# Advice 1: How to find the base of the triangle

Often in problems in plane geometry and trigonometry required to find the base of the triangle. For this operation there is even a few methods.
You will need
• Calculator
Instruction
1
The strict definition of the base of the triangle" in geometry does not exist. Typically, this term denotes the side of the triangle, which opposite vertices held perpendicular (lowered height). Also this term called "the unequal side of an equilateral triangle. Therefore, select from the variety of examples, known in mathematics under the term "solution of triangles", the options in which meet the height and equilateral triangles.
If you know the height and the area of a triangle, to find the base of the triangle (the length of a side, which lowered the height), we use the formula for finding the area of a triangle, claiming that the area of any triangle can calculate, by multiplying half the base length to the length of the height:
S=1/2*c*h, where:
S - the area of the triangle

- the length of its base,

h - the length of the height of the triangle.
From this formula we find:
s=2*S/h.
For example, if the area of the triangle is 20 sq. cm, and the length of the height - 10 cm, the base of the triangle will be:
C=2*20/10=4 (cm).
2
If you know the sidewall and the perimeter an equilateral triangle, the length of the base can be calculate by the following formula:
C=R-2*a, where:
P - perimeter of the triangle

a - the length of the sides of the triangle

- the length of its base.
3
If you know the side opposite and the magnitude of the base angle of the equilateral triangle, the length of the base can be calculate by the following formula:
C=a*√(2*(1-cosC)), where:
C - the value of the opposite base angle of the equilateral triangle

a - the length of the sides of the triangle.

- the length of its base.
(Formula is a direct consequence of the theorem of cosines)
There is a more compact recording of the formula:
C=2*a*sin(B/2)
4
If you know the sidewall and the value of the adjacent base angle of an equilateral triangle, the length of the base could be calculated in the following easy-to-remember formula:
C=2*a*cosA
A - the value of the adjacent base angle of an equilateral triangle,

a - the length of the sides of the triangle.

- the length of its base.
This formula is a consequence of the theorem about projections.
5
If you know the radius of the circumscribed circle and the value of the opposite base angle of the equilateral triangle, the length of the base can be calculate by the following formula:
C=2*R*sinC, where:
C - the value of the opposite base angle of the equilateral triangle

R - radius of a triangle circumscribed around a circle

- the length of its base.
This formula is a direct consequence of the theorem of sines.
Note
To start obstreperous from the particulars and see how to find the base of the triangle that is neither equilateral, nor isosceles, nor rectangular. Since the basis in this figure can serve as any party to begin, select some face and "will consider" its base. Accordingly, turn the triangle so that he was standing and look for the length.
Useful advice
How to find isosceles triangle? Looking in this triangle. If an equilateral triangle given a side and angle, which is opposite the base, you can hold this angle the height of the triangle. As a result, the property of an equilateral triangle, you get two equal rectangle.

# Advice 2 : How to find the area of an equilateral triangle

Is called an equilateral triangle having three equal sides and three equal angles. This triangle is also called the correct. The altitude drawn from the vertex to the base is also the angle bisector and median, which implies that this line divides the angle into two equal angles, and the base, which falls into two equal segments. These properties of the triangle will calculate its areaequal to half the height works on any of its sides.
You will need
• to know what the height and its properties
• - know what a right triangle
• to know what the hypotenuse is and sides,
• - be able to solve equations with one variable with parentheses
Instruction
1
If in a right triangle is known to at least one side and its height, to determine the area of a shape, multiply the height by the length of the side and divide the resulting number by two.
2
To compute the area of a triangle with known height and known side first, find the height. For this we consider one of the equal right-angled triangles formed by the altitude.
3
Side opposite the right angle is the hypotenuse, and the other two - legs. So, the height of the equilateral triangle will be one of the smaller sides of the rectangular triangle. The second leg will be half the side of the large triangle, as the height of the right rectangle divides it in half, as a median.
4
According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Therefore, in order to know the height from the square of the hypotenuse (that is, from square one of the sides of an equilateral triangle) subtract the square of the side formed by the half side of the equilateral triangle, and after that from the result of this calculation, extract the square root.
5
Now when the height is known, find the area of a shape by multiplying the height by the length of a side and dividing the resulting value by two.
6
If you only know the height, then again consider one of the right triangles formed when carrying out height, which bisects the angle and side of a regular polygon. Based on the Pythagorean theorem, write down the equation a2 = c2-(1/2*s)2 where a2 is the height, c2 is the side of equilateral triangle. In this equation find the value of the variable a.
7
Knowing the height, calculate the area of the right triangle. To do this, multiply the height of the side of the triangle and divide after multiplying the result in half.
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