of the special triangles, is calculated as the area of the equilateral triangle. If we say “a” to the sides opposite the 30º angle; the height is “a / 2” and the base length is “a√3”. In this case, the area ​\( \displaystyle\frac{a^2.\sqrt{3}}{4} \)​.

The area formula of the triangle whose length and an angle of two sides are known;

Area formula of a triangle with known three side lengths (Heron’s Formula);

Three edge lengths are usually given when calculating the area of ​​triangles which all sides have been different lengths. In such questions, no height information is given. The area only needs to be calculated based on the length of the edges. In order to calculate the area of ​​the triangle;

– The triangle’s perimeter is found (sum of all three sides) ⇒ a + b + c
– The found perimeter is divided by 2 (This value is named an “u” value) ⇒ ​\( u = ​ \displaystyle\frac{a+b+c}{2} \)
– And the area information of the triangle is calculated with the area formula below. 🙂

Area = ​\( \sqrt{u. (u-a).(u-b).(u-c)} \)

 

The area formula for a triangle with known perimeter and radius of the inscribed circle in a triangle;

If “O” is a centre of an inscribed circle in the triangle, ‘r’ is a radius of this circle and perimeter of the triangle is equal “a+b+c”, the area of the triangle is found with the formula which is below.

\( u = ​ \displaystyle\frac{a+b+c}{2} \)

Area = u.r

NOTE:

In a triangle, there is a relationship like given below between the radius of the inscribed circle and the edge heights of the triangle.

\( \displaystyle\frac{1}{r} = \frac{1}{h_a} +\frac{1}{h_b} +\frac{1}{h_c} \)

To find the area of a triangle with knowing all sides and outer circumcircle radius;

 

The area of ​​a right-angle triangle;

The area of ​​a right-angle triangle is equal half of the multiply of perpendicular sides.

 

On the hypotenuse of the right tangent circle of a right triangle, the part lengths are m and n is the area with m.n.

 

In the triangle 15 – 75 – 90, which is one of the special right triangles, the height lowered from the right angle is equal to the length of the hypotenuse ¼.

 

Area of ​​triangles with equal height;

The ratio of the areas of the triangles with equal heights is equal to the ratio of the lengths of the floors from which the height is lowered.

! Since the bases of the ABC and ACD triangles are on the same line and the peaks are at the same point, their heights are equal.

 

Area of ​​triangles whose base lengths are equal;

The ratio of the areas of the triangles whose bases are equal is equal to the ratio of their height of the floor lengths that are equal.

! The bases of the ABC and DBC triangles are equal and overlapping.

 

When asked about the area of ​​the region whose bases are equal and between heights of two different triangles and shown as shaded, the area is calculated as follows.

 

Area of ​​equilateral triangles (Similar triangles);

Lines that are parallel to each other and base in a triangle and divide the edges they cut into equal lengths among themselves, divide the triangle proportionally into areas. The areas of the regions between these parallel lines are in direct proportion to the odd numbers. The main reason for this occurrence is similarity.

 

The area of ​​the triangle between two parallel lines;

The distance between two parallel lines is the height of all triangles drawn with the corners on these lines. In this case, for the triangle with a point A on the line d1 in the figure; Regardless of where point A (points D, E) is, the base [BC] does not change at all, and since the height remains the same, the areas of the resulting triangles are equal.

 

Açıortayın oluşturduğu üçgenin alanı;

The proportion of the areas where the bisector divides the triangle drawn from one corner of the triangle is equal to the ratio of the edges adjacent to the corner where this bisector is located.

 

Kenarortay kullanarak üçgenin alanı;

The formula used when calculating area in triangles given Kenarortay lengths is called “Heron Formula”. As per the Kenarortay Heron formula;

Areas created by Kenarortaylar;

 

Kenar uzunluklarına göre üçgenin alanı;

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