Formula for finding an isosceles triangle. How to find the area of a triangle (formulas)

In order to help their child with homework, parents must know many things themselves. How to find the area of an isosceles triangle, how does the participial phrase differ from the participial phrase, what is the acceleration of gravity?

Your son or daughter may have problems with any of these questions, and they will turn to you for clarification. In order not to fall on your face and maintain your authority in children's eyes, it is worth brushing up on some elements of the school curriculum.

Let's take the question of an isosceles triangle as an example. Geometry at school is difficult for many people, and after school it is forgotten most quickly.

But when your children enter 8th grade, you will have to remember the formulas regarding geometric shapes. An isosceles triangle is one of the simplest figures in terms of finding its parameters.

If everything you once taught about triangles has been forgotten, let's remember. An isosceles triangle is one in which two sides have the same length. These equal edges are called the lateral sides of an isosceles triangle. The third side is its foundation.

There is an option in which all 3 sides are equal. It is called an equilateral triangle. All formulas applied to an isosceles apply to it, and, if necessary, any of its sides can be called a base.

To find the area we need to divide the base in half. A straight line descended to the resulting point from the vertex connecting the sides will intersect the base at a right angle.

This is the property of such triangles: the median, that is, the straight line from the vertex to the middle of the opposite side, in an isosceles triangle is its bisector (a straight line dividing the angle in half) and its altitude (perpendicular to the opposite side).

To find the area of an isosceles triangle, you need to multiply its height by its base, and then divide this product in half.

To find the area of a triangle, the formula is simple: S=ah/2, where a is the length of the base, h is the height.

This can be clearly explained as follows. Cut out a similar shape from paper, find the middle of the base, draw a height to this point and carefully cut along this height. You will get two right triangles.

If we place them next to each other with their hypotenuses (long sides), we will create a rectangle, one side of which will be equal to the height of our figure, and the other to half of its base. That is, the formula will be confirmed.

Visual demonstration is very important. If your child learns not to mindlessly memorize formulas, but to understand their meaning, geometry will no longer seem like a difficult subject to him.

The best student in the class is not the student who memorizes, but the student who thinks and, most importantly, understands.

How to find the area of a figure if one angle is right?

It may turn out that the angle between the sides of a given triangular figure is 90°. Then this triangle will be called a right triangle, its sides will be called legs, and its base will be called the hypotenuse.

The area of such a figure can be calculated using the above method (find the middle of the hypotenuse, draw the height to it, multiply it by the hypotenuse, divide it in half). But the problem can be solved much simpler.

Let's start with clarity. A right isosceles triangle is exactly half a square when cut diagonally. And if the area of a square is found by simply raising its side to the second power, then the area of the figure we need will be half as large.

S=a 2 /2, where a is the length of the leg.

The area of an isosceles right triangle is equal to half the square of its side. The problem turned out to be not as serious as it seemed at first glance.

Solving geometric problems does not require superhuman efforts and may well be useful not only for children, but also for you when finding answers to any practical questions.

Geometry is an exact science. If you delve into its basics, there will be few difficulties with it, and the logic of the evidence can greatly captivate your child. You just need to help him a little. No matter how good a teacher he gets, parental help will not be superfluous.

And in the case of studying geometry, the method mentioned above will be very useful - clarity and simplicity of explanation.

At the same time, we must not forget about the accuracy of the formulations, otherwise we can make this science much more complex than it actually is.

Depending on the type of triangle, there are several options for finding its area. For example, to calculate the area of a right triangle, use the formula S= a * b / 2, where a and b are its legs. If you want to find out the area of an isosceles triangle, then you need to divide the product of its base and height by two. That is, S= b*h / 2, where b is the base of the triangle, and h is its height.

Next, you may need to calculate the area of an isosceles right triangle. Here the following formula comes to the rescue: S = a* a / 2, where the legs “a” and “a” must necessarily have the same values.

Also, we often have to calculate the area of an equilateral triangle. It is found by the formula: S= a * h/ 2, where a is the side of the triangle, and h is its height. Or according to this formula: S= √3/ 4 *a^2, where a is the side.

How to find the area of a right triangle

Do you need to find the area of a right triangle, but the problem statement does not indicate the dimensions of two of its legs at once? Then we cannot use this formula (S= a * b / 2) directly.

Let's consider several possible solutions:

If you do not know the length of one leg, but the dimensions of the hypotenuse and the second leg are given, then we turn to the great Pythagoras and, using his theorem (a^2+b^2=c^2), we calculate the length of the unknown leg, then use it to calculate the area of the triangle.
If the length of one leg and the degree slope of the angle opposite it are given: we find the length of the second leg using the formula - a=b*ctg(C).
Given: the length of one leg and the degree slope of the angle adjacent to it: to find the length of the second leg, we use the formula - a=b*tg(C).
And lastly, given: the angle and length of the hypotenuse: we calculate the length of both of its legs using the following formulas - b=c*sin(C) and a=c*cos(C).

How to find the area of an isosceles triangle

The area of an isosceles triangle can be very easily and quickly found using the formula S= b*h / 2, but if one of the indicators is missing, the task becomes much more complicated. After all, it is necessary to perform additional actions.

Possible task options:

Given: the length of one of the sides and the length of the base. Using the Pythagorean theorem, we find the height, that is, the length of the second leg. Provided that the length of the base divided by two is the leg, and the initially known side is the hypotenuse.
Given: the base and the angle between the side and the base. We calculate the height using the formula h=c*ctg(B)/2 (do not forget to divide side “c” by two).
Given: the height and the angle that was formed by the base and side: we use the formula c=h*tg(B)*2 to find the height, and multiply the result by two. Next we calculate the area.
Known: the length of the side and the angle formed between it and the height. Solution: we use the formulas - c=a*sin(C)*2 and h=a*cos(C) to find the base and height, after which we calculate the area.

How to find the area of an isosceles right triangle

If all the data is known, then using the standard formula S= a* a / 2 we calculate the area of an isosceles right triangle, but if some indicators are not indicated in the problem, then additional actions are performed.

For example: we do not know the lengths of both sides (we remember that in an isosceles right triangle they are equal), but the length of the hypotenuse is given. Let's apply the Pythagorean theorem to find the same sides "a" and "a". Pythagorean formula: a^2+b^2=c^2. In the case of an isosceles right triangle, it transforms into this: 2a^2 = c^2. It turns out that to find leg “a”, you need to divide the length of the hypotenuse by the root of 2. The result of the solution will be the length of both legs of an isosceles right triangle. Next we find the area.

How to find the area of an equilateral triangle

Using the formula S= √3/ 4*a^2 you can easily calculate the area of an equilateral triangle. If the radius of the triangle's circumscribed circle is known, then the area can be found using the formula: S= 3√3/ 4*R^2, where R is the radius of the circle.

Find out how to find the area of a parallelogram. Squares and rectangles are parallelograms, like any other four-sided figure in which opposite sides are parallel. The area of a parallelogram is calculated by the formula: S = bh, where “b” is the base (the bottom side of the parallelogram), “h” is the height (the distance from the top to the bottom side; the height always intersects the base at an angle of 90°).

In squares and rectangles, the height is equal to the side because the sides intersect the top and bottom at right angles.

Compare triangles and parallelograms. There is a simple connection between these figures. If any parallelogram is cut diagonally, you get two equal triangles. Similarly, if you add two equal triangles together, you get a parallelogram. Therefore, the area of any triangle is calculated by the formula: S = ½bh, which is half the area of the parallelogram.

Find the base of the isosceles triangle. Now you know the formula for calculating the area of a triangle; It remains to find out what “base” and “height” are. The base (denoted as "b") is the side that is not equal to the other two (equal) sides.
Lower the perpendicular to the base. Make this from the vertex of the triangle, which is opposite to the base. Remember that a perpendicular intersects the base at a right angle. This perpendicular is the height of the triangle (denoted as “h”). Once you find the value of "h", you can calculate the area of the triangle.
- In an isosceles triangle, the altitude intersects the base exactly in the middle.
Look at half of an isosceles triangle. Notice that the altitude has divided the isosceles triangle into two equal right triangles. Look at one of them and find its sides:
- The short side is equal to half the base: b 2 (\displaystyle (\frac (b)(2))).
- The second side is the height “h”.
- The hypotenuse of a right triangle is the lateral side of an isosceles triangle; Let's denote it as "s".
Use the Pythagorean theorem. If two sides of a right triangle are known, its third side can be calculated using the Pythagorean theorem: (side 1) 2 + (side 2) 2 = (hypotenuse) 2. In our example, the Pythagorean theorem will be written like this: .
- Most likely, you know the Pythagorean theorem in the following notation: a 2 + b 2 = c 2 (\displaystyle a^(2)+b^(2)=c^(2)). We use the words side 1, side 2, and hypotenuse to prevent confusion with the example variables.
Calculate the value of "h". Remember that in the formula for calculating the area of a triangle, there are variables "b" and "h", but the value of "h" is unknown. Rewrite the formula to calculate "h":
- (b 2) 2 + h 2 = s 2 (\displaystyle ((\frac (b)(2)))^(2)+h^(2)=s^(2))
  h 2 = s 2 − (b 2) 2 (\displaystyle h^(2)=s^(2)-((\frac (b)(2)))^(2))
  .
Substitute known values into the formula and calculate “h”. This formula can be applied to any isosceles triangle whose sides are known. Substitute the value of the base for "b" and the value of the side for "s" to find the value of "h".
- In our example: b = 6 cm; s = 5 cm.
- Substitute the values into the formula:
  h = (s 2 − (b 2) 2) (\displaystyle h=(\sqrt (())s^(2)-((\frac (b)(2)))^(2)))
  h = (5 2 − (6 2) 2) (\displaystyle h=(\sqrt (())5^(2)-((\frac (6)(2)))^(2)))
  h = (25 − 3 2) (\displaystyle h=(\sqrt (())25-3^(2)))
  h = (25 − 9) (\displaystyle h=(\sqrt (())25-9))
  h = (16) (\displaystyle h=(\sqrt (())16))
  h = 4 (\displaystyle h=4) cm.
Plug the base and height values into the formula to calculate the area of the triangle. Formula: S = ½bh; Substitute the values of “b” and “h” into it and calculate the area. Be sure to write square units in your answer.
- In our example, the base is 6 cm and the height is 4 cm.
- S = ½bh
  S = ½(6 cm)(4 cm)
  S = 12 cm 2.
Let's look at a more complex example. In most cases, you will be given a more difficult task than the one discussed in our example. To calculate the height, you need to take the square root, which, as a rule, is not taken entirely. In this case, write the height value as a simplified square root. Here's a new example:
- Calculate the area of an isosceles triangle whose sides are 8 cm, 8 cm, 4 cm.
- For the base “b”, select the side that is 4 cm.
- Height: h = 8 2 − (4 2) 2 (\displaystyle h=(\sqrt (8^(2)-((\frac (4)(2)))^(2))))
  = 64 − 4 (\displaystyle =(\sqrt (64-4)))
  = 60 (\displaystyle =(\sqrt (60)))
- Simplify the square root using factors: h = 60 = 4 ∗ 15 = 4 15 = 2 15 . (\displaystyle h=(\sqrt (60))=(\sqrt (4*15))=(\sqrt (4))(\sqrt (15))=2(\sqrt (15)).)
- S = 1 2 b h (\displaystyle =(\frac (1)(2))bh)
  = 1 2 (4) (2 15) (\displaystyle =(\frac (1)(2))(4)(2(\sqrt (15))))
  = 4 15 (\displaystyle =4(\sqrt (15)))
- The answer can be written with the root or extract the root on a calculator and write the answer as a decimal fraction (S ≈ 15.49 cm 2).

The letter designations of the sides and angles in the above figure correspond to the designations indicated in the formulas. So this will help you match them with the elements of an isosceles triangle. From the conditions of the problem, determine which elements are known, find their designations in the drawing and select the appropriate formula.

Formula for the area of an isosceles triangle

The following are formulas for finding the area of an isosceles triangle: through the sides, the side and the angle between them, through the side, the base and the angle at the apex, through the side of the base and the angle at the base, etc. Just find the most suitable one in the picture on the left. For the most curious, the text on the right explains why the formula is correct and how exactly it can be used to find the area.

can be found knowing its side and basis. This expression was obtained by simplifying a more general, universal formula. If we take Heron's formula as a basis, and then take into account that the two sides of the triangle are equal to each other, then the expression simplifies to the formula presented in the picture.
An example of using such a formula is given in the example of solving the problem below.
The second formula allows you to find its area through the sides and the angle between them is half the square of the side, multiplied by the sine of the angle between the sides
If we mentally lower the height to the side of an isosceles triangle, we note that its length will be equal to a * sin β. Since the length of the lateral side is known to us, the height dropped onto it is now known, half of their product will be equal to the area of the given isosceles triangle (Explanation: the full product gives the area of the rectangle, which is obvious. The height divides this rectangle into two small rectangles, with the sides of the triangle are their diagonals, which divide them exactly in half. Thus, the area of an isosceles triangle will be equal to half the product of the lateral side and the height). See also Formula 5
The third formula shows finding the area through the side, base and apex angle.
Strictly speaking, knowing one of the angles of an isosceles triangle, you can find the others, so using this or the previous formula is a matter of taste (by the way, this is why you can remember only one of them).
The third formula also has another interesting feature - the product a sin α will give us the length of the height lowered to the base. As a result, we get a simple and obvious formula 5.
Area of an isosceles triangle can also be found through the side of the base and the corner at the base(the angles at the base are equal) as the square of the base divided by four tangents of half the angle formed by its sides. If you look closely, it becomes obvious that half the base (b/2) multiplied by tan(β/2) gives us the height of the triangle. Since the height in an isosceles triangle is, at the same time, a bisector and a median, then tg(β/2) is the ratio of half the base (b/2) to the height - tg(β/2) = (b/2)/h. Whence h = b / (2 tan(β/2)). As a result, the formula will again be reduced to the simpler Formula 5, which is quite obvious.
Of course area of an isosceles triangle can be found by dropping the height from the top to the base, resulting in two right triangles. Further - everything is obvious. Half the product of the height and the base and there is the required area. For an example of using this formula, see the problem below (2nd solution method)
This formula is obtained if you try to find the area of an isosceles triangle using the Pythagorean theorem. To do this, we express the height from the previous formula, which is at the same time the leg of a right triangle formed by the side, half of its base and height, through the Pythagorean theorem. The lateral side is the hypotenuse, therefore, from the square of the lateral side (a) we subtract the square of the second leg. Since it is equal to half the base (b/2), its square will be equal to b 2 /4. Extracting the root from this expression will give us the height. As can be seen in Formula 6. If the numerator and denominator are multiplied by two, and then the two of the numerator is entered under the root sign, we get the second version of the same formula, which is written through the equal sign.
By the way, the smartest ones can see that if you open the brackets in Formula 1, it will turn into Formula 6. Or vice versa, the difference of the squares of two numbers, factored, will give us the original, first one.

Designations, which were applied in the formulas in the figure:

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

b- base length

α - the size of one of two equal angles at the base

β - the size of the angle between the equal sides of the triangle and the one opposite to its base

h- the length of the height lowered from the vertex of an isosceles triangle to the base

Important. Pay attention to the variable designations! Don't get confused α And β, and a And b!

Note. This is part of a lesson with geometry problems (section area of an isosceles triangle). Here are problems that are difficult to solve. If you need to solve a geometry problem that is not here, write about it in the forum. To indicate the action of extracting a square root in problem solutions, the symbol √ or sqrt() is used, with the radical expression indicated in parentheses.

Task

The side of an isosceles triangle is 13 cm and the base is 10 cm. Find the area isosceles triangle.

Solution.

1st method. Let's apply Heron's formula. Since the triangle is isosceles, it will take a simpler form (see formula 1 in the list of formulas above):

where a is the length of the sides, and b is the length of the base.
Substituting the values of the lengths of the sides of the triangle from the problem statement, we obtain:
S = 1/2 * 10 * √ ((13 + 5)(13 - 5)) = 5 √ (18 * 8) = 60 cm 2

2nd method. Let's apply the Pythagorean theorem
Let's assume that we don't remember the formula used in the first solution. Therefore, let us lower height BK from vertex B to base AC.
Since the height of an isosceles triangle divides its base in half, the length of half the base will be equal to
AK = AC / 2 = 10 / 2 = 5 cm.

The height with half the base and the side of the isosceles triangle forms a right triangle ABK. In this triangle we know the hypotenuse AB and the leg AK. Let us express the length of the second leg through the Pythagorean theorem.

Mathematics is an amazing science. However, such a thought comes only when you understand it. To achieve this, you need to solve problems and examples, draw diagrams and pictures, prove theorems.

The path to understanding geometry lies through solving problems. An excellent example would be tasks in which you need to find the area of an isosceles triangle.

What is an isosceles triangle, and how is it different from others?

In order not to be intimidated by the terms “height”, “area”, “base”, “isosceles triangle” and others, you will need to start with the theoretical foundations.

First about the triangle. This is a flat figure, which is formed from three points - vertices, in turn, connected by segments. If two of them are equal to each other, then the triangle becomes isosceles. These sides were called lateral, and the remaining one became the base.

There is a special case of an isosceles triangle - equilateral, when the third side is equal to two lateral ones.

Shape Properties

They turn out to be faithful assistants in solving problems that require finding the area of an isosceles triangle. Therefore, it is necessary to know and remember them.

The first of them: the angles of an isosceles triangle, one side of which is the base, are always equal to each other.
The property about additional constructions is also important. The height, median and bisector drawn to the unpaired side coincide.
The same segments drawn from the corners at the base of the triangle are equal in pairs. This also often makes it easier to find a solution.
Two equal angles in it always have a value less than 90º.
And lastly: the inscribed and circumscribed circles are constructed in such a way that their centers lie at a height to the base of the triangle, and therefore the median and bisector.

How to recognize an isosceles triangle in a problem?

If, when solving a task, the question arises of how to find the area of an isosceles triangle, then you first need to understand that it belongs to this group. And certain signs will help with this.

Two angles or two sides of a triangle are equal.
The bisector is also the median.
The altitude of a triangle turns out to be the median or bisector.
The two heights, medians, or bisectors of a figure are equal.

Designations of quantities adopted in the formulas under consideration

To simplify how to find the area of an isosceles triangle using formulas, the replacement of its elements with letters has been introduced.

Attention! It is important not to confuse “a” with “A” and “b” with “B”. These are different quantities.

Formulas that can be used in different tasks

The lengths of the sides are known, and you need to find the area of an isosceles triangle.

In this case, you need to square both values. Multiply the number obtained from changing the side by 4 and subtract the second from it. Take the square root of the resulting difference. Divide the length of the base by 4. Multiply the two numbers. If you write these actions in letters, you get the following formula:

Let it be recorded under No. 1.

Find the area of an isosceles triangle using the side values. A formula that some may find simpler than the first.

The first step is to find half of the base. Then find the sum and difference of this number with the side. Multiply the last two values and take the square root. The last step is to multiply everything by half the base. Literal equality will look like this:

This is formula No. 2.

A way to find the area of an isosceles triangle if the base and height to it are known.

One of the shortest formulas. In it you need to multiply both given quantities and divide them by 2. This is how it will be written:

The number of this formula is 3.

In the task, the sides of the triangle and the value of the angle lying between the base and the side are known.

Here, in order to find out what the area of an isosceles triangle will be equal to, the formula will consist of several factors. The first one is the value of the sine of the angle. The second is equal to the product of the side and the base. The third is a fraction of ½. General mathematical notation:

The serial number of the formula is 4.

The problem is given: the lateral side of an isosceles triangle and the angle lying between its lateral sides.

As in the previous case, the area is found using three factors. The first is equal to the value of the sine of the angle specified in the condition. The second is the square of the side. And the last one is also equal to half one. As a result, the formula will be written like this:

Her number is 5.

A formula that allows you to find the area of an isosceles triangle if its base and the angle opposite it are known.

First you need to calculate the tangent of half the known angle. Multiply the resulting number by 4. Square the length of the side, which is then divided by the previous value. Thus, we get the following formula:

The last formula number is 6.

Sample problems

First task: it is known that the base of an isosceles triangle is 10 cm and its height is 5 cm. We need to determine its area.

To solve it, it is logical to choose formula number 3. Everything in it is known. Plug in the numbers and count. It turns out that the area is 10 * 5 / 2. That is, 25 cm 2.

Second task: an isosceles triangle is given a side and a base that are equal to 5 and 8 cm, respectively. Find its area.

First way. According to formula No. 1. When squaring the base, the result is 64, and the quadruple square of the side is 100. Subtracting the first from the second, the result is 36. The root is perfectly extracted from this, which is equal to 6. The base divided by 4 is equal to 2. The final value is determined as the product of 2 and 6, that is, 12. This is the answer: the required area is 12 cm 2.

Second way. According to formula No. 2. Half of the base is equal to 4. The sum of the side and the found number gives 9, their difference is 1. After multiplication, it turns out 9. Extracting the square root gives 3. And the last action, multiplying 3 by 4, which gives the same 12 cm 2.

By solving geometry problems and determining how to find the area of an isosceles triangle, you can gain invaluable experience. The more different variants of tasks are completed, the easier it is to find the answer in a new situation. Therefore, regular and independent completion of all tasks is the path to successful learning of the material.

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