Areas of triangles.
In order to help their own child with lessons, the ancestors themselves must know a huge number of things. How to find the area of an isosceles triangle, how does a participial phrase differ from a participial phrase, what is the acceleration of gravity?
Math Lesson 8 Area of a Triangle
Your son or daughter may have difficulties with any of these questions, and they will specifically turn to you for clarification. In order not to fall face first into the mud and maintain your own authority in the eyes of children, it is worth refreshing your memory of certain 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 most quickly forgotten.
But when your kids go to 8 Class, you will have to remember the formulas regarding geometric shapes. An isosceles triangle is one of the most common figures in terms of finding its characteristics.
Let's start by clarifying the definitions.
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 3rd side is its base.
There is an option in which all 3 sides are equal. It is called an equilateral triangle. All formulas used for an isosceles apply to it, and, if necessary, each of its sides can be called a base.
To find the area, we will need to divide the base in half. A flat one, lowered to the acquired point from the top connecting the sides, will intersect the base at a right angle.
This is the property of similar triangles: the median, in other words, equal from the top to the middle of the reverse side, in an isosceles triangle is its bisector (a straight line dividing the angle in half) and its altitude (perpendicular to the reverse 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 ordinary: 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), then a rectangle will be formed, one side of which will be equal to the height of our figure, and the other to half of its base. In other words, the formula will be confirmed.
The best student in the class is not the student who memorizes, but the student who thinks and, most importantly, understands.
How find Area of a figure if one angle is right?
It may turn out that the angle between the sides of this 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.
Square Such a figure can be calculated using the above method (we 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 even more simply.
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 ordinary construction to the second power of its side, then the area of the figure suitable for us 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 severe as it seemed at first glance.
Geometry is a precise science. If you think about its bases, then there will be few problems 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 unnecessary.
And in the case of studying geometry, the method mentioned above will be very useful - clarity and ease of explanation.
With all this, we must not forget about the accuracy of the formulations; otherwise, this science can be made even more complex than it is in essence.
Abstracts
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As you may remember from your school geometry curriculum, a triangle is a figure formed from three segments connected by three points that do not lie on the same straight line. A 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 angles, the answer will also be correct. Triangles are divided according to the number of equal sides and the size of the angles in the figures. Thus, triangles are distinguished as isosceles, equilateral and scalene, as well as rectangular, acute and obtuse, respectively.
There are a lot of formulas for calculating the area of a triangle. Choose how to find the area of a triangle, i.e. Which formula to use is up to you. But it is worth noting only some of the notations that are 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 be useful to you if you completely forgot your geometry course. Below are the most understandable and uncomplicated options for calculating the unknown and mysterious area of a triangle. It is not difficult and will be useful both for your household needs and for helping your children. Let's remember how to calculate the area of a triangle as easily as possible:
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 in which one angle is equal to 90 degrees (hence called right). A right angle is formed by two perpendicular lines (in the case of a triangle, two perpendicular segments). In a right triangle there can only be one right angle, because... the sum of all angles of any one triangle is equal to 180 degrees. It turns out that 2 other angles should divide the remaining 90 degrees, for example 70 and 20, 45 and 45, etc. So, you remember the main thing, all that remains is to find out how to find the area of a right triangle. Let's imagine that we have such a right triangle in front of us, and we need to find its area S.
1. The simplest way to determine the area of a right triangle is calculated using the following formula:
In our case, the area of the right triangle is: S = 2.5 cm * 3 cm / 2 = 3.75 sq. cm.
In principle, there is no longer any need to verify the area of the triangle in other ways, because Only this one will be useful and will help in everyday life. 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 area of a right triangle that can still be used:
We decided to use the first formula and with some minor blots (we drew it in a notebook and used an old ruler and protractor), but we got the correct calculation:
S = (2.5*2.5)/(2*0.9)=(3*3)/(2*1.2). We got the following results: 3.6=3.7, but taking into account the shift of cells, we can forgive this nuance.
Isosceles triangle and its area.
If you are faced with the task of calculating the formula for an isosceles triangle, then the easiest way is to use the main and what is considered to be the classical formula for the area of a triangle.
But first, before finding the area of an isosceles triangle, let’s find out what kind of figure it is. An isosceles triangle is a triangle in which two sides have the same length. These two sides are called lateral, the third side is called the base. Do not confuse an isosceles triangle with an equilateral triangle, i.e. a regular 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.
What is area? Strange question - isn't it? In ordinary life, we are accustomed to the fact that all sorts of flat figures (such as the surface of a table, a chair, the floor of our apartments, etc.) have not only length and width, but also some other characteristic that we, without thinking , we call it area. Now let’s think about it: what is an area anyway?
Let's start with the simplest thing. The basis is the fact that:
In other words, we consider the area of a square with a side one meter to be one “meter of area.”
Look carefully at the picture and make sure that it really is drawn there - “square meter”! And remember the designation.
Now here’s a tricky question: what is it? Area of a square with side? But no!
Look: a square with a side.
And to get square meters (that is,), we must draw, for example, like this:
How to get, say, ? Well, for example like this:
And in general, if we take a rectangle whose sides are equal to meters and meters, then in this rectangle:
Fits exactly square meters. Look carefully: we have “layers”, each of which is exactly square meters.
This means that a rectangle of size x contains a total of square meters. This number, how many square meters fit in a rectangle, is its square.
What if the figure is not a rectangle at all, but some kind of abracadabra?
I’ll surprise you - there are such terrible abracadabras for which it is absolutely impossible to determine how many square meters there are. Even approximately! Unfortunately, it is impossible to draw such figures.
But they exist! They look like, for example, a “comb” with very fine teeth.
And so, for normal figures, you can intuitively (that is, for yourself) assume that the area of a figure is the number of square units (meters, centimeters, etc.) that “fit” in this figure. A more strict, “real” definition area, see the following levels of theory.
And just imagine, mathematicians have learned to express areas for many figures through some linear (those that can be measured with a ruler) elements of the figures. These expressions are called "area formulas". There are quite a lot of these formulas - mathematicians have been trying for a long time. Try to remember the simplest and most basic formulas first, and then the more complex ones.
Area formulas
Square
Rectangle
Right triangle
Triangle (free)
There are several area formulas for a triangle.
Basic formula
Second basic formula
Third formula
Which formula should you choose for your problem? The main ones are formulas 1 and 2. The third formula must be applied if everything is given to you: three sides and the radius of the inscribed circle. But that doesn’t happen, right? That's why we use formula 3, rather the opposite, to find the radius of the inscribed circle. Then you need to find the area using one of formulas 1, 2 or 4, and then the radius: .
Well, formula 4 allows you to find the area on both sides using lengthy arithmetic. And don’t make mistakes in arithmetic when you apply Heron’s formula!
Arbitrary quadrilateral
For an arbitrary quadrilateral there is nothing more, but for “good” quadrilaterals there are other formulas.
Parallelogram
Basic formula
Second formula
Rhombus
A rhombus has diagonals that are perpendicular, so basic for him it becomes formula:
Second formula
And the additional formula becomes
Trapezoid
Basic formula
Second formula
"Tricky questions about area"
In addition to problems that simply ask you to find the area, there are also all sorts of questions. Well, for example:
Let's answer this question in two ways. The first method is formal: we use the formula for the area of a square. So, it was, which means the area has increased several times!
In the case of squares, there is a second way to “touch” and be convinced directly of this number.
Let's draw:
If you don’t have a square, then all that remains is to substitute new values into the formulas - and don’t be surprised if the numbers suddenly turn out to be quite large.
AREA OF TRIANGLE AND QUADAGON. BRIEFLY ABOUT THE MAIN THINGS
Right triangle
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If you need to find the area of a triangle, don’t worry that you have long forgotten everything that teachers put into your head at school. Our article will tell you how to solve this issue, and in different ways.
To begin with, remember that a triangle is a figure that is formed by the intersection of three straight lines. The three points where the lines intersect are the vertices of the figure, and the segments opposite them are the edges of the triangle. There are several particular types of triangles (isosceles, right, equilateral), the areas of which we will also look for.
How to calculate the area of a triangle using the general formula
For the most general case, the area of a given geometric figure is calculated using the following formula: Area = ½ the length of one of the sides of the figure, multiplied by the length of the height lowered to this side.
Find the area of a triangle if we know all three of its sides
If you know all three sides of the triangle, then you can find its area using Heron's formula. First, let's find the semi-perimeter of the triangle by adding the lengths of all three of its sides and dividing by two. Then we find the square of the area, according to the following formula: SS = p (p-a)(p-b)(p-c), where a, b, c are the lengths of the sides of the figure, and p is the half perimeter. To find the area, simply take the square root of the resulting value.
Find the area of a triangle if we know its hypotenuse, leg and the angle formed by them
To do this, we will use a trigonometric table and the following formula:
S=1/2*a*b*sinB, where a and b are the leg with the hypotenuse, and B is the angle formed at their intersection.
Using this formula, we can find the area of an ordinary triangle, an equilateral triangle, an isosceles triangle, and a rectangular triangle.
Find the area of a triangle if we know the side and the angle opposite to it
We apply the formula: S=1/2(a*a)/(2tgB), where a is the known leg, and B is the angle opposite to it.
Find the area of a triangle if we only know the hypotenuse and leg
First, let's find the value FF=1/2(в*в – а*а). Then we extract the root (F) from this number and substitute it into the formula to find the area of the triangular figure: S=a*F. Here a is the leg, b is the hypotenuse.
Find the area of a triangle if we know one of the acute angles and the hypotenuse
We substitute the values known from the conditions of the problem into the formula: S=1/2(в*в)* cosA*sinA*. Here the acute angle is A, and B is the hypotenuse.
Find the area of a triangle using the coordinates of the vertices
If, according to the conditions of the problem, you are given the coordinates of three points, which are the vertices of a triangular figure, then you can also calculate the area.
So, you are given vertices A (x1, y1), B (x2, y2), C (x3, y3). To find the area, we use the following formula: S=1/2((x1-x3)(y2-y3) - (x2-x3)(y1-y3)). At the same time, remember that the module must be taken from the value that you calculate in brackets, because some points may have coordinates with a minus sign.
You can also do things differently.
Method 1. First find the lengths of all sides of the triangular figure, and then use Heron’s formula, which was described above. First, we find the squares of the sides using the following formulas:
AB*AB=(x1-x2)(x1-x2) + (y1-y2)(y1-y2);
BV*BV=(x2-x3)(x2-x3) + (y2-y3)(y2-y3);
BA*BA=(x3-x1)(x3-x1) + (y3-y1)(y3-y1).
Find the half perimeter of a triangular figure:
p=1\2(AB+ BV+ VA)
Now we substitute the values into the formula:
SS=p(p-AB)(p-BV)(p-VA). This is the squared area. We extract the root from the meaning and finally find what we were looking for.
By the way, for the sake of curiosity, you can calculate the area from coordinates using the two methods described above. Then you will know that the totals will vary slightly. This happens because the result obtained during the first calculation will have a rounded value than the result obtained using Heron's formula. Thus, to obtain more accurate data, it is recommended to use the second method.
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