Area of figures on checkered paper. Complete instructions (2020)
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. where S is the area of the triangle,
- lengths of the sides of the triangle,
- height of the triangle,
- the angle between the sides and,
- radius of the inscribed circle,
R - radius of the circumscribed 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 where S is the area of the square,
- length of the side of the square,
- length of the diagonal of the square.
Rectangle area formula
- Area of a rectangle equal to the product of the lengths of its two adjacent sides
where S is the area of the rectangle,
- lengths of the sides of the 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
where S is the area of the parallelogram,
- lengths of the sides of the parallelogram,
- length of parallelogram height,
- the angle between the sides of the parallelogram.
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. where S is the area of the rhombus,
- length of the side of the rhombus,
- length of the height of the rhombus,
- the angle between the sides of the rhombus,
1, 2 - lengths of 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,
Maintaining your privacy is important to us. For this reason, we have developed a Privacy Policy that describes how we use and store your information. Please review our privacy practices and let us know if you have any questions.
Collection and use of personal information
Personal information refers to data that can be used to identify or contact a specific person.
You may be asked to provide your personal information at any time when you contact us.
Below are some examples of the types of personal information we may collect and how we may use such information.
What personal information do we collect:
- When you submit an application on the site, we may collect various information, including your name, telephone number, email address, etc.
How we use your personal information:
- The personal information we collect allows us to contact you with unique offers, promotions and other events and upcoming events.
- From time to time, we may use your personal information to send important notices and communications.
- We may also use personal information for internal purposes, such as conducting audits, data analysis and various research in order to improve the services we provide and provide you with recommendations regarding our services.
- If you participate in a prize draw, contest or similar promotion, we may use the information you provide to administer such programs.
Disclosure of information to third parties
We do not disclose the information received from you to third parties.
Exceptions:
- If necessary - in accordance with the law, judicial procedure, in legal proceedings, and/or on the basis of public requests or requests from government bodies in the Russian Federation - to disclose your personal information. We may also disclose information about you if we determine that such disclosure is necessary or appropriate for security, law enforcement, or other public importance purposes.
- In the event of a reorganization, merger, or sale, we may transfer the personal information we collect to the applicable successor third party.
Protection of personal information
We take precautions - including administrative, technical and physical - to protect your personal information from loss, theft, and misuse, as well as unauthorized access, disclosure, alteration and destruction.
Respecting your privacy at the company level
To ensure that your personal information is secure, we communicate privacy and security standards to our employees and strictly enforce privacy practices.
If you are planning to do the renovation yourself, then you will need to make an estimate for construction and finishing materials. To do this, you will need to calculate the area of the room in which you plan to carry out renovation work. The main assistant in this is a specially developed formula. The area of the room, namely its calculation, will allow you to save a lot of money on building materials and direct the freed-up financial resources in a more appropriate direction.
Geometric shape of the room
The formula for calculating the area of a room directly depends on its shape. The most typical for domestic buildings are rectangular and square rooms. However, during redevelopment, the standard form may be distorted. The rooms are:
- Rectangular.
- Square.
- Complex configuration (for example, round).
- With niches and projections.
Each of them has its own calculation features, but, as a rule, the same formula is used. The area of a room of any shape and size, one way or another, can be calculated.
Rectangular or square room
To calculate the area of a rectangular or square room, just remember your school geometry lessons. Therefore, it should not be difficult for you to determine the area of the room. The calculation formula looks like:
S rooms=A*B, where
A is the length of the room.
B is the width of the room.
To measure these values you will need a regular tape measure. To get the most accurate calculations, it is worth measuring the wall on both sides. If the values do not agree, take the average of the resulting data as a basis. But remember that any calculations have their own errors, so the material should be purchased with a reserve.
A room with a complex configuration
If your room does not fit the definition of “typical”, i.e. has the shape of a circle, triangle, polygon, then you may need a different formula for calculations. You can try to roughly divide the area of a room with this characteristic into rectangular elements and make calculations using the standard method. If you do not have this opportunity, then use the following methods:
- Formula for finding the area of a circle:
S room=π*R 2, where
R is the radius of the room.
- Formula for finding the area of a triangle:
S room = √ (P(P - A) x (P - B) x (P - C)), where
P is the semi-perimeter of the triangle.
A, B, C are the lengths of its sides.
Hence P=A+B+C/2
If you have any difficulties during the calculation process, then it is better not to torture yourself and turn to professionals.
Area of the room with projections and niches
Often the walls are decorated with decorative elements in the form of various niches or projections. Also, their presence may be due to the need to hide some unaesthetic elements of your room. The presence of ledges or niches on your wall means that the calculation should be carried out in stages. Those. First, the area of a flat section of the wall is found, and then the area of the niche or protrusion is added to it.
The area of the wall is found by the formula:
S walls = P x C, where
P - perimeter
C - height
You also need to consider the presence of windows and doors. Their area must be subtracted from the resulting value.
Room with multi-level ceiling
A multi-level ceiling does not complicate the calculations as much as it seems at first glance. If it has a simple design, then calculations can be made based on the principle of finding the area of walls complicated by niches and projections.
However, if your ceiling design has arched and wave-like elements, then it is more appropriate to determine its area using the floor area. To do this you need:
- Find the dimensions of all straight sections of walls.
- Find the floor area.
- Multiply the length and height of the vertical sections.
- Sum the resulting value with the floor area.
Step-by-step instructions for determining the general
room area
- Clear the room of unnecessary things. During the measurement process, you will need free access to all areas of your room, so you need to get rid of anything that might interfere with this.
- Visually divide the room into regular and irregular shaped areas. If your room has a strictly square or rectangular shape, then you can skip this step.
- Make a random layout of the room. This drawing is needed so that all the data is always at hand. Also, it will not give you the opportunity to get confused in numerous measurements.
- Measurements must be taken several times. This is an important rule to avoid errors in calculations. Also, if you use it, make sure that the beam lies flat on the wall surface.
- Find the total area of the room. The formula for the total area of a room is to find the sum of all areas of individual sections of the room. Those. S total = S walls+S floor+S ceiling
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:
![](https://i1.wp.com/elhow.ru/images/articles/25/254/25452/inner/01.jpg)
- 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:
![](https://i1.wp.com/elhow.ru/images/articles/25/254/25452/inner/02.jpg)
- 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
![](https://i2.wp.com/elhow.ru/images/articles/25/254/25452/inner/03.jpg)
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
![](https://i1.wp.com/elhow.ru/images/articles/25/254/25452/inner/04.jpg)
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:
![](https://i1.wp.com/elhow.ru/images/articles/25/254/25452/inner/05.jpg)
- 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:
![](https://i2.wp.com/elhow.ru/images/articles/25/254/25452/inner/06.jpg)
- 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:
![](https://i0.wp.com/elhow.ru/images/articles/25/254/25452/inner/07.jpg)
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.
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.
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 |