Area formulas for all geometrics. How to calculate the area of a figure. Trapezium area formulas

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, an off-system unit of area is a hundred, a hectare. This is the case if the measured surface is a piece of land. The system unit of area is the square of length. In the SI system, it is customary to consider that the unit of area of a flat surface is a square meter. In the CGS, the unit of area is expressed in square centimetres.

Geometry and area formulas are inextricably linked. This connection lies in the fact that the calculation of the areas of flat figures is based precisely on their application. For many figures, several options are derived, according to which their square sizes are calculated. Based on the data from the problem statement, we can determine the simplest way to solve it. This facilitates the calculation and reduces the probability of calculation errors to a minimum. To do this, consider the main area of \u200b\u200bfigures in geometry.

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

1) The area of a triangle is calculated from the base a and the height h. The base is 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 exactly the same way if the hypotenuse is considered the base. If, however, the leg is taken as the base, then the area of \u200b\u200bthe right-angled 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 value of the sine is found in the tables. It can also be found using a calculator. Then the area of the triangle is:

According to this equality, you can also make sure that the area of \u200b\u200ba right triangle is determined through the lengths of the legs. Because the 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, in which 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 the calculation is:

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

Square

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

The proof follows from the definition that a rectangle is called a square. All sides forming a square have the same dimensions. Therefore, the calculation of the area of such a rectangle is reduced 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 a 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 the 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 is 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 mathematical operations with diagonals. The proof relies on the fact that the diagonal segments at d1 and d2 intersect at right angles. The table of sines shows that for a right angle, this function is equal to one. Therefore, the area of a rhombus is calculated as follows:

The area of a rhombus can also be found in another way. It is also not difficult to prove this, 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:

Trapeze

How to find the area of a trapezoid through the bases (a and b), if their lengths are indicated in the problem? 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 size of a rectangular trapezoid can also be calculated in the same way. At the same time, 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 instead of the height.

Cylinder and parallelepiped

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. 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 are consistent with a particular pair. Faces that are opposite have the same parameters. First find S(1), S(2), S(3) - square dimensions of unequal faces. Then the surface area of the parallelepiped:

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 one, which calculates the area of the ring, contains larger R and smaller r radii. More often they are called external and internal. In the second expression, the ring area is calculated using the larger D and smaller d diameters. Thus, the area of the ring according to 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 correct? There is no general formula for the area of such figures. But if it is depicted on a coordinate plane, for example, it can 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 integer coordinates, then only them are taken into account. To then find out what the area is, use the formula proved by Pick. It is necessary to add the number of points located inside the polyline with half the points lying on it, and subtract one, i.e. it is calculated in this way:

where C, D - the number of points located inside and on the entire polyline, respectively.

To solve problems in geometry, you need to know formulas - such as the area of a triangle or the area of \u200b\u200ba parallelogram - as well as simple tricks, which we will talk about.

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 exam in mathematics, other formulas for the area of a triangle are also 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 break this figure into those that we all know about, and find its area - as the sum of the areas of these figures.

Divide this quadrilateral by a horizontal line into two triangles with a common base equal to . The heights of these triangles are 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 \u200b\u200bthe figure can be represented as the difference of any areas.

It is not so easy to calculate what the base and height in 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-angled triangles. See them in the picture? We get: .

Answer: .

3. Sometimes in a task it is necessary to find the area not of the whole figure, but of its part. Usually we are talking about the area of \u200b\u200bthe sector - part of the circle. Find the area of \u200b\u200bthe sector of the circle of radius , whose arc length is equal to .

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

Knowledge of how to measure the Earth appeared in antiquity and gradually took shape in the science of geometry. From the Greek language, this word is translated as “land surveying”.

The measure of the length of a flat area of the Earth in length and width is the 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 by which they can be easily calculated. Such figures are a triangle, a quadrilateral, a polygon, a circle.

To find the area of a complex flat figure, it is broken down into many simple shapes such as triangles, trapezoids, or rectangles. Then mathematical methods derive a formula 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 shape - 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's recall the theorems of sines and cosines known from the school mathematics course. Letting go of all calculations, we arrive at the following formulas:

S=√ - Heron's formula known to all, where p=(a+b+c)/2 - half-perimeter of a 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 the side of the triangle).

Quadrilateral

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, it is necessary to divide it by a diagonal into two triangles whose areas S1 and S2 are generally not equal.

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

S=(a+c) h/2=e h, if the quad is a trapezoid (here a and c are the bases, e is the middle line 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 lowered 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 corners, 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 triangles, find the area of each of them, and then add them up. But if the polygon belongs to the class of regular ones, then the formula is used:

S \u003d a n h / 2 \u003d a² n / \u003d 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. the segment drawn from the center of the polygon to one of its sides at an angle of 90°.

A 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 polygon area formula 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'll 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 have translated the degree measure to radian, using the ratio 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

System and non-system units of measurement are applied. System units are referred to as SI (System International). This is a square meter (square meter, m²) and units derived from it: mm², cm², km².

In square millimeters (mm²), for example, they measure the cross-sectional area of \u200b\u200bwires in electrical engineering, in square centimeters (cm²) - the cross section of a beam in structural mechanics, in square meters (m²) - an apartment or house, in square kilometers (km²) - a territory in geography .

However, non-systemic units of measurement are sometimes used, such as: weaving, ar (a), hectare (ha) and acre (ac). We give the following ratios:

1 weave \u003d 1 a \u003d 100 m² \u003d 0.01 ha;
1 ha = 100 a = 100 acres = 10000 m² = 0.01 km² = 2.471 as;
1 ac = 4046.856 m² = 40.47 a = 40.47 acres = 0.405 ha.

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

Positive - Area cannot be less than zero;
Normalization - a square with a side of unity has an area of 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 shapes.

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 semiperimeter.
Circle sector. The area of a sector of a circle is equal to the product of its arc and half the radius.
circle segment. To get 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 an ellipse is equal to the product of the lengths of the major and minor semiaxes of the ellipse times pi.
Ellipse. Another option how to calculate the area of an ellipse is through its two radii.
Triangle. Through base and height. The formula for the area of a circle in terms of 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 diagonal. The area of a square is 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

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

Triangle area formulas

Triangle area formula for 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
The formula for the area of a triangle given three sides and the radius of the circumscribed circle
The formula for the area of a triangle given three sides and the radius of an inscribed circle
Area of a triangle is equal to the product of the half-perimeter of the triangle and the radius of the inscribed circle.

Square area formulas

The formula for the area of a square given the length of a side
square area is equal to the square of its side length.
The formula for the area of a square given the length of the diagonal
square area equal to half the square of the length of its diagonal.
S= 1 2
2

Rectangle area formula

Rectangle area

Formulas for the area of a parallelogram

Parallelogram area formula for side length and height
Parallelogram area
The formula for the area of a parallelogram given two sides and the angle between them
Parallelogram area is equal to the product of the lengths of its sides multiplied by the sine of the angle between them.
a b sinα

Formulas for the area of a rhombus

Rhombus area formula given side length and height
Rhombus area is equal to the product of the length of its side and the length of the height lowered to this side.
The formula for the area of a rhombus given the length of the side and the angle
Rhombus area 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.
The formula for the area of a rhombus from the lengths of its diagonals
Rhombus area is equal to half the product of the lengths of its diagonals.

Trapezium area formulas

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

Rectangle

Square

Rhombus

Trapeze

Cylinder and parallelepiped

Ring

Polygon

Calculation formulas

Triangle

Quadrilateral

Polygon

A circle

Units

Triangle area formulas

Square area formulas

Rectangle area formula

Formulas for the area of ​​a parallelogram

Formulas for the area of ​​a rhombus

Trapezium area formulas

Read also

Formulas for the area of a parallelogram

Formulas for the area of a rhombus