You will need

- geometric formula to find the area of an isosceles triangle ABC:
- S = 1/2 x b x h, where:
- - S - the area of the triangle ABC,
- - b - the length of its base AC,
- - h - the length of its height.

Instruction

1

Measure the length of the base AC of an isosceles

**triangle**ABC, usually the length of the base**of the triangle**is given in terms of the problem. Let the length of the base is equal to 6 cm Measure the height of an isosceles**triangle**. The height is a segment drawn from the vertex**of the triangle**perpendicular to its base. Let the conditions of the problem height h = 10 cm.2

Calculate

**area**of isosceles**triangle**formula. To do this, divide the length of the base AC in half: 6/2=3 cm, so 1/2b=3 cm. Multiply half the base length as**the triangle**on the length of the height h: 3 x 10=30 cm, you found**the area of the**isosceles**triangle**ABC the length of its base and height. If the length of the altitude is unknown, but given the length of a side**of a triangle**, then we first find the length of altitude of an isosceles**triangle**by the formula h = 1/2·√(4a2 – b2).3

Calculate the length of the height of an isosceles

**triangle**by length of sides and base. Let a be the length of any side of an isosceles**triangle**, in terms of the problem it is equal to 10 cm Substituting values of the lengths of the sides and base of an isosceles**triangle**in the formula, find the length of its height h=1/2√(4x100 – 36) =10 cm Calculate the height of an isosceles**triangle**, to continue the calculation, substituting the found values to the specified formula for finding the area**of a triangle**by its height and base.Note

In an isosceles triangle the altitude is also the median and the angle bisector of the triangle.

Two angles of an isosceles triangle are equal.

Two angles of an isosceles triangle are equal.

# Advice 2: 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.

# Advice 3: How to calculate side of isosceles triangle

Isosceles, or isosceles called triangle whose lengths of two sides of the same. If necessary, calculate the length of one of the sides of such figures it is possible to use knowledge of angles at its vertices, in combination with the length of one side or radius of the circumscribed circle. These parameters of the polygon are linked by a theorem of sines, cosines, and some other permanent ratio.

Instruction

1

To calculate the length of the sides of an isosceles triangle (b) according to the known conditions of the length of its base (a) and the value of the adjacent angle (α) use the theorem of cosines. It implies that you should divide the length of the known sides for twice the cosine is given in terms of angle: b = a/(2*cos(α)).

2

The same theorem and apply for the reverse operation - calculate the length of the base (a) at a known length of sides (b) and value of angle (α) between these two parties. In this case, the theorem allows to obtain the equality, the right part of which contains twice the product of the lengths of the known sides into the cosine of the angle: a = 2*b*cos(α).

3

If in addition the lengths of the sides (b) in terms of the magnitude of the angle between them (β), to calculate the length of the base (a) use the theorem of sines. It implies a formula, according to which should be twice the length of the side multiplied by the sine of half of the known angle: a = 2*b*sin(β /2).

4

The theorem of sines can be used to find the length of the lateral side (b) of an isosceles triangle if the known base length (a) and opposite him the value of angle (β). In this case, double the sine of half of the known angle divided by the resulting value of the length of the base: b = a/(2*sin(β/2)).

5

If the isosceles triangle circumscribed circle, the radius of which (R) is known, to calculate the lengths of the sides need to know the measure of the angle at one vertex of the shape. If the conditions given information about the angle between the sides (β), calculate the length of the base (a) of the polygon by doubling the product of the radius by the sine of this angle: a = 2*R*sin(β). If given the value of the base angle (α) to find the length of the lateral side (b) just replace the angle in the formula: b = 2*R*sin(α).