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Area of ​​a triangle

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This service will help you find the area of ​​a triangle online.

A triangle is a geometric figure formed by three points that do not lie on the same line and are interconnected. The length of each side of the triangle does not exceed the sum of the lengths of the other two sides.

In any triangle, the sum of the angles is 180 ° or π radians. Also, in any triangle, equal angles lie against equal sides, and a larger angle lies against the larger side.

The area of ​​the triangle can be found by different formulas.

If at least one of the sides is known and the height omitted on this side, then the area is as half the product of these quantities:

The formula of the area of ​​a triangle according to the known side and height

If at least one of the sides is known and the height omitted on this side, then the area is as half the product of these quantities:

$$ S = <1 over 2> cdot a cdot h_a = <1 over 2> cdot b cdot h_b = <1 over 2> cdot c cdot h_c. $$

If two sides are known and the angle between them, then the area of ​​the triangle can be found by multiplying these values ​​and dividing by 2:

$$ S = <1 over 2> cdot a cdot b cdot sin (C) = <1 over 2> cdot a cdot c cdot sin (B) = <1 over 2> cdot b cdot c cdot sin (A) $$

This online service uses the Heron formula to calculate the area of ​​a triangle (you need to know the lengths of three sides):

The area of ​​the triangle along the radius of the inscribed circle and three sides


The area of ​​the triangle is equal to half the sum of all three sides of the triangle times the radius of the inscribed circle. or in another way one can say: "The area of ​​a triangle is equal to half the perimeter of the triangle times the radius of the inscribed circle."

Area of ​​a right-angled triangle according to the Heron formula


Heron's formula for a right-angled triangle, where p is the semiperimeter of the triangle, calculated by the formula

Triangle - This is a geometric figure formed by three segments that connect three points that do not lie on one straight line. The line segments are called the sides of the triangle, and the points are called the vertices of the triangle.

Square Is a numerical characteristic characterizing the size of a plane bounded by a closed geometric figure.

Area is measured in square units: km 2, m 2, cm 2, mm 2, etc.

Depending on the type of triangle and its known source data, triangle area can be calculated:

The area of ​​the triangle through two sides and the angle between them

<2>cdot a cdot b cdot sin (alpha)>

The formula for finding the area of ​​a triangle through 2 sides and an angle:

<2> cdot a cdot b cdot sin ( alpha)>, where a, b - side of the triangle, α - the angle between them.

The area of ​​the triangle through the radius of the circumscribed circle and 3 sides

The formula for finding the area of ​​a triangle through the described circle and sides:

<4 cdot R >>, where a, b, c - side of the triangle, R - radius of the circumscribed circle.

The area of ​​the triangle through the radius of the inscribed circle and 3 sides

The formula for finding the area of ​​a triangle through an inscribed circle and sides:

<2>> where a, b, c - side of the triangle, r - radius of the inscribed circle.

The formula can be rewritten differently, given that < dfrac<2>> is the semiperimeter of the triangle. In this case, the formula will look like this: S = where p - the semiperimeter of the triangle.

The area of ​​the triangle across the side and two adjacent corners

The formula for finding the area of ​​a triangle across a side and 2 adjacent corners:

<2> cdot dfrac> where a - side of the triangle, α and β - adjacent corners, γ - the opposite angle, which can be found by the formula:

The area of ​​a triangle according to the Heron formula

The formula for finding the area of ​​a triangle according to the Heron formula (if 3 sides are known):

> where a, b, c - sides of the triangle, p Is the semiperimeter of the triangle, which can be found by the formula p = < dfrac<2>>

Area of ​​a right triangle through hypotenuse and acute angle

<4>cdot c^2 cdot sin (2 alpha)>

The formula for finding the area of ​​a right triangle by hypotenuse and acute angle:

<4> cdot c ^ 2 cdot sin (2 alpha)>, where c - hypotenuse of a triangle, α - any of the adjacent sharp corners.

Area of ​​a right-angled triangle through a leg and an adjacent corner

The formula for finding the area of ​​a right-angled triangle along a leg and an adjacent corner:

<2> cdot a ^ 2 cdot tg ( alpha)>, where a - a triangle leg, α - adjacent corner.

Area of ​​a right-angled triangle through the radius of the inscribed circle and the hypotenuse

The formula for finding the area of ​​a right triangle by the radius of the inscribed circle and hypotenuse:

where c - hypotenuse of a triangle, r - radius of the inscribed circle.

The area of ​​the isosceles triangle through the base and side

The formula for the area of ​​an isosceles triangle through the base and side:

<><4> sqrt <4 cdot a ^ 2-b ^ 2 >>, where a - the side of the triangle, b - base of the triangle

The area of ​​the isosceles triangle through the base and corner

<2>cdot a cdot b cdot sin( alpha)>

The formula for the area of ​​an isosceles triangle through the base and angle:

<2> cdot a cdot b cdot sin ( alpha)>, where a - the side of the triangle, b - the base of the triangle, α - the angle between the base and the side.

The area of ​​the isosceles triangle through the base and height

The formula for the area of ​​an isosceles triangle through the base and height:

<2> cdot b cdot h>, where b - the base of the triangle, h - height drawn to the base.

The area of ​​the isosceles triangle through the sides and the angle between them

The formula for the area of ​​an isosceles triangle through the sides and the angle between them:

<2> cdot a ^ 2 cdot sin ( alpha)>, where a - the side of the triangle, α - the angle between the sides.

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