LovesTheTeam.com

Instructions

1

The basic formula of the area of ​​an arbitrary triangle ABC is calculated as follows: S =? * C * h, where c is the base triangle, H is the height drawn to this base.

2

The formula for calculating the area through the product of the sides and the sin angle between them is: S =? * A * b * sin ?.

3

Let a circle of radius r be inscribed in the triangle, then the area formula triangle will have the form: S =? * P * r, where P is the perimeter triangleI.e. S =? * (A + b + c) * r.

4

Let around triangle A circle of radius R is described. The area formula triangle Through the radius of the circumscribed circle and the length of the sides triangle: S = (a * b * c) / (4 * R). The formula of the area triangle Through the radius of the circumscribed circle and the angles triangle: S = 2 * R ^ 2 * sin? * Sin? * Sin ?.

5

There is the Heron formula for the square triangle, named after the ancient Greek mathematician Heron of Alexandria, who lived at the very beginning of our era. This formula gives the definition of the area through the lengths of all sides triangle: S =? * V ((a + b + c) * (b + c - a) * (a + c - b) * (a + b - c)) The formula with the introduction of the semiperimeter concept is simplified: S = V (p * (p - a) * (p - b) * (p - c)), where p = (a + b + c) / 2 is the semiperimeter.

6

Area formula triangle through the length of the side and the angles triangle: S = a ^ 2 * sin? * Sin? / (2 * sin?), Where? And? - adjacent corners, eh? - Opposite angle to side a.

7

For a rectangular triangle The area formula is simplified and looks like this: S =? * A * b, i.e. area Rectangular triangle Is equal to half the product of the lengths of the legs.

8

Area formula for equilateral triangle: S = (a ^ 2 * v3) / 4.

9

The area formula for an isosceles rectangular triangle: S =? * (A ^ 2 + b ^ 2), where a and b are the legs triangle. In addition, for any triangle The following inequality holds: S <* * (a ^ 2 + b ^ 2).

Instructions

1

If the lengths of the legs (a and b) of a rectangularTriangle are given explicitly under the conditions of the problem, the formula for calculating the area (S) of the figure will be very simple - multiply these two quantities, and divide the result in half: S = ½ * a * b. For example, if the lengths of the two short sides of such a triangle are 30 cm and 50 cm, its area should be ½ * 30 * 50 = 750 cm².

2

If the triangle is placed in a two-dimensionalorthogonal coordinate system and given the coordinates of its vertices A (X₁, Y₁), B (X₂, Y₂) and C (X₃, Y₃), start by calculating the lengths of the legs themselves. To do this, consider triangles composed of each side and its two projections onto the coordinate axes. The fact that these axes are perpendicular makes it possible to find the side length by the Pythagorean theorem, since it is a hypotenuse in such an auxiliary triangle. The lengths of the side projections (the legs of the auxiliary triangle) are found by subtracting the corresponding coordinates of the points forming the side. The side lengths must be equal to | AB | = √ ((X₁-X₂) ² + (Y₁-Y₂) ²), | BС | = √ ((X₂-X₃) ² + (Y₂-Y₃) ²), | CA | = √ ((X₃-X₁) ² + (Y₃-Y₁) ²).

3

Determine which pair of sides are the legs- this can be done from the lengths obtained in the previous step. The legs must be shorter than the hypotenuse. Then use the formula from the first step - find half the product of the calculated values. Provided that the legs are AB and BC, the general formula can be written as follows: S = ½ * (√ ((X₁-X₂) ² + (Y₁-Y₂) ²) * √ ((X₂-X₃) ² + (Y₂-Y₃) ²).

4

If a rectangular triangle is placed inthree-dimensional coordinate system, the sequence of operations will not change. Just add the third coordinates of the corresponding points to the formulas for calculating the lengths of the sides: | AB | = √ ((X₁-X₂) ² + (Y₁-Y₂) ² + (Z₁-Z₂) ²), | BС | = √ ((X₂-X₃) ² + (Y₂-Y₃) ² + (Z₂-Z₃) ²), | CA | = √ ((X₃-X₁) ² + (Y₃-Y₁) ² + (Z₃-Z₁) ²). The final formula in this case should look like this: S = ½ * (√ ((X₁-X₂) ² + (Y₁-Y₂) ² + (Z₁-Z₂) ²) * √ ((X₂-X₃) ² + (Y₂- Y₃) ² + (Z₂-Z₃) ²).

You will need

• - Notebook;
• - ruler;
• - a pencil;
• - a pen;
• - calculator.

Instructions

1

A rectangle is a quadrilateral with all corners straight. A special case rectangle is a square.Area rectangle Is a quantity equal to the product of its length and width. And the square of the square is equal to its length of its side, raised to the second degree. If only width, then you must first find the length, and then calculate the area.

2

For example, given a rectangle ABCD (Fig. 1), where AB = 5 cm, BO = 6.5 cm. Find the area rectangle AVCD.

3

Because ABCD - rectangle, AO = OS, BO = OD (as diagonals rectangle). Consider the triangle ABC. AB = 5 (by convention), AC = 2AO = 13 cm, angle ABC = 90 (since ABCD is a rectangle). Hence ABC is a right-angled triangle in which AB and BC are legs, and AC is the hypotenuse (because it is opposite to the right angle).

4

The Pythagorean theorem says: the square of the hypotenuse is equal to the sum of the squares of the legs. By the Pythagorean theorem, find the catheter BCBC2 = AC2 - AB2BC2 = 13 ^ 2 - 5 ^ 2BC2 = 169 - 25BC2 = 144BC = √144BC = 12

5

Now you can find the area rectangle AVCD.S = AB * BCS = 12 * 5S = 60.

6

A variant is also possible, where width will be known in part. For example, a rectangle ABCD is given, where AB = 1 / 4AD, OM is the median of the triangle AOD, OM = 3, AO = 5. Find the area rectangle AVCD.

7

Consider the triangle AOD. The angle OAD is equal to the angle ODA (because AC and BD are diagonals rectangle). Consequently, the triangle A0D is isosceles. And in an isosceles triangle, the median OM is simultaneously a bisectrix and a height. Hence, the triangle AOM is rectangular.

8

In the triangle AOM, where OM and AM are the legs, find what is equal to OM (hypotenuse). By the Pythagorean theorem, AM ^ 2 = AO2 - OM2AM = 25-9AM = 16AM = 4

9

Now calculate the area rectangle AVCD. AM = 1 / 2AD (because OM, being a median, divides AD in half). Therefore, AD = 8.AB = 1 / 4AD (by assumption). Hence AB = 2.S = AB * ADS = 2 * 8S = 16