In many areas of everyday life, geometry helps people answer important questions and solve problems of life. At least 4 thousand years ago, this knowledge was already used, for example, in ancient Egypt for land management. And today, many professions, from fashion designers to architects, need basic geometric knowledge to know how to calculate the area.
Body surface and its area
This is a measure of how much space is inside a flat shape. In the general case, the surface area is the sum of all areas of geometric shapes covering the surface of an object. Calculating the surface area of the body is often required in everyday life, for example, to find out how much paint you need to buy to cover the wall, or slate to repair the roof of the house.
Since ancient times, people have learned to determine the area of flat geometric shapes using the grid method. It lies in the fact that a scaled grid of simple squares, for example, 1x1 cm, is applied to the measured figure. After that, you can easily calculate the square area by counting the number of grid squares inside the form. In this case, each square of the grid has a width of 1 cm and a height of 1 cm, and the area of this square of the grid is one square centimeter.
Using a grid to count squares in a shape is a very simple way to determine the area, but it cannot be used to determine the area of complex shapes. The area of such complex objects can be calculated using simple mathematical formulas. The simplest and most frequently used calculations in life are squares and rectangles, and you need to know how to calculate the area in meters.
Often, in reality, calculations can be more complicated. For example, a typical floor plan of a room may not consist of a simple rectangle or square. In this case, before calculating the total area, it is necessary to divide the measured complex surface into several simple geometric figures.
Simple rectangle calculation
If you look carefully around, you can see many examples of rectangles. By definition, a rectangle is a four-sided polygon whose angles are at a right angle of 90 degrees. Calculating the surface area of a rectangle’s body is a simple mathematical operation that is most often used by a person in everyday life. Why is it important to know the area formula? Many objects and furnishings surrounding a person have a rectangular shape: house, walls, floor, roof. And very often you need to know their area for construction or repair.
If the rectangle has a length b and a width h, we can find the area S by multiplying the width by its length. Therefore: S = bxh.
Example. How to calculate the area of a rectangle, if the side and width are known, for example, 4 cm long and 3 cm wide, then: S = 4x3 = 12.
Answer: 12 cm 2 .
A square is a kind of rectangle with equal angles and sides.
S = bxb = b 2 .
Example. If the square has sides of 3 cm, we can find S by squaring the value of the side. Therefore, we have: S = 3x3 = 9.
Answer: 9 cm 2 .
Parallelogram formulas
A parallelogram is a four-sided polygon with two pairs of parallel sides of the same length. By definition, a rectangle is also a type of parallelogram, but with equal angles. The parallelogram area is calculated in the same way as for a rectangle (height × width), but it is important to understand that height does not mean the length of the vertical sides, but the distance between the sides.
It can be seen from the figure that the height is the distance between two parallel sides of the parallelogram, located at right angles between them. S = ADxh. S = bxh, where AD = b is the base, h is the height.
Example. If the parallelogram has a base of 3 cm and a height of 2 cm, then the area S is equal to the product of the base by the height. Therefore, we have: S = 3x2 = 6.
Answer: 6 cm 2 .
Trapezoid base
Consider how to correctly calculate the area of the trapezoid. A trapezoid is a four-sided polygon with one pair of parallel sides. If two non-parallel sides have the same length, the shape is called an isosceles or ordinary trapezoid. If the non-parallel sides have different lengths, it is called non-isosceles. However, despite this additional difficulty in determining, the area of the irregular trapezoid can be calculated using a simple formula.
Measurements to calculate the area of the trapezoid:
- Align the straight edge of the protractor along the shorter of the two parallel sides.
- Use a protractor to draw a line perpendicular from the base of the trapezoid to the opposite parallel side.
- Measure the height distance using a ruler.
- Measure the length of the shorter parallel side.
- Measure the length of the longer parallel side.
- To find the area of the trapezoid, first you need to calculate the average value of its two parallel sides: (a + b) / 2.
- The area of the isosceles (or any) trapezoid is equal to the product of the average length of the base and top with height.
- Trapezoid area: S = 1/2 × h × (a + b).
It should be noted that the height of the trapezoid is always perpendicular to the base, just like the height of the parallelogram. Example: a = 3 cm, b = 5 cm, h = 4 cm. S = 4x (3 + 5) / 2 = 16.
Answer: 16 cm 2 .
Types of Triangles
A triangle is a polygon that has three sides and can be classified into the following types:
- An equilateral triangle has equal sides and equal angles.
- An isosceles triangle has two equal sides and two equal angles.
- A versatile triangle has three unequal sides and three unequal angles.
- A right triangle has one right angle of 90 degrees.
- An acute-angled triangle has all angles less than 90 degrees.
- An obtuse triangle has one angle greater than 90 degrees.
The area of any triangle is determined by the formulas.
1. How to calculate the area of a triangle, if the height and base of the triangle are known:
- S = 1⁄2 × a × h, where: h is the height, a is the base.
- S = 1⁄2xa × b × sinα, where: a, b are any two sides, α is the angle between them.
- S = p × r, where: p = (a + b + c) / 2 is the half-perimeter, a, b, c are three sides, r is the radius of the circle.
Equilateral Triangle Area:
- S = a 2 x √ 3 ⁄4, where a = b = c.
The area of the isosceles triangle:
- S = 1⁄4xbx√ (4a 2 -b 2 ).
2. How to calculate the area of a triangle if two sides are specified and the angle between them:
- S = 1⁄2xaxbxsinC = 1⁄2xbxcxsinA = 1⁄2xaxcxsinB.
Example 1: Find the S triangle, whose side is 14 cm and the height is 10 cm.
Solution: b = 14 cm, h = 10 cm, A = 1⁄21410 = 70
Answer: 70 cm 2 .
Example 2. Find the region of the triangle whose sides and the angle between them are defined as follows: a = 5 cm and b = 7 cm, C = 45 degrees.
Solution: Area of the triangle = 1⁄2xaxbxsin 45.
Area = 1⁄2 × 5 × 7 × 0.707 (since sin45 = 0.707)
Area = 1⁄2 × 24.745 = 12.3725
Answer: 12.3725 cm 2 .
Example 3. Find the area (in m 2 ) of an isosceles triangle whose sides are 10 m and the base is 12 m.
Solution: The area of an isosceles triangle is determined by:
A = 1⁄4xbx√ (4a 2 -b 2 ) A = 1⁄4x12x√ (4x (10) 2 - (12) 2 ) A = 48
Answer: 48 m 2 .
Example 4. Find the area of a triangle whose sides are 8, 9 and 11, respectively. All units are in meters (m).
Solution: Parties a = 8, b = 9 and c = 11. According to Heron’s formula, the area of a triangle can be determined by the following formula: A = √ (sx (sa) x (sb) x (sc)). First of all, we need to determine s, which is the semiperimeter of the triangle: s = 1⁄2x (a + b + c) = 1⁄2x (8 + 9 + 11) = 14.
Now, having inserted the half-perimeter value into the Heron formula, we can determine the area of the triangle: A = √ (sx (sa) x (sb) x (sc)). A = √ (14x (14-8) x (14-9) x (14-11)). A = √ (1260) = 35.50
Answer: 35.50 m 2 .
Diamond Area Measurement
A rhombus is a special type of parallelogram that has equal sides and equal opposite angles. The rhombus area can be determined using three methods.
1. The method of height of the base. First, select one side as the base, as they are the same length. Then determine the height - the perpendicular distance from the selected base to the opposite side.
- The area is a product of these two quantities and is determined by the formula: S = a × h, where: S is the area of the rhombus, h is the height of the rhombus, AB = BC = AD = DC = a = the side of the rhombus.
2. The method of diagonals. Another simple formula for the area of a rhombus is when the lengths of the diagonals are known. The area is half the product of the diagonals.
- As a formula: S = 1 / 2xACxBD, where: S is the area of the rhombus, AC is the large diagonal, BD is the smaller diagonal.
3. The use of trigonometry. In trigonometry, there is a convenient formula when the length of the side and any angle are known:
- S = a2 × sin α, where: S is the area of the rhombus, B = BC = AD = DC = a is the side of the rhombus, α is an acute angle, β is an obtuse angle.
Circle surface
A circle is a shape consisting of a closed curved line. Each part of the line is at the same distance from the center of the area, called the radius. Since ancient times, it is known how to calculate the area of a circle if a radius is specified. The area of the circle is calculated by the formula S = πxr 2 , where: S is the area of the circle,
π is the number pi (3.1415), r is the radius of the circle.
To find the area of a circle, perform the following steps. Record the given radius or diameter of the quantity as r or d, respectively. How to calculate the area of a circle if a diameter is specified? This is not difficult at all, you need to calculate the radius by dividing the diameter by 2, and multiply the data using a calculator or manually. The response will be in square units.
Task: Find the area of a circle with a radius of 10 cm.
Solution: We have a circle radius = 10 cm. Circle area = 3.1416 × 10 × 10 = 314.16.
Answer: 314.16 cm 2 .
Find the area of a circle with a diameter of 15 cm.
Solution: We have a circle diameter = 15 cm. Radius = 15/2 = 7.5 cm. Circle area = 3.14x7.5x7.5 = 176.625 = 176.63 (round up to 2 decimal places).
Answer: 176.63 cm 2 .
Simple geometric shapes of roofs.
Before performing roofing, you need to know how to calculate the roof area in order to determine how much material is required. It is always necessary to take its quantity with a margin and add at least 10 percent of the total roof area to account for construction waste.
Previously, before calculating, the roof scheme is divided into simple geometric figures, in our example these are two trapezoids and two triangles. How to calculate the roof area for trapezoidal elements? The area is calculated by the following formula: S = (a + b) xh / 2, where: a - width of the lower overhang - 10 m, b - width along the ridge - 7 m, h - height - 5 m.
For triangular elements, the formula is used: S = axh / 2, where: a - the slope width along the lower overhang is 7 m, h is the slope height - 3 m.
Measurement order:
- Measure the length, width and height of each geometric shape of the roof, including wind windows. This information may be available in the original floor plan of the building, or if the roof surface is relatively low and flat, you can measure it yourself. If the owner of the house himself cannot safely climb the roof, the calculation can be performed by external measurements of the building.
- Multiply the length and width of each triangular or trapezoidal plane separately.
- Calculate the area for symmetrical triangular planes by multiplying the length of the base of the triangle (the longest side) by its height (distance from the middle of the longest side to the opposite corner).
- Then divide the total by 2 to get the result in square meters. S = axh / 2 = 7x3 / 2 = 10.5 m 2 .
- Calculate the area for the trapezoid by multiplying the width of the lower overhang plus the width along the ridge to its height (distance from the middle of the longest side to the opposite corner).
- Then divide the total by 2 to get the result in square meters.
- Multiply the area by 0.1 to get a 10 percent premium for the stock of roofing material S = (a + b) xh / 2 = (10 + 7) * 5/2 = 42.5 m 2 .
- Put the squares of all the pieces together. S = 10.5 + 10.5 + 42.5 + 42.5 = 106 m 2 .
- The result is a total roof area of 106 m 2 , with a margin of 116 m 2 .
Instrumental measurements at home
To measure the area of the house, you will need tools to very accurately perform calculations, which can become the basis for repairs, sales, or home insurance. Before calculating the area, you need to take a tape measure, a pencil and a notebook on which to draw the simplest plan of the house plan. It can be taken from the developer’s passport or other project documents. With the last source you need to be careful, these numbers may not always be accurate, for example, some repair work may not be taken into account. Therefore, it would be more correct to measure the area yourself.
How to calculate the area of a house manually? If you need to measure the floor area manually, it is best to measure the external walls, not forgetting the various building recesses, utility rooms, upper floors, individual buildings or garages. When simple basic measurements are taken, the area is calculated by multiplying the length of the house by the width.
Depending on the shape of the construction plan, you may need to break it into simple geometric shapes. In this example, the house has 9 meters by 12 meters, giving us 108 square meters. The garage is 6 meters by 3 meters, which is 18 square meters, the total area is 126 square meters.
Pre-repair floor measurements
How to calculate the floor area before repair work, for example, linoleum replacement or painting? For a square or rectangular room, you will first need to measure the length and width of the room. Then multiply the length and width, we get the length x width = area. Thus, if a room is 3 meters wide and 5 meters long, the total area will be 15 square meters.
This measurement can be used in calculating the required amount of tile mortar, sealant, linoleum, which the owner plans to use for his project. To calculate the area for the selection of materials, as a rule, you need to add a 10% safety factor: just multiply the area by 1.1, and then round to an integer value.
In the example, when the total area is 15 m 2 , you will need to order an additional amount of tiles and mortar for 16.5 square meters. If the room is not rectangular, you need to divide it into two or more elementary geometric shapes in order to calculate the total area.
The calculator for the wrong figure
Very often the measured space has a very complex shape, which is not always possible to break into simple elements.
To simply determine such an area, you should use the SketchAndCalc online application. It is a square calculator for irregular shapes for any form of image. This is the only area calculator that can calculate from uploaded images, it has a unique function that allows the user to set the drawing scale of any image before drawing the perimeter. Thus, the angles or curves of an irregular figure are easily calculated.
Simply put, if there is an image that can be downloaded, or the address of a map to search for, you can calculate the area of the irregular shape no matter how complicated it is, just by drawing the perimeter of the area. The calculator can even summarize the calculations of several areas together by drawing layers. After calculating the first area, you can add a new drawing layer, which allows you to perform an unlimited number of area calculations.
Area calculator results are displayed in inches and meters, increasing its utility and eliminating the need for conversion. This, along with accurate drawing and zooming tools, ensures that the areas of each irregular shape are calculated accurately. It can also place regular polygon shapes with fixed angles and precise lines.
A tool with a limited pattern is snapped to common angles, and the length line can be edited manually using the keyboard. The application is useful if the measured area has a straight side or length. Another unique feature of SketchAndCalcTM is that it has an advanced curve drawing tool for irregular shapes. Some area calculator applications allow you to search on a map.
SketchAndCalc does this very accurately using longitude and latitude searches. Regardless of whether the measured area is located on agricultural land or at sea, the user will spend less time searching and more time calculating the area. This is a universal utility used in many industries, in construction, gardening. It is also used by enthusiasts to improve their home and house territory. A landscape calculator or land area calculator has also found its users among land surveyors. Now they know how to calculate the land area easily and quickly.
However, in addition to these general applications, many working in the field of education, medicine, science and research need to calculate the area of irregular shapes, such as cell membranes or other objects found in biology, and enjoy using this application.
To apply mathematics in everyday life, it is not enough to be able to count one plus one. An essential aspect of the environment is geometric structures, that is, the representation of everyday objects in a rectangular, square, round or triangular shape. And you need to be able to calculate the desired area.
In addition, geometric shapes are also used in the construction of diagrams, schemes, presentations. That is why it is so important to be able to do various calculations, including calculating the area.