How to calculate the area of different shapes. How to find the area of a figure? Geometric shape of the room

How to find the area of a figure?

Knowing and being able to calculate the areas of various figures is necessary not only for solving simple geometric problems. You cannot do without this knowledge when drawing up or checking estimates for repairs of premises, calculating the amount of necessary consumables. So let's figure out how to find the areas of different shapes.

The part of the plane contained within a closed contour is called the area of this plane. Area is expressed by the number of square units contained in it.

To calculate the area of basic geometric shapes, you must use the correct formula.

Area of a triangle

Designations:

If h, a are known, then the area of the required triangle is determined as the product of the lengths of the side and the height of the triangle lowered to this side, divided in half: S=(a h)/2
If a, b, c are known, then the required area is calculated using Heron’s formula: the square root taken from the product of half the perimeter of the triangle and three differences of half the perimeter and each side of the triangle: S = √(p (p - a) (p - b)·(p - c)).
If a, b, γ are known, then the area of the triangle is determined as half the product of 2 sides, multiplied by the value of the sine of the angle between these sides: S=(a b sin γ)/2
If a, b, c, R are known, then the required area is determined as dividing the product of the lengths of all sides of the triangle by four radii of the circumscribed circle: S=(a b c)/4R
If p, r are known, then the required area of the triangle is determined by multiplying half the perimeter by the radius of the circle inscribed in it: S=p·r

Square area

Designations:

If the side is known, then the area of a given figure is determined as the square of the length of its side: S=a 2
If d is known, then the area of the square is determined as half the square of the length of its diagonal: S=d 2 /2

Area of a rectangle

Designations:

S - determined area,
a, b - lengths of the sides of the rectangle.

If a, b are known, then the area of a given rectangle is determined by the product of the lengths of its two sides: S=a b
If the lengths of the sides are unknown, then the area of the rectangle must be divided into triangles. In this case, the area of a rectangle is determined as the sum of the areas of its constituent triangles.

Area of a parallelogram

Designations:

S is the required area,
a, b - side lengths,
h is the length of the height of a given parallelogram,
d1, d2 - lengths of two diagonals,
α is the angle between the sides,
γ is the angle between the diagonals.

If a, h are known, then the required area is determined by multiplying the lengths of the side and the height lowered to this side: S=a h
If a, b, α are known, then the area of the parallelogram is determined by multiplying the lengths of the sides of the parallelogram and the sine of the angle between these sides: S=a b sin α
If d 1 , d 2 , γ are known, then the area of the parallelogram is determined as half the product of the lengths of the diagonals and the sine of the angle between these diagonals: S=(d 1 d 2 sinγ)/2

Area of a rhombus

Designations:

S is the required area,
a - side length,
h - height length,
α is the smaller angle between the two sides,
d1, d2 - lengths of two diagonals.

If a, h are known, then the area of the rhombus is determined by multiplying the length of the side by the length of the height that is lowered to this side: S=a h
If a, α are known, then the area of the rhombus is determined by multiplying the square of the side length by the sine of the angle between the sides: S=a 2 sin α
If d 1 and d 2 are known, then the required area is determined as half the product of the lengths of the diagonals of the rhombus: S=(d 1 d 2)/2

Area of trapezoid

Designations:

If a, b, c, d are known, then the required area is determined by the formula: S= (a+b) /2 *√.
With known a, b, h, the required area is determined as the product of half the sum of the bases and the height of the trapezoid: S=(a+b)/2 h

Area of a convex quadrilateral

Designations:

If d 1 , d 2 , α are known, then the area of a convex quadrilateral is determined as half the product of the diagonals of the quadrilateral, multiplied by the sine of the angle between these diagonals: S=(d 1 · d 2 · sin α)/2
For known p, r, the area of a convex quadrilateral is determined as the product of the semi-perimeter of the quadrilateral and the radius of the circle inscribed in this quadrilateral: S=p r
If a, b, c, d, θ are known, then the area of a convex quadrilateral is determined as the square root of the product of the difference in the semi-perimeter and the length of each side minus the product of the lengths of all sides and the square of the cosine of half the sum of two opposite angles: S 2 = (p - a )(p - b)(p - c)(p - d) - abcd cos 2 ((α+β)/2)

Area of a circle

Designations:

If r is known, then the required area is determined as the product of the number π and the squared radius: S=π r 2

If d is known, then the area of the circle is determined as the product of the number π by the square of the diameter divided by four: S=(π d 2)/4

Area of a complex figure

Complex ones can be broken down into simple geometric shapes. The area of a complex figure is defined as the sum or difference of its component areas. Consider, for example, a ring.

Designation:

S - ring area,
R, r - radii of the outer circle and inner circle, respectively,
D, d are the diameters of the outer and inner circles, respectively.

In order to find the area of the ring, you need to subtract the area from the area of the larger circle smaller circle. S = S1-S2 = πR 2 -πr 2 = π (R 2 -r 2).

Thus, if R and r are known, then the area of the ring is determined as the difference in the squares of the radii of the outer and inner circles, multiplied by pi: S=π(R 2 -r 2).

If D and d are known, then the area of the ring is determined as a quarter of the difference in the squares of the diameters of the outer and inner circles, multiplied by pi: S= (1/4)(D 2 -d 2) π.

Patch area

Let's assume that inside one square (A) there is another (B) (of a smaller size), and we need to find the shaded cavity between the figures "A" and "B". Let's say, the "frame" of a small square. For this:

Find the area of figure "A" (calculated using the formula for finding the area of a square).
Similarly, we find the area of figure "B".
Subtract area "B" from area "A". And thus we get the area of the shaded figure.

Now you know how to find the areas of different shapes.

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.

In contact with

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 not equal in the general case.

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

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 proven 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:

where B, G - the number of points located inside and on the entire broken line, respectively.

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,

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How to calculate the area of ​​different shapes. How to find the area of ​​a figure? Geometric shape of the room

Area of ​​a triangle

Designations:

Square area

Designations:

Area of ​​a rectangle

Designations:

Area of ​​a parallelogram

Designations:

Area of ​​a rhombus

Designations:

Area of ​​trapezoid

Designations:

Area of ​​a convex quadrilateral

Designations:

Area of ​​a circle

Designations:

Area of ​​a complex figure

Designation:

Patch area

Calculation formulas

Triangle

Quadrangle

Polygon

Circle

Units

Rectangle

Square

Rhombus

Trapezoid

Cylinder and parallelepiped

Ring

Polygon

Triangle area formulas

Square area formulas

Rectangle area formula

Parallelogram area formulas

Formulas for the area of ​​a rhombus

Trapezoid area formulas

How to calculate the area of different shapes. How to find the area of a figure? Geometric shape of the room

Area of a triangle

Area of a rectangle

Area of a parallelogram

Area of a rhombus

Area of trapezoid

Area of a convex quadrilateral

Area of a circle

Area of a complex figure

Formulas for the area of a rhombus