Find the area of a triangle if two sides are known. How to calculate the area of a triangle

21.10.2019

As you may remember from the school curriculum in geometry, a triangle is a figure formed from three segments connected by three points that do not lie on one straight line. The triangle forms three angles, hence the name of the figure. The definition may be different. A triangle can also be called a polygon with three corners, the answer will be just as true. Triangles are divided according to the number of equal sides and the size of the angles in the figures. So distinguish such triangles as isosceles, equilateral and scalene, as well as rectangular, acute-angled and obtuse-angled, respectively.

There are many formulas for calculating the area of a triangle. Choose how to find the area of a triangle, i.e. which formula to use, only you. But it is worth noting only some of the notation that is used in many formulas for calculating the area of a triangle. So remember:

S is the area of the triangle,

a, b, c are the sides of the triangle,

h is the height of the triangle,

R is the radius of the circumscribed circle,

p is the semi-perimeter.

Here are the basic notations that may come in handy if you have completely forgotten the course of geometry. The most understandable and not complicated options for calculating the unknown and mysterious area of \u200b\u200bthe triangle will be given below. It is not difficult and will come in handy both for your household needs and for helping your children. Let's remember how to calculate the area of a triangle as easy as shelling pears:

In our case, the area of the triangle is: S = ½ * 2.2 cm. * 2.5 cm. = 2.75 sq. cm. Remember that area is measured in square centimeters (sqcm).

Right triangle and its area.

A right triangle is a triangle with one angle equal to 90 degrees (therefore called a right triangle). A right angle is formed by two perpendicular lines (in the case of a triangle, two perpendicular segments). In a right triangle, there can be only one right angle, because the sum of all the angles of any one triangle is 180 degrees. It turns out that 2 other angles should divide the remaining 90 degrees among themselves, for example, 70 and 20, 45 and 45, etc. So, you remembered the main thing, it remains to find out how to find the area right triangle. Imagine that we have such a right triangle in front of us, and we need to find its area S.

1. The easiest way to determine the area of a right triangle is calculated using the following formula:

In our case, the area of a right triangle is: S = 2.5 cm * 3 cm / 2 = 3.75 sq. cm.

In principle, it is no longer necessary to verify the area of a triangle in other ways, since in everyday life it will come in handy and only this one will help. But there are also options for measuring the area of a triangle through acute angles.

2. For other calculation methods, you must have a table of cosines, sines and tangents. Judge for yourself, here are some options for calculating the areas of a right-angled triangle that you can still use:

We decided to use the first formula and with small blots (we drew in a notebook and used an old ruler and protractor), but we got the right calculation:

S \u003d (2.5 * 2.5) / (2 * 0.9) \u003d (3 * 3) / (2 * 1.2). We got such results 3.6=3.7, but taking into account the cell shift, we can forgive this nuance.

Isosceles triangle and its area.

If you are faced with the task of calculating the formula of an isosceles triangle, then the easiest way is to use the main one and, as is considered the classic formula for the area of a triangle.

But first, before we find the area of an isosceles triangle, we will find out what kind of figure it is. An isosceles triangle is a triangle whose two sides are the same length. These two sides are called the sides, the third side is called the base. Do not confuse an isosceles triangle with an equilateral one, i.e. an equilateral triangle with all three sides equal. In such a triangle, there are no special tendencies to the angles, or rather to their size. However, the angles at the base in an isosceles triangle are equal, but different from the angle between equal sides. So, you already know the first and main formula, it remains to find out what other formulas for determining the area of an isosceles triangle are known:

From the opposite vertex) and divide the resulting product by two. In form it looks like this:

S = ½ * a * h,

where:
S is the area of the triangle,
a is the length of its side,
h is the height lowered to this side.

Side length and height must be presented in the same units. In this case, the area of \u200b\u200bthe triangle will turn out in the corresponding "" units.

Example.
On one of the sides of a scalene triangle 20 cm long, a perpendicular from the opposite vertex 10 cm long is lowered.
The area of the triangle is required.
Solution.
S = ½ * 20 * 10 = 100 (cm²).

If you know the lengths of any two sides of a scalene triangle and the angle between them, then use the formula:

S = ½ * a * b * sinγ,

where: a, b are the lengths of two arbitrary sides, and γ is the angle between them.

In practice, for example, when measuring land, the use of the above formulas is sometimes difficult, since it requires additional constructions and measurement of angles.

If you know the lengths of all three sides of a scalene triangle, then use Heron's formula:

S = √(p(p-a)(p-b)(p-c)),

a, b, c are the lengths of the sides of the triangle,
р – semi-perimeter: p = (a+b+c)/2.

If, in addition to the lengths of all sides, the radius of the circle inscribed in the triangle is known, then use the following compact formula:

where: r is the radius of the inscribed circle (p is the semi-perimeter).

To calculate the area of a scalene triangle of the circumscribed circle and the length of its sides, use the formula:

where: R is the radius of the circumscribed circle.

If the length of one of the sides of the triangle and three angles is known (in principle, two are enough - the value of the third is calculated from the equality of the sum of the three angles of the triangle - 180º), then use the formula:

S = (a² * sinβ * sinγ)/2sinα,

where α is the value of the angle opposite to side a;
β, γ are the values of the remaining two angles of the triangle.

The need to find various elements, including area triangle, appeared many centuries before our era among astronomers Ancient Greece. Square triangle can be calculated different ways using different formulas. The calculation method depends on which elements triangle known.

Instruction

If from the condition we know the values of the two sides b, c and the angle formed by them?, then the area triangle ABC is found by the formula:
S = (bcsin?)/2.

If from the condition we know the values of the two sides a, b and the angle not formed by them?, then the area triangle ABC is found as follows:
Finding the angle?, sin? = bsin? / a, further on the table we determine the angle itself.
Finding an angle? = 180°-?-?.
Find the area itself S = (absin?)/2.

If from the condition we know the values of only three sides triangle a, b and c, then the area triangle ABC is found by the formula:
S = v(p(p-a)(p-b)(p-c)) , where p is the semiperimeter p = (a+b+c)/2

If from the condition of the problem we know the height triangle h and the side to which this height is lowered, then the area triangle ABC by formula:
S = ah(a)/2 = bh(b)/2 = ch(c)/2.

If we know the values of the sides triangle a, b, c and the radius of the circumscribed near the given triangle R, then the area of this triangle ABC is determined by the formula:
S = abc/4R.
If three sides a, b, c and the radius of the inscribed in are known, then the area triangle ABC is found by the formula:
S = pr, where p is the semiperimeter, p = (a+b+c)/2.

If ABC is equilateral, then the area is found by the formula:
S = (a^2v3)/4.
If triangle ABC is isosceles, then the area is determined by the formula:
S = (cv(4a^2-c^2))/4, where c is triangle.
If triangle ABC is a right triangle, then the area is determined by the formula:
S = ab/2, where a and b are legs triangle.
If triangle ABC is a right isosceles triangle, then the area is determined by the formula:
S = c^2/4 = a^2/2, where c is the hypotenuse triangle, a=b - leg.

Tip 3: How to find the area of a triangle if you know the angle

Knowing only one parameter (the value of the angle) is not enough to find the area tre square . If there are any additional dimensions, then to determine the area, you can choose one of the formulas in which the angle value is also used as one of the known variables. A few of the most commonly used formulas are listed below.

Instruction

If, in addition to the angle (γ) formed by the two sides tre square , the lengths of these sides (A and B) are also known, then square(S) figures can be defined as half the product of the side lengths and the sine of this known angle: S=½×A×B×sin(γ).

To determine the area of a triangle, you can use different formulas. Of all the methods, the easiest and most often used is the multiplication of the height by the length of the base, and then dividing the result by two. but this method far from the only one. Below you can read how to find the area of a triangle using different formulas.

Separately, we will consider methods for calculating the area of specific types of triangle - rectangular, isosceles and equilateral. We accompany each formula with a short explanation that will help you understand its essence.

Universal ways to find the area of a triangle

The formulas below use special notation. We will decipher each of them:

a, b, c are the lengths of the three sides of the figure we are considering;
r is the radius of a circle that can be inscribed in our triangle;
R is the radius of the circle that can be described around it;
α - the value of the angle formed by the sides b and c;
β is the angle between a and c;
γ - the value of the angle formed by the sides a and b;
h is the height of our triangle, lowered from angle α to side a;
p is half the sum of sides a, b and c.

It is logically clear why you can find the area of a triangle in this way. The triangle is easily completed to a parallelogram, in which one side of the triangle will act as a diagonal. The area of a parallelogram is found by multiplying the length of one of its sides by the value of the height drawn to it. The diagonal divides this conditional parallelogram into 2 identical triangles. Therefore, it is quite obvious that the area of our original triangle should be equal to half the area of this auxiliary parallelogram.

S=½ a b sin γ

According to this formula, the area of a triangle is found by multiplying the lengths of its two sides, that is, a and b, by the sine of the angle they form. This formula is logically derived from the previous one. If we lower the height from angle β to side b, then, according to the properties of a right triangle, when multiplying the length of side a by the sine of angle γ, we get the height of the triangle, that is, h.

The area of the figure under consideration is found by multiplying half the radius of the circle, which can be inscribed in it, by its perimeter. In other words, we find the product of the semiperimeter and the radius of the mentioned circle.

S= a b c/4R

According to this formula, the value we need can be found by dividing the product of the sides of the figure by 4 radii of the circle circumscribed around it.

These formulas are universal, as they make it possible to determine the area of any triangle (scalene, isosceles, equilateral, right-angled). This can be done with the help of more complex calculations, which we will not dwell on in detail.

Areas of triangles with specific properties

How to find the area of a right triangle? A feature of this figure is that its two sides are simultaneously its heights. If a and b are legs, and c becomes the hypotenuse, then the area is found as follows:

How to find the area of an isosceles triangle? It has two sides with length a and one side with length b. Therefore, its area can be determined by dividing by 2 the product of the square of the side a by the sine of the angle γ.

How to find the area of an equilateral triangle? In it, the length of all sides is a, and the value of all angles is α. Its height is half the product of the length of side a times the square root of 3. To find the area of a regular triangle, you need the square of side a multiplied by the square root of 3 and divided by 4.

The triangle is one of the most common geometric shapes, which we are familiar with already in primary school. The question of how to find the area of a triangle is faced by every student in geometry lessons. So, what are the features of finding the area of \u200b\u200ba given figure can be distinguished? In this article, we will consider the basic formulas necessary to complete such a task, and also analyze the types of triangles.

Types of triangles

You can absolutely find the area of a triangle different ways, because in geometry there is more than one type of figure containing three angles. These types include:

obtuse.
Equilateral (correct).
Right triangle.
Isosceles.

Let's take a closer look at each of existing types triangles.

Such a geometric figure is considered the most common in solving geometric problems. When it becomes necessary to draw an arbitrary triangle, this option comes to the rescue.

In an acute triangle, as the name implies, all angles are acute and add up to 180°.

Such a triangle is also very common, but is somewhat less common than an acute-angled one. For example, when solving triangles (that is, you know several of its sides and angles and need to find the remaining elements), sometimes you need to determine whether the angle is obtuse or not. Cosine is a negative number.

In the value of one of the angles exceeds 90°, so the remaining two angles can take small values (for example, 15° or even 3°).

To find the area of a triangle of this type, you need to know some of the nuances, which we will talk about next.

Regular and isosceles triangles

A regular polygon is a figure that includes n angles, in which all sides and angles are equal. This is the right triangle. Since the sum of all the angles of a triangle is 180°, each of the three angles is 60°.

The right triangle, due to its property, is also called an equilateral figure.

It is also worth noting that only one circle can be inscribed in a regular triangle and only one circle can be circumscribed around it, and their centers are located at one point.

In addition to the equilateral type, one can also distinguish an isosceles triangle, which differs slightly from it. In such a triangle, two sides and two angles are equal to each other, and the third side (to which equal angles adjoin) is the base.

The figure shows an isosceles triangle DEF, the angles D and F of which are equal, and DF is the base.

Right triangle

A right triangle is so named because one of its angles is a right angle, i.e. equal to 90°. The other two angles add up to 90°.

The largest side of such a triangle, lying opposite an angle of 90 °, is the hypotenuse, while the other two of its sides are the legs. For this type of triangles, the Pythagorean theorem is applicable:

The sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

The figure shows a right triangle BAC with hypotenuse AC and legs AB and BC.

To find the area of a triangle with a right angle, you need to know numerical values his legs.

Let's move on to the formulas for finding the area of \u200b\u200ba given figure.

Basic formulas for finding the area

In geometry, two formulas can be distinguished that are suitable for finding the area of most types of triangles, namely for acute-angled, obtuse-angled, regular and isosceles triangles. Let's analyze each of them.

By side and height

This formula is universal for finding the area of the figure we are considering. To do this, it is enough to know the length of the side and the length of the height drawn to it. The formula itself (half the product of the base and the height) is as follows:

where A is the side of the given triangle and H is the height of the triangle.

For example, to find the area of an acute-angled triangle ACB, you need to multiply its side AB by the height CD and divide the resulting value by two.

However, it is not always easy to find the area of a triangle in this way. For example, to use this formula for an obtuse-angled triangle, you need to continue one of its sides and only then draw a height to it.

In practice, this formula is used more often than others.

Two sides and a corner

This formula, like the previous one, is suitable for most triangles and in its meaning is a consequence of the formula for finding the area by the side and height of a triangle. That is, the formula under consideration can be easily deduced from the previous one. Its wording looks like this:

S = ½*sinO*A*B,

where A and B are the sides of the triangle and O is the angle between sides A and B.

Recall that the sine of an angle can be viewed in a special table named after the outstanding Soviet mathematician V. M. Bradis.

And now let's move on to other formulas that are suitable only for exceptional types of triangles.

Area of a right triangle

In addition to the universal formula, which includes the need to draw a height in a triangle, the area of \u200b\u200ba triangle containing a right angle can be found from its legs.

So, the area of a triangle containing a right angle is half the product of its legs, or:

where a and b are the legs of a right triangle.

right triangle

This type of geometric figures is distinguished by the fact that its area can be found with the specified value of only one of its sides (since all sides of a regular triangle are equal). So, having met with the task of “find the area of a triangle when the sides are equal”, you need to use the following formula:

S = A 2 *√3 / 4,

where A is the side of an equilateral triangle.

Heron's formula

Last option To find the area of a triangle, this is Heron's formula. In order to use it, you need to know the lengths of the three sides of the figure. Heron's formula looks like this:

S = √p (p - a) (p - b) (p - c),

where a, b and c are the sides of the given triangle.

Sometimes the task is given: "the area of \u200b\u200ba regular triangle is to find the length of its side." In this case, you need to use the formula already known to us for finding the area of a regular triangle and derive from it the value of the side (or its square):

A 2 \u003d 4S / √3.

Exam problems

There are many formulas in the tasks of the GIA in mathematics. In addition, quite often it is necessary to find the area of a triangle on checkered paper.

In this case, it is most convenient to draw the height to one of the sides of the figure, determine its length by cells and use the universal formula for finding the area:

So, after studying the formulas presented in the article, you will not have problems finding the area of a triangle of any kind.