Instruction

1

Method 1. Classic.

Area of isosceles triangle can be calculated by the classic formula: polypropelene the base of the triangle to its height.

S=1/2bh

b is the length of the base of the triangle;

h - the length of the height of treugolnika.

Area of isosceles triangle can be calculated by the classic formula: polypropelene the base of the triangle to its height.

S=1/2bh

b is the length of the base of the triangle;

h - the length of the height of treugolnika.

2

Method 2. Heron's Formula.

a - length one of the equal sides of the triangle;

b - length of base of triangle.

a - length one of the equal sides of the triangle;

b - length of base of triangle.

3

Method 3. It follows from the formula of method 1.

α is the angle between the side and the base;

γ - the angle between the equal sides.

α is the angle between the side and the base;

γ - the angle between the equal sides.

Note

There are signs of an isosceles triangle:

1) an isosceles triangle has 2 equal angles;

2) the height of the triangle coincides with its median;

3) the Height of a triangle coincides with its bisector;

4) angle Bisector of a triangle coincides with its median;

5) the isosceles triangle the median is equal to 2;

6) isosceles triangle 2 equal height;

7) an isosceles triangle 2 equal bisectors.

1) an isosceles triangle has 2 equal angles;

2) the height of the triangle coincides with its median;

3) the Height of a triangle coincides with its bisector;

4) angle Bisector of a triangle coincides with its median;

5) the isosceles triangle the median is equal to 2;

6) isosceles triangle 2 equal height;

7) an isosceles triangle 2 equal bisectors.

# Advice 2 : How to solve the problem about the area of a triangle

One of the figures covered math and geometry is a triangle. Triangle - a polygon that has 3 vertices (corners) and 3 sides; part of the plane bounded by the three points United in pairs in three segments. There are many tasks associated with finding the different values of this figure. One of them is

**square**. Depending on the initial data of the problem there are several formulas for determining the area**of a triangle**.Instruction

1

If you are aware of the length of the side and held her by the height h

**of the triangle**, use the formula S= ?h*a.2

In a right triangle

a) if we know the length of the legs a and b, the formula is S= a*b / 2;

b) if there are inscribed in a rectangle rectangle circle and a circumscribed circle, also known as their radii, then use the formula S=r2 + 2rR.

**, the area**can be found these ways:a) if we know the length of the legs a and b, the formula is S= a*b / 2;

b) if there are inscribed in a rectangle rectangle circle and a circumscribed circle, also known as their radii, then use the formula S=r2 + 2rR.

3

The task is to determine the area

**of a triangle**in which the lengths of all sides scalene**triangle**, is solved using properiter. First find out the perimeter**of the triangle**by the formula p=?(a+b+c). Next, use the formula S=vp*(p-a)*(p-b)*(p-c).4

The task can be specified only the length of one side

**of the triangle**, but it is equiangular, then you'll need the formula S=a2 v3 / 4.5

In terms of the problem known values of angles and lengths of the adjoining sides. For these applications there are of the formula:

a) S=?a*b*sin? - if you know the angle and length of the two sides adjacent thereto;

b) S=c2 / 2*(ctg ? + ctg ?) – here it is necessary to know the length of sides and magnitude of the two angles adjacent to this side;

in) S=c2 *sin ? * sin ? / 2 sin * (? + ?) – if you know the length of a side and the adjacent angles.

g) If you specify only angles and one side, find

a) S=?a*b*sin? - if you know the angle and length of the two sides adjacent thereto;

b) S=c2 / 2*(ctg ? + ctg ?) – here it is necessary to know the length of sides and magnitude of the two angles adjacent to this side;

in) S=c2 *sin ? * sin ? / 2 sin * (? + ?) – if you know the length of a side and the adjacent angles.

g) If you specify only angles and one side, find

**the area**according to the following formula S=A2 *sin ? * sin ? / 2 sin ?, where a is the side opposite the corner ?.6

For tasks where the lengths of all sides and radius of the circumscribed circle, select a formula S= a*b*C / 4R.

7

The task of finding square you are aware of all the angles, and the radius of the circumscribed circle. For this task, use the formula S=2R2 *sin ? * sin ? * sin ?.

8

In addition to the described and inscribed in a circle of triangles is related to one of the sides of the circle. Area in such problems is given by S=(p-b) * rb , where R – properiter

**of triangle**b – side**of the triangle**, rb is the radius of the circle on the b side.# Advice 3 : How to find the formula area of isosceles triangle

This is called an isosceles triangle with two sides equal to each other. All formulas are designed to determine the

**area**of an arbitrary**triangle**, just to isosceles. However, the formula**of the area**of an isosceles**triangle**have a simpler form and are sometimes more convenient in calculations.You will need

- trigonometric ratios

Instruction

1

Under the height of an isosceles

**triangle**usually mean the length of the perpendicular on the "unequal" side, and under the base is the length of this side. For finding the**area**of an isosceles**triangle,**label the length of its equal sides by a, the length of the base – through with, and the length of the altitude through C. In this case, the formula to calculate**square**(N) will look as follows:P = ½ * C *2

To find

*the formula for***area**of an isosceles**triangle**using the base and the length of the equal sides, use the Pythagorean theorem and the fact that the ground divided by the height in half. We get the following expression for the height:h = √(A2 - C2/4), substituting it into the above*formula*, we get:P = ½ * C * V(A2 - C2/4).3

For finding

**area**of isosceles**triangle**based on Heron formula, substitute into it the lengths of the sides of an isosceles**triangle**given the fact that two of them are equal. After a number of reductions will be:P = ½ * C * √[(a - C)*(a + C)].It is easy to see that both formulas are identical since the difference of the squares of the first formula is simply decomposed into the product of the sum and difference.4

To find

α - the angle between the equal sides and the base;

γ - the angle between the equal sides.Then, using basic trigonometric ratios, we get:P = ½ * a * s * cos(γ/2),P = ½ * C * a * sin(α/2),P = ½ * S2 / tg(γ/2),P = ½ * S2 * tg(α/2),N = A2 * sin(γ/2) * cos(γ/2),N = A2 * sin(α/2) * cos(α/2),

*the formula for***area**of an isosceles**triangle**using the values of catch, label:α - the angle between the equal sides and the base;

γ - the angle between the equal sides.Then, using basic trigonometric ratios, we get:P = ½ * a * s * cos(γ/2),P = ½ * C * a * sin(α/2),P = ½ * S2 / tg(γ/2),P = ½ * S2 * tg(α/2),N = A2 * sin(γ/2) * cos(γ/2),N = A2 * sin(α/2) * cos(α/2),

5

The above formulas cover all the main options calculate

P = ½ * C *

marking height on marking medians or bisectors.

**area**isosceles**triangle**. However, if we consider that the height of the isosceles**triangle**is at the same time its bisector and median, you can "take" a couple of formulas, replacingP = ½ * C *

marking height on marking medians or bisectors.