How to calculate the area of ​​an isosceles triangle. How to find the area of ​​a triangle (formulas)

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.

1. Compare triangles and parallelograms. There is a simple connection between these figures. If any parallelogram is cut diagonally, you get two equal triangle. 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 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.

2. 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.

3. Look at half of an isosceles triangle. Note 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”.
   • Hypotenuse right triangle is the lateral side of an isosceles triangle; let's denote it as "s".

4. 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.

5. 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))
     .

6. 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.

7. 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. Don't forget to write in your answer square units measurements.

   • 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.

8. 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 extract Square root, which, as a rule, is not completely extracted. 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 in the form decimal(S ≈ 15.49 cm2).

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's called 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 similar 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 (the 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.

Solution 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 - 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. Whichever good teacher no matter what 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 a leg, and the initially known side– 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.