Formulas for simple figures. How to calculate the area of a figure

Knowledge of how to measure the Earth appeared in ancient times and gradually took shape in the science of geometry. This word is translated from Greek as “land surveying”.

The measure of the extent of a flat section of the Earth in length and width is area. In mathematics, it is usually denoted by the Latin letter S (from the English “square” - “area”, “square”) or the Greek letter σ (sigma). S denotes the area of a figure on a plane or the surface area of a body, and σ is the cross-sectional area of a wire in physics. These are the main symbols, although there may be others, for example, in the field of strength of materials, A is the cross-sectional area of the profile.

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Calculation formulas

Knowing the areas of simple figures, you can find the parameters of more complex ones.. Ancient mathematicians developed formulas that can be used to easily calculate them. Such figures are triangle, quadrangle, polygon, circle.

To find the area of a complex plane figure, it is broken down into many simple figures such as triangles, trapezoids or rectangles. Then, using mathematical methods, a formula is derived for the area of this figure. A similar method is used not only in geometry, but also in mathematical analysis to calculate the areas of figures bounded by curves.

Triangle

Let's start with the simplest figure - a triangle. They are rectangular, isosceles and equilateral. Take any triangle ABC with sides AB=a, BC=b and AC=c (∆ ABC). To find its area, let us recall the sine and cosine theorems known from the school mathematics course. Letting go of all calculations, we arrive at the following formulas:

S=√ - Heron’s formula, known to everyone, where p=(a+b+c)/2 is the semi-perimeter of the triangle;
S=a h/2, where h is the height lowered to side a;
S=a b (sin γ)/2, where γ is the angle between sides a and b;
S=a b/2, if ∆ ABC is rectangular (here a and b are legs);
S=b² (sin (2 β))/2, if ∆ ABC is isosceles (here b is one of the “hips”, β is the angle between the “hips” of the triangle);
S=a² √¾, if ∆ ABC is equilateral (here a is a side of the triangle).

Quadrangle

Let there be a quadrilateral ABCD with AB=a, BC=b, CD=c, AD=d. To find the area S of an arbitrary 4-gon, you need to divide it by the diagonal into two triangles, the areas of which S1 and S2 are in general not equal.

Then use the formulas to calculate them and add them, i.e. S=S1+S2. However, if a 4-gon belongs to a certain class, then its area can be found using previously known formulas:

S=(a+c) h/2=e h, if the tetragon is a trapezoid (here a and c are the bases, e is the midline of the trapezoid, h is the height lowered to one of the bases of the trapezoid;
S=a h=a b sin φ=d1 d2 (sin φ)/2, if ABCD is a parallelogram (here φ is the angle between sides a and b, h is the height dropped to side a, d1 and d2 are diagonals);
S=a b=d²/2, if ABCD is a rectangle (d is a diagonal);
S=a² sin φ=P² (sin φ)/16=d1 d2/2, if ABCD is a rhombus (a is the side of the rhombus, φ is one of its angles, P is the perimeter);
S=a²=P²/16=d²/2, if ABCD is a square.

Polygon

To find the area of an n-gon, mathematicians break it down into the simplest equal figures - triangles, find the area of each of them and then add them. But if the polygon belongs to the class of regular, then use the formula:

S=a n h/2=a² n/=P²/, where n is the number of vertices (or sides) of the polygon, a is the side of the n-gon, P is its perimeter, h is the apothem, i.e. a segment drawn from the center of the polygon to one of its sides at an angle of 90°.

Circle

A circle is a perfect polygon with an infinite number of sides. We need to calculate the limit of the expression on the right in the formula for the area of a polygon with the number of sides n tending to infinity. In this case, the perimeter of the polygon will turn into the length of a circle of radius R, which will be the boundary of our circle, and will become equal to P=2 π R. Substitute this expression into the above formula. We will get:

S=(π² R² cos (180°/n))/(n sin (180°/n)).

Let's find the limit of this expression as n→∞. To do this, we take into account that lim (cos (180°/n)) for n→∞ is equal to cos 0°=1 (lim is the sign of the limit), and lim = lim for n→∞ is equal to 1/π (we converted the degree measure into a radian, using the relation π rad=180°, and applied the first remarkable limit lim (sin x)/x=1 at x→∞). Substituting the obtained values into the last expression for S, we arrive at the well-known formula:

S=π² R² 1 (1/π)=π R².

Units

Systemic and non-systemic units of measurement are used. System units belong to the SI (System International). This is a square meter (sq. meter, m²) and units derived from it: mm², cm², km².

In square millimeters (mm²), for example, they measure the cross-sectional area of wires in electrical engineering, in square centimeters (cm²) - the cross-section of a beam in structural mechanics, in square meters (m²) - in an apartment or house, in square kilometers (km²) - in geography .

However, sometimes non-systemic units of measurement are used, such as: weave, ar (a), hectare (ha) and acre (as). Let us present the following relations:

1 hundred square meters=1 a=100 m²=0.01 hectares;
1 ha=100 a=100 acres=10000 m²=0.01 km²=2.471 ac;
1 ac = 4046.856 m² = 40.47 a = 40.47 acres = 0.405 hectares.

Area formula is necessary to determine the area of a figure, which is a real-valued function defined on a certain class of figures of the Euclidean plane and satisfying 4 conditions:

Positivity - Area cannot be less than zero;
Normalization - a square with side unit has area 1;
Congruence - congruent figures have equal area;
Additivity - the area of the union of 2 figures without common internal points is equal to the sum of the areas of these figures.

Formulas for the area of geometric figures.

Geometric figure	Formula	Drawing
The result of adding the distances between the midpoints of opposite sides of a convex quadrilateral will be equal to its semi-perimeter.
Circle sector. The area of a sector of a circle is equal to the product of its arc and half its radius.
Circle segment. To obtain the area of segment ASB, it is enough to subtract the area of triangle AOB from the area of sector AOB.	S = 1 / 2 R(s - AC)
The area of the ellipse is equal to the product of the lengths of the major and minor semi-axes of the ellipse and the number pi.
Ellipse. Another option for calculating the area of an ellipse is through two of its radii.
Triangle. Through the base and height. Formula for the area of a circle using its radius and diameter.
Square . Through his side. The area of a square is equal to the square of the length of its side.
Square. Through its diagonals. The area of a square is equal to half the square of the length of its diagonal.
Regular polygon. To determine the area of a regular polygon, it is necessary to divide it into equal triangles that would have a common vertex at the center of the inscribed circle.	S= r p = 1/2 r n a

To solve geometry problems, you need to know formulas - such as the area of a triangle or the area of a parallelogram - as well as simple techniques that we will cover.

First, let's learn the formulas for the areas of figures. We have specially collected them in a convenient table. Print, learn and apply!

Of course, not all geometry formulas are in our table. For example, to solve problems in geometry and stereometry in the second part of the profile Unified State Exam in mathematics, other formulas for the area of a triangle are used. We will definitely tell you about them.

But what if you need to find not the area of a trapezoid or triangle, but the area of some complex figure? There are universal ways! We will show them using examples from the FIPI task bank.

1. How to find the area of a non-standard figure? For example, an arbitrary quadrilateral? A simple technique - let's divide this figure into those that we know everything about, and find its area - as the sum of the areas of these figures.

Divide this quadrilateral with a horizontal line into two triangles with a common base equal to . The heights of these triangles are equal And . Then the area of the quadrilateral is equal to the sum of the areas of the two triangles: .

Answer: .

2. In some cases, the area of a figure can be represented as the difference of some areas.

It is not so easy to calculate what the base and height of this triangle are equal to! But we can say that its area is equal to the difference between the areas of a square with a side and three right triangles. Do you see them in the picture? We get: .

Answer: .

3. Sometimes in a task you need to find the area of not the entire figure, but part of it. Usually we are talking about the area of a sector - part of a circle. Find the area of a sector of a circle of radius whose arc length is equal to .

In this picture we see part of a circle. The area of the entire circle is equal to . It remains to find out which part of the circle is depicted. Since the length of the entire circle is equal (since ), and the length of the arc of a given sector is equal , therefore, the length of the arc is several times less than the length of the entire circle. The angle at which this arc rests is also a factor of less than a full circle (that is, degrees). This means that the area of the sector will be several times smaller than the area of the entire circle.

Area of a geometric figure- a numerical characteristic of a geometric figure showing the size of this figure (part of the surface limited by the closed contour of this figure). The size of the area is expressed by the number of square units contained in it.

Triangle area formulas

Formula for the area of a triangle by side and height
Area of a triangle equal to half the product of the length of a side of a triangle and the length of the altitude drawn to this side
Formula for the area of a triangle based on three sides and the radius of the circumcircle
Formula for the area of a triangle based on three sides and the radius of the inscribed circle
Area of a triangle is equal to the product of the semi-perimeter of the triangle and the radius of the inscribed circle.

Square area formulas

Formula for the area of a square by side length
Square area equal to the square of the length of its side.
Formula for the area of a square along the diagonal length
Square area equal to half the square of the length of its diagonal.
S= 1 2
2

Rectangle area formula

Area of a rectangle

Parallelogram area formulas

Formula for the area of a parallelogram based on side length and height
Area of a parallelogram
Formula for the area of a parallelogram based on two sides and the angle between them
Area of a parallelogram is equal to the product of the lengths of its sides multiplied by the sine of the angle between them.
a b sin α

Formulas for the area of a rhombus

Formula for the area of a rhombus based on side length and height
Area of a rhombus equal to the product of the length of its side and the length of the height lowered to this side.
Formula for the area of a rhombus based on side length and angle
Area of a rhombus is equal to the product of the square of the length of its side and the sine of the angle between the sides of the rhombus.
Formula for the area of a rhombus based on the lengths of its diagonals
Area of a rhombus equal to half the product of the lengths of its diagonals.

Trapezoid area formulas

Heron's formula for trapezoid
Where S is the area of the trapezoid,
- lengths of the bases of the trapezoid,
- lengths of the sides of the trapezoid,

The areas of geometric figures are numerical values characterizing their size in two-dimensional space. This value can be measured in system and non-system units. So, for example, a non-systemic unit of area is a hundredth, a hectare. This is the case if the surface being measured is a piece of land. The system unit of area is the square of length. In the SI system, the unit of flat surface area is the square meter. In the GHS, the unit of area is expressed as a square centimeter.

Geometry and area formulas are inextricably linked. This connection lies in the fact that the calculation of the areas of plane figures is based precisely on their application. For many figures, several options are derived from which their square dimensions are calculated. Based on the data from the problem statement, we can determine the simplest possible solution. This will facilitate the calculation and reduce the likelihood of calculation errors to a minimum. To do this, consider the main areas of figures in geometry.

Formulas for finding the area of any triangle are presented in several options:

1) The area of a triangle is calculated from the base a and height h. The base is considered to be the side of the figure on which the height is lowered. Then the area of the triangle is:

2) The area of a right triangle is calculated in the same way if the hypotenuse is considered the base. If we take the leg as the base, then the area of the right triangle will be equal to the product of the legs halved.

The formulas for calculating the area of any triangle do not end there. Another expression contains the sides a,b and the sinusoidal function of the angle γ between a and b. The sine value is found in the tables. You can also find it out using a calculator. Then the area of the triangle is:

Using this equality, you can also make sure that the area of a right triangle is determined through the lengths of the legs. Because angle γ is a right angle, so the area of a right triangle is calculated without multiplying by the sine function.

3) Consider a special case - a regular triangle, whose side a is known by condition or its length can be found when solving. Nothing more is known about the figure in the geometry problem. Then how to find the area under this condition? In this case, the formula for the area of a regular triangle is applied:

Rectangle

How to find the area of a rectangle and use the dimensions of the sides that have a common vertex? The expression for calculation is:

If you need to use the lengths of the diagonals to calculate the area of a rectangle, then you will need a function of the sine of the angle formed when they intersect. This formula for the area of a rectangle is:

Square

The area of a square is determined as the second power of the side length:

The proof follows from the definition that a square is a rectangle. All sides that form a square have the same dimensions. Therefore, calculating the area of such a rectangle comes down to multiplying one by the other, i.e., to the second power of the side. And the formula for calculating the area of a square will take the desired form.

The area of a square can be found in another way, for example, if you use the diagonal:

How to calculate the area of a figure that is formed by a part of a plane bounded by a circle? To calculate the area, the formulas are:

Parallelogram

For a parallelogram, the formula contains the linear dimensions of the side, height and mathematical operation - multiplication. If the height is unknown, then how to find the area of the parallelogram? There is another way to calculate. A certain value will be required, which will be taken by the trigonometric function of the angle formed by adjacent sides, as well as their length.

The formulas for the area of a parallelogram are:

Rhombus

How to find the area of a quadrilateral called a rhombus? The area of a rhombus is determined using simple math with diagonals. The proof is based on the fact that the diagonal segments in d1 and d2 intersect at right angles. The table of sines shows that for a right angle this function is equal to unity. Therefore, the area of a rhombus is calculated as follows:

The area of a rhombus can also be found in another way. This is also not difficult to prove, given that its sides are the same in length. Then substitute their product into a similar expression for a parallelogram. After all, a special case of this particular figure is a rhombus. Here γ is the interior angle of the rhombus. The area of a rhombus is determined as follows:

Trapezoid

How to find the area of a trapezoid through the bases (a and b), if the problem indicates their lengths? Here, without a known value of the height length h, it will not be possible to calculate the area of such a trapezoid. Because this value contains the expression for calculation:

The square dimension of a rectangular trapezoid can also be calculated in the same way. It is taken into account that in a rectangular trapezoid the concepts of height and side are combined. Therefore, for a rectangular trapezoid, you need to specify the length of the side side instead of the height.

Cylinder and parallelepiped

Let's consider what is needed to calculate the surface of the entire cylinder. The area of this figure is a pair of circles called bases and a side surface. The circles forming circles have radius lengths equal to r. For the area of a cylinder the following calculation takes place:

How to find the area of a parallelepiped that consists of three pairs of faces? Its measurements match the specific pair. Opposite faces have the same parameters. First, find S(1), S(2), S(3) - the square dimensions of the unequal faces. Then the surface area of the parallelepiped is:

Ring

Two circles with a common center form a ring. They also limit the area of the ring. In this case, both calculation formulas take into account the dimensions of each circle. The first of them, calculating the area of the ring, contains the larger R and smaller r radii. More often they are called external and internal. In the second expression, the ring area is calculated through the larger D and smaller d diameters. Thus, the area of the ring based on known radii is calculated as follows:

The area of the ring, using the lengths of the diameters, is determined as follows:

Polygon

How to find the area of a polygon whose shape is not regular? There is no general formula for the area of such figures. But if it is depicted on a coordinate plane, for example it could be checkered paper, then how to find the surface area in this case? Here they use a method that does not require approximately measuring the figure. They do this: if they find points that fall into the corner of the cell or have whole coordinates, then only them are taken into account. To then find out what the area is, use the formula proved by Peake. It is necessary to add the number of points located inside the broken line with half the points lying on it, and subtract one, i.e. it is calculated this way: