How to find the area of ​​a right triangle without height. How to find the area of ​​a triangle (formulas)

In geometry classes in high school we were all taught about triangles. However, within the framework of the school curriculum, we receive only the most necessary knowledge and learn the most common and standard methods of calculation. Are there unusual ways to find this value?

As an introduction, let's recall which triangle is considered a right triangle, and also denote the concept of area.

A right triangle is a closed geometric figure, one of the angles of which is equal to 90 0. The integral concepts in the definition are the legs and the hypotenuse. The legs are two sides that form a right angle at the connection point. The hypotenuse is the side opposite the right angle. A right triangle can be isosceles (two of its sides will be the same size), but never equilateral (all sides are the same length). The definitions of height, median, vectors and other mathematical terms will not be analyzed in detail. They are easy to find in reference books.

Area of ​​a right triangle. Unlike rectangles, the rule about

the product of the parties in the definition is not valid. Speaking in a dry language of terms, then the area of ​​a triangle is understood as the property of this figure to occupy a part of the plane, expressed by a number. Quite difficult to understand, you see. We will not try to delve deeply into the definition, our goal is not this. Let's move on to the main thing - how to find the area of ​​a right triangle? We will not perform the calculations themselves, we will indicate only the formulas. To do this, let's define the notation: A, B, C - sides of the triangle, legs - AB, BC. Angle ACB is straight. S is the area of ​​the triangle, h n n is the height of the triangle, where nn is the side on which it is lowered.

Method 1. How to find the area of ​​​​a right triangle if the size of its legs is known

Method 2. Find the area of ​​an isosceles right triangle

Method 3. Calculating the area through a rectangle

We complete the right-angled triangle to a square (if the triangle

isosceles) or rectangle. We get a simple quadrangle made up of 2 identical right triangles. In this case, the value of the area of ​​one of them will be equal to half the area of ​​the resulting figure. S of a rectangle is calculated by the product of the sides. We denote this value by M. The desired value of the area will be equal to half of M.

Method 4. "Pythagorean pants." The famous Pythagorean theorem

We all remember her formulation: "the sum of the squares of the legs ...". But not everyone can

say, and here some "pants". The fact is that initially Pythagoras studied the relationship built on the sides of a right triangle. Having identified patterns in the ratio of the sides of the squares, he was able to derive the formula known to all of us. It can be used when the value of one of the sides is unknown.

Method 5. How to find the area of ​​a right triangle using Heron's formula

It's also a pretty simple calculation. The formula involves the expression of the area of ​​a triangle in terms of the numerical values ​​of its sides. For calculations, you need to know the magnitude of all sides of the triangle.

S = (p-AC)*(p-BC), where p = (AB+BC+AC)*0.5

In addition to the above, there are many other ways to find the size of such a mysterious figure as a triangle. Among them: calculation by the method of an inscribed or circumscribed circle, calculation using the coordinates of the vertices, the use of vectors, absolute values, sines, tangents.

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 learn how to find the area of ​​a 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:

Instruction

Task 1.
Find the lengths of all sides of the triangle if it is known that one leg is 1 cm longer than the other, and the triangle is 28 cm.

Decision.
Write down the basic area formula S = (a*b)/2 = 28. It is known that b = a + 1, substitute this value into the formula: 28 = (a*(a+1))/2.
Open the brackets, get a quadratic equation with one unknown a^2 + a - 56 = 0.
Find this by calculating the discriminant D = 1 + 224 = 225. The equation has two solutions: a_1 = (-1 + √225)/2 = (-1 + 15)/2 = 7 and a_2 = (-1 - √ 225)/2 = (-1 - 15)/2 = -8.
The second one doesn't make sense, since the length of a segment cannot be negative, so a = 7 (cm).
Find the length of the second leg b = a + 1 = 8 (cm).
The length of the third side remains. According to the Pythagorean theorem for a right triangle c^2 = a^2 + b^2 = 49 + 64, hence c = √(49 + 64) = √113 ≈ 10.6 (cm).

Task 2.
Find the lengths of all sides of a right triangle if its area is known to be 14 cm and angle ACB is 30°.

Decision.
Write down the basic formula S = (a*b)/2 = 14.
Now express the lengths of the legs through the product of the hypotenuse and trigonometric functions according to the property of a right triangle:
a = c*cos(ACB) = c*cos(30°) = c*(√3/2) ≈ 0.87*c.
b = c*sin(ACB) = c*sin(30°) = c*(1/2) = 0.5*c.

Substitute the obtained values ​​into the area formula:
14 = (0.87*0.5*c^2)/2, from where:
28 ≈ 0.435*s^2 → c = √64.4 ≈ 8 (cm).
You have found the length of the hypotenuse, now find the lengths of the other two sides:
a = 0.87*c = 0.87*8 ≈ 7 (cm), b = 0.5*c = 0.5*8 = 4 (cm).

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First, let's agree on notation. The leg is called the side of a right triangle, which is adjacent to the right angle (that is, it makes an angle of 90 degrees with the other side). We will agree to denote the lengths of the legs a and b. The values ​​of the acute angles of a right-angled triangle opposite the legs will be called A and B, respectively. The hypotenuse is the side of a right triangle that is opposite the right angle (that is, it is opposite the right angle, forming acute angles with other sides of the triangle). Let us denote the length of the hypotenuse by s. Denote the required area by S.

Instruction

Apply the formula S = (a ^ 2) / (2 * tg (A)) if you are given only one of the legs (a), but the angle opposite to this leg (A) is also known. The sign "^2" denotes squaring.

Use the formula S=(a^2)*tg(B)/2 d if you are given only one of the legs (a), but you also know the angle adjacent to this leg (B).

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Sources:

• "Manual in Mathematics for Applicants to Universities", ed. G.N. Yakovleva, 1982.

The relationship between the sides and angles of a right triangle is covered in a branch of mathematics called trigonometry. To find the sides of a right triangle, it is enough to know the Pythagorean theorem, the definitions of trigonometric functions, and have some means for finding the values ​​of trigonometric functions, for example, a calculator or Bradis tables. Consider below the main cases of problems of finding the sides of a right triangle.

You will need

• Calculator, Bradis tables.

Instruction

If one of the acute angles is given, for example, A, and one of the legs, for example, a, then the hypotenuse and the other leg are calculated from the relations: b=a*tg(A), c=a*sin(A).

Helpful advice

In the event that you do not know the value of the sine or cosine of one of the angles necessary for calculating, you can use the Bradis tables, they provide the values ​​​​of trigonometric functions for a large number of angles. In addition, most modern calculators are able to calculate the sines and cosines of angles.

Sources:

• how to calculate the side of a right triangle in 2019

Tip 4: How to find the base of a right triangle

In such a figure as a right triangle, there is necessarily a clear ratio of sides relative to each other. Knowing two of them, you can always find the third. How this can be done, you will learn from the instructions below.

You will need

• - calculator.

Instruction

Square both legs, and add them together a2 + b2. The result is the hypotenuse ( basis) in the square c2. Next, you just need to extract the root from the latter, and the hypotenuse is found. This method is simple and easy to use on . The main thing in the process of finding parties triangle thus - do not forget to extract the root from the preliminary result in order to avoid the most common mistake. The formula was derived thanks to the most famous Pythagorean theorem in the world, which in all sources looks like: a2+b2 = c2.

Divide one of the legs a by the sine of its opposite angle sin α. In the event that the sides and sines are known in the condition, this option for finding the hypotenuse will be acceptable. The formula in this case will have a very simple form: c=a/sin α. Be careful with all calculations.

Multiply side a by two. The hypotenuse has been calculated. This is perhaps the most elementary way of finding us a hand. But, unfortunately, this method is used only in one case - if the side that lies opposite the angle in degree measure is equal to the number thirty. If there is one, you can be sure that it will always be exactly half of the hypotenuse. Accordingly, you just have to double it and you're done.

Divide leg a by the cosine of the angle cos α adjacent to it. This method is suitable only if you know one of the legs and the cosine of the angle adjacent to it. This method resembles the one already presented to you earlier, in which the leg is also used, but instead of the cosine, the sine of the opposite angle. Only in this case it will have a slightly different appearance: с=a/ cos α. That's all.

Advice 5: How to find the angle if the sides of a right triangle are known

Tre square, one of the angles of which is right (equal to 90 °), is called a right angle. Its longest side always lies opposite the right angle and is called the hypotenuse, and the other two sides are called skates. If the lengths of these three sides are known, then find the values ​​of all angles of the triangle square and not difficult, since in fact you only need to calculate one of the angles. This can be done in several ways.

Instruction

Use to calculate the quantities (α, β, γ) of the definition of trigonometric functions in terms of a rectangular tri. Such, for example, for the sine of an acute angle as the ratio of the length of the opposite leg to the length of the hypotenuse. So, if the lengths of the legs (A and B) and the hypotenuse (C), then you can find, for example, the sine of the angle α, which lies opposite the leg A, by dividing the length sides And for the length sides C (hypotenuse): sin(α)=A/C. Having learned the value of the sine of this angle, you can find its value in degrees using the inverse function of the sine - the arcsine. That is, α=arcsin(sin(α))=arcsin(A/C). In the same way, you can find the value of an acute angle in a triangle. square e, but it's not necessary. Since the sum of all angles is square a is 180°, and in tre square If one of the angles is equal to 90°, then the value of the third angle can be calculated as the difference between 90° and the value of the found angle: β=180°-90°-α=90°-α.

Instead of defining the sine, you can use the definition of the cosine of an acute angle, which is formulated as the ratio of the length of the leg adjacent to the desired angle to the length of the hypotenuse: cos(α)=B/C. And here, use the inverse trigonometric function (arccosine) to find the value of the angle in degrees: α=arccos(cos(α))=arccos(B/C). After that, as in the previous step, it remains to find the value of the missing angle: β=90°-α.

You can use a similar tangent - it is expressed by the ratio of the length of the leg opposite the desired angle to the length of the adjacent leg: tg(α)=A/B. The angle value in degrees is again determined through the inverse trigonometric function - : α=arctg(tg(α))=arctg(A/B). The missing angle formula will remain unchanged: β=90°-α.

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Tip 6: How to find the length of a side of a right triangle

A triangle is considered a right triangle if one of its corners is a right angle. Side triangle located opposite the right angle is called the hypotenuse, and the other two sides- catheters. To find the lengths of the sides of a rectangle triangle, can be used in several ways.

Instruction

1. The values ​​of two legs are known

In this case, the area of ​​a right triangle is calculated by the formula:
S=0.5ab

2. One leg and hypotenuse are known

Under such conditions, it is most logical to use the Pythagorean theorem and the above formula:
S = 0.5∙sqrt(c^2-a^2)∙a,
where sqrt is the square root, c^2-a^2 is the radical expression denoting the difference between the square of the hypotenuse and the leg.

3. Given the values ​​of all sides of the triangle

For such problems, you can use the Heron formula:
S = (p-a)(p-b),
where p is the semi-perimeter, which is found by the following expression: p = 0.5∙ (a+b+c)

4. One leg and angle are known

Here it is worth turning to trigonometric functions. For example, tg(1) = 1/сtg(1) = b/a. That is, thanks to this ratio, it is possible to determine the value of the unknown leg. Then the problem is reduced to the first point.

5. Known hypotenuse and angle

In this case, the trigonometric functions of sine and cosine are also used: cos(2)=1/sin(2) = b/c. Then the solution of the problem is reduced to the second paragraph of the article.

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Tip 11: What are the names of the sides of a right triangle

definition similar to the first. A right triangle is a triangle that has two sides that are perpendicular.

Hypotenuse and legs

In acute and obtuse triangles, the segments connecting the vertices of the angles are simply called sides. The side has other names. Those that are adjacent to the right angle are called legs. The side opposite the right angle is called the hypotenuse. Translated from Greek, the word "hypotenuse" means "stretched", and "leg" - "perpendicular".

Relationships between the hypotenuse and legs

The sides of a right-angled triangle are interconnected by certain ratios, which greatly facilitate calculations. For example, knowing the dimensions of the legs, you can calculate the length of the hypotenuse. This ratio, named after the one who discovered it, was called the Pythagorean theorem and it looks like this:

c2=a2+b2, where c is the hypotenuse, a and b are legs. That is, the hypotenuse will be equal to the square root of the sum of the squares of the legs. To find any of the legs, it is enough to subtract the square of the other leg from the square of the hypotenuse and extract the square root from the resulting difference.

Adjacent and opposite leg

Draw a right triangle ACB. The letter C is used to denote the vertex of a right angle, A and B are the vertices of acute angles. The sides opposite each corner are conveniently called a, b, and c, after the names of the angles opposite them. Consider angle A. Leg a for it will be opposite, leg b - adjacent. The ratio of the opposite leg to the hypotenuse is called . This trigonometric function can be calculated using the formula: sinA=a/c. The ratio of the adjacent leg to the hypotenuse is called the cosine. It is calculated by the formula: cosA=b/c.

Thus, knowing the angle and one of the sides, it is possible to calculate the other side using these formulas. Both legs are also connected by trigonometric relations. The ratio of the opposite to the adjacent is called the tangent, and the ratio of the adjacent to the opposite is called the cotangent. These ratios can be expressed by the formulas tgA=a/b or ctgA=b/a.

A right triangle is found in reality on almost every corner. Knowledge of the properties of this figure, as well as the ability to calculate its area, will undoubtedly be useful to you not only for solving problems in geometry, but also in life situations.

triangle geometry

In elementary geometry, a right triangle is a figure that consists of three connected segments that form three angles (two acute and one straight). A right triangle is an original figure, characterized by a number of important properties that form the foundation of trigonometry. Unlike an ordinary triangle, the sides of a rectangular figure have their own names:

• The hypotenuse is the longest side of a triangle that lies opposite the right angle.
• Legs - segments that form a right angle. Depending on the angle under consideration, the leg can be adjacent to it (forming this angle with the hypotenuse) or opposite (lying opposite the angle). There are no legs for non-rectangular triangles.

It is the ratio of the legs and hypotenuse that forms the basis of trigonometry: sines, tangents and secants are defined as the ratio of the sides of a right triangle.

Right triangle in reality

This figure is widely used in reality. Triangles are used in design and technology, so the calculation of the area of ​​\u200b\u200bthe figure has to be done by engineers, architects and designers. The bases of tetrahedra or prisms have the shape of a triangle - three-dimensional figures that are easy to meet in everyday life. In addition, a square is the simplest representation of a "flat" right triangle in reality. A square is a locksmith, drawing, construction and carpentry tool that is used to build corners by both schoolchildren and engineers.

Area of ​​a triangle

The area of ​​a geometric figure is a quantitative estimate of how much of the plane is bounded by the sides of a triangle. The area of ​​an ordinary triangle can be found in five ways, using Heron's formula or operating in calculations with such variables as the base, side, angle and radius of the inscribed or circumscribed circle. The simplest area formula is expressed as:

where a is the side of the triangle, h is its height.

The formula for calculating the area of ​​a right triangle is even simpler:

where a and b are legs.

Working with our online calculator, you can calculate the area of ​​a triangle using three pairs of parameters:

• two legs;
• leg and adjacent angle;
• leg and opposite angle.

In tasks or everyday situations, you will be given different combinations of variables, so this form of calculator allows you to calculate the area of ​​a triangle in several ways. Let's look at a couple of examples.

Real life examples

Ceramic tile

Let's say you want to line the walls of the kitchen with ceramic tiles, which have the shape of a right triangle. In order to determine the consumption of tiles, you must find out the area of ​​\u200b\u200bone element of the cladding and the total area of ​​\u200b\u200bthe surface to be treated. Suppose you need to process 7 square meters. The length of the legs of one element is 19 cm each, then the area of ​​\u200b\u200bthe tile will be equal to:

This means that the area of ​​one element is 24.5 square centimeters or 0.01805 square meters. Knowing these parameters, you can calculate that to finish 7 square meters of a wall you will need 7 / 0.01805 = 387 facing tiles.

school task

Suppose that in a school geometry problem it is required to find the area of ​​a right triangle, knowing only that the side of one leg is 5 cm, and the value of the opposite angle is 30 degrees. Our online calculator is accompanied by an illustration showing the sides and angles of a right triangle. If side a = 5 cm, then its opposite angle is the angle alpha, equal to 30 degrees. Enter this data into the calculator form and get the result:

Thus, the calculator not only calculates the area of ​​a given triangle, but also determines the length of the adjacent leg and hypotenuse, as well as the value of the second angle.

Conclusion

Rectangular triangles are found in our lives literally on every corner. Determining the area of ​​such figures will be useful to you not only when solving school assignments in geometry, but also in everyday and professional activities.

A triangle is a flat geometric figure with one angle equal to 90 °. At the same time, in geometry it is often required to calculate the area of ​​such a figure. How to do this, we will tell further.

The simplest formula for determining the area of ​​a right triangle

Initial data, where: a and b are the sides of the triangle coming out of the right angle.

That is, the area is equal to half the product of the two sides that come out of the right angle. Of course, there is Heron's formula used to calculate the area of ​​an ordinary triangle, but to determine the value, you need to know the length of three sides. Accordingly, you will have to calculate the hypotenuse, and this is extra time.

Find the area of ​​a right triangle using Heron's formula

This is a well-known and original formula, but for this you will have to calculate the hypotenuse along two legs using the Pythagorean Theorem.

In this formula: a, b, c are the sides of the triangle, and p is the semi-perimeter.

Find area of ​​right triangle given hypotenuse and angle

If none of the legs is known in your problem, then you will not be able to use the simplest method. To determine the value, you need to calculate the length of the legs. This is done simply by the hypotenuse and the cosine of the included angle.

b=c×cos(α)

Knowing the length of one of the legs, using the Pythagorean theorem, you can calculate the second side coming out of the right angle.

b 2 \u003d c 2 -a 2

In this formula, c and a are the hypotenuse and leg, respectively. Now you can calculate the area using the first formula. In the same way, one of the legs can be calculated, given the second and the angle. In this case, one of the desired sides will be equal to the product of the leg and the tangent of the angle. There are other ways to calculate the area, but knowing the basic theorems and rules, you can easily find the desired value.

If you do not have any of the sides of the triangle, but only the median and one of the angles, then you can calculate the length of the sides. To do this, use the properties of the median to divide a right triangle by two. Accordingly, it can act as a hypotenuse if it comes out of an acute angle. Use the Pythagorean theorem to find the length of the sides of a triangle that emerge from a right angle.

As you can see, knowing the basic formulas and the Pythagorean Theorem, you can calculate the area of ​​a right triangle, having only one of the angles and the length of one of the sides.