In this article, we will learn about a summary of the most important
laws of geometric shapes, where we will learn about the laws of area
and perimeter for all shapes such as rhombus, square, circle,
rectangle, trapezoid, sphere, parallelogram and cuboid, and solve
explanatory questions on all these shapes, and we will learn about
the definition of perimeter and area. We will learn about all this and
more on the Math page in English
square rules
Square Perimeter formula
The perimeter of any closed shape is defined as the distance
surrounding the shape, and the perimeter of the square can be found
by calculating the sum of the lengths of its four sides, and because
all the sides of the square are equal in length, its perimeter can be
found using the following law
- When knowing the length of the side: perimeter of the square = 4 x side length
- When knowing the length of the diagonal: If the length of one of the diagonals of the square is known, its perimeter can be found using the following relationship
- Perimeter of the square = 2 x square root of 2 x length of the diagonal
Examples of calculating the perimeter of a square
- ? First example: A square with side length 5 cm, what is its perimeter
Solution: Perimeter of the square = 4 x side length, therefore:
Perimeter of square = 4 x 5 = 20 cm.
- ? Second example: If the perimeter of the square is 92 cm, what is the length of its side
Solution: Perimeter of the square = 4 x side length, so the side length
circumference of the square/4, and from it: side length = 92/4 = 23
cm.
- Third example: What is the perimeter of the square if one of its diagonals is 20 cm
Solution: Perimeter of the square = 2 x diameter x 2√. Hence: the
perimeter of the square = 2×20×2√= 2√40 cm
square area formula
Area can be defined as the amount of space covered by a two-
dimensional figure, and there are several laws by which the area of a
square can be found, which are:
- Using one of the side lengths: area = (side length)², and the length of any of the four sides can be substituted, because the lengths of the sides of the square are equal
- Using the length of the diagonal of the square: area = 1/2 x the square of the length of the diagonal,
Examples of calculating the area of a square
- ? First example: A square has two sides of 5 cm and 5 cm, what is its area
Solution: area of square = (side length)²; That is: the side × the side,
and from it: the area of the square = 5 × 5 = 25 cm²
- Second example: If the area of the square is 625 square meters, what is the side length of the square
Solution: The area of the square = side length x side length, and from
it: 625 = (side length)², and by taking the square root of both sides,
we will get:
- ? Third example: What is the area of a square whose diagonal length is 10 cm
Solution: The area of the square = 1/2 x the square of the diagonal length = 1/2 x 10² = 50 cm².
rectangle laws
Perimeter of a rectangle
- Perimeter of the rectangle = 2 x (length + width);
- When one of its dimensions and area is known, the perimeter of a rectangle can be calculated using the following law: From the relationship above, which relates the area of the rectangle and its perimeter, the area of the rectangle = (perimeter x length - 2 x square length)/2, or the area of the rectangle = (perimeter x width - 2 x squared width)/2, can be rearranged so that the perimeter of the rectangle = 2 x the area of the rectangle + 2 x the square of the length) / length, or the area of the rectangle = (2 x the area of rectangle + 2 x the square of the width) / width,
- When knowing the length of the diameter and one of its dimensions: perimeter of the rectangle = 2 x (length or width + (square of diameter - square of length or square of width) √),
Examples of calculating the perimeter of a rectangle
- Example 1: What is the perimeter of a rectangle whose length is 7 cm and width is 4 cm
Solution: Applying the law: Perimeter of a rectangle = 2 x (length +
width) = 2 x (7 + 4) = 22 cm
- The second example: a rectangle with a length of 12 cm and a width of 7 cm, what is its perimeter.
Solution: Applying the law: the perimeter of the rectangle = 2 x
(length + width) = 2 x (12 + 7) = 38 cm.
rectangle area
The area of a rectangle can be measured using several laws
according to specific cases, as follows
- When knowing the length and width, the area of a rectangle can be calculated using the following simple formula: Area of the rectangle = length x width,
- When knowing the diameter and one of the dimensions, the area of the rectangle can be calculated using the following formula: Area of the rectangle = length or width x square root of (diameter squared - length squared or width squared),
- When one of its dimensions and perimeter are known, the area of a rectangle can be calculated using the following formula: Area of a rectangle = (perimeter x length - 2 x square length)/2, or area of a rectangle = (perimeter x width -2 x squared width)/2.
- When knowing the smaller angle between the two diagonals, and the length of the diagonal, the area of a rectangle can be calculated using the following formula: The area of the rectangle = (square of the length of the diagonal x sin (the smallest angle between the two diagonals)/2),
Examples of calculating the area of a rectangle
- Example 1: Calculate the area of a rectangle of length 7 cm and width 4 cm.
Solution: According to the law: Area = Length x Width = 7 x 4 = 28 cm².
- Second example: If the area of a rectangular picture frame is 56 cm², and its length is 7 cm, what is its width.
Solution: According to the law: Area = Length x Width = 7 x Width
= 56 cm², including width = 8 cm.
Circuit laws
circumference of a circle
The circumference of a circle can be defined as the linear distance
surrounding the edges of a circle, and the circumference of a circle
is related to the diameter and radius usually according to the
following relationship:
- circumference = π×diameter = 2×π×radius,
- Example 1: A circle has a radius of 8 cm, what is its circumference
Solution: Substitute the value of the radius which is equal to r = 8 cm
in the circumference of the circle: circumference of the circle = 2 x
π x s = 2 x 3.14 x 8 = 50.24 cm
- Second example: What is the circumference of the roof of a circular tower, if the distance from the center of the tower to the outside is 10 m?
Solution: Substitute the value of the radius, which is equal to r = 10
m, in the law of the circumference of a circle: circumference of the
circle = 2 x π x s = 2 x 3.14 x 10 = 62.8 m, which is the
circumference of the circular surface from the outside.
Circle area
As for the area of a circle, it is the amount of space that a circular
body occupies on a flat surface, and it can be expressed by one of
the following mathematical relations
- The area of the circle = π×radius²,
- Area of a circle = (π/4) x diameter², if the diameter of the circle is known.
- The area of a circle = the circumference of the circle² / (4×π), if the circumference of the circle is known.
Examples of the area of a circle
- Example 1: A circle has a radius of 3 cm, what is its area?
Solution: Substitute the value of the radius, which is n = 3 cm, into
the formula for the area of a circle: Area of the circle = π × radius² =
3.14 × (3)² = 28.26 cm².
- Second example: A circle has a diameter of 8 cm, what is its area
Substitute the value of the diameter, which is equal to: length of the
diameter = 8 cm in the formula for the area of circle: area of circle =
(π/4) x length of diameter² = (3.14/4) x (8)² = 50.24 cm².
Example 3 What is the circumference of a semicircle with a radius of 2 cm
- Solution: Substitute the value of the radius, which is 2 cm, into the formula for circumference of a semicircle = radius x (π+2),
- Hence the circumference of the semicircle = 2 (3.14 + 2) = 10.28 cm.
ball circumference
The law of the circumference of the ball
The sphere is defined as a symmetrical three-dimensional, circular
body, and the line connecting the center and the boundaries of the
circle is called the radius, and the longest straight line passing
through the center of the sphere is called the diameter of the sphere,
and equal to twice the length of the radius of the sphere. To find the
circumference of the sphere, the following formula is used:
circumference of the sphere = 2 x π x radius,
There are several other laws of football, namely:
- Diameter = 2 x radius
- Sphere's surface area = 4 × π × radius 3
- Volume of the sphere = 4/3 x π x radius 3.
The relationship between the circumference of a circle and the circumference of a sphere
The relationship between the circumference of the circle and the
circumference of the sphere can be clarified through the following
example: A circle with a diameter of 8.5 cm, what is its
circumference? To solve this question, the following steps are to be
followed: Since the circumference of the circle is the same as the
circumference of the sphere, the circumference of the circle is equal
to: diameter × π and therefore the circumference of the circle = 3.14
x 8.5 so the circumference of the circle = 26.69 cm, rounded to 26.7
cm. Note: π or pi is a mathematical constant that relates the
circumference and diameter of a circle, which is an irrational
number so it does not have a decimal representation, and it is worth
noting that most people use 3.14 or 3.14159 in arithmetic, or
sometimes it is rounded by the fraction 7/22
Sphere surface area formula
The formula for the surface area law of a sphere was discovered two
thousand years ago by the Greek philosopher Archimedes, who
found that the surface area of a sphere is equal to the area of the
curved wall of the smallest cylinder that can contain the sphere, and
that the surface area of a sphere is four times the area of a circle of
the same radius (the area of the circle). = π×radius²), and
accordingly, the surface area of a sphere can be calculated using the following law:
- Sphere's surface area = 4×π×radius²; whereas:
- π is a scalar constant of 3.14 or 22/7.
The curved surface area of the hemisphere can also be found through
the following law: The curved surface area of the hemisphere = ½ ×
4 × π × radius² = 2 × π × radius², and because the hemisphere
consists of two surfaces; The first is the curved part and the other is
the circular flat base, finding the total area of the hemisphere surface
requires adding the curved surface area with the circular flat base area, to result in that
- Total surface area of a hemisphere = 2 x π x radius² + π x radius² = 3 x π x radius².
Examples of calculating the surface area of a sphere
- Example) 1: Find the surface area of a sphere with a radius of 5.5 m )
Solution: Substitute the value of the radius equal to 5.5 m in the
formula for the surface area of the sphere = 4 × π × radius², from
which it follows that: The surface area of the sphere = 4 × 3.14 ×
5.5² = 379.94 m²
- Second example: a sphere has an area of 2464 cm², find its
- radius to two decimal places
Solution: Substitute the value of the sphere’s area of 2464 cm² into the formula for the surface area of the sphere = 4 × π × radius², from which it is obtained that: 2464 = 4 × 3.14 × radius², and by dividing both sides by 4, it results in: 616 = 3.14 × radius² , then by dividing both sides by 3.14, it results in: radius ² = 196.17, then by taking the square root of both sides, it results in: radius = 14.00 cm.
- Example 3: A hemisphere has a radius of 8.3 cm, find the surface area of the hemisphere without the circular base
Solution: Substitute the value of the radius of 8.3 cm into the formula for the curved surface area of a hemisphere = 2 x π x radius², and from that it follows that: the curved surface area of a hemisphere = 2 x 3.14 x 8.3² = 432.6 cm².
- Fourth example: Find the value of the area of the surface of a sphere with a radius of 6 cm
Solution: Substitute the value of the radius of 6 cm into the formula for the surface area of the sphere = 4 × π × radius², from which it is obtained that: the surface area of the sphere = 4 × 3.14 × 6² = 452.16 cm²
Parallelogram laws
The area of a parallelogram
The area of a parallelogram can be calculated in several ways
The first method: This method is used if the base length and height are known, and the law is:
- Area = base length x height, and it is worth noting that the height of the parallelogram must be perpendicular to the base, and it represents the length of the straight line connecting the base and the opposite side. adjacent to it or complementary to it)
The second method: This method is used if the two sides of a parallelogram and the included angle between them are known, and the law is:
- Area = 1st side x 2nd side x sin (an angle of the parallelogram), where all two adjacent angles are supplementary in the parallelogram; So their sum is 180°, and sin (angle) = sin (-180); Any sine of an angle is its supplement.
The third method: This method is used if the length of the diagonal of the parallelogram and the angle between them are known, and the law is:
- Area = 1/2 x (first diameter x second diameter x sin (the angle between the two diagonals)),
- Example 1: A parallelogram with base length 10 and height 8 What is its area?
Solution: Applying the parallelogram area law, the area = 8 x 10 = 80 square units.
- Second example: A parallelogram with base length 3 and height 6 What is its area?
Solution: By applying the parallelogram area law, the area = 6 x 3 = 18 square units.
Parallelogram perimeter laws
The perimeter of a geometric figure generally expresses the distance surrounding it from the outside, and the perimeter of a parallelogram, like other geometric shapes, is equal to the sum of the lengths of its four sides, so it can be expressed using the following law: Perimeter of a parallelogram (ABC D) = A + B + C + D , or perimeter of a parallelogram (ab c d) = 2 x (length of the base or upper side + length of one of the sides); Where a, b, c, d are the lengths of the sides of the parallelogram, this can be illustrated using the following example:
Example of a perimeter of a parallelogram
- A parallelogram has sides: 10 cm and 6 cm, what is its perimeter?
Since all opposite sides of a parallelogram are equal, The lengths of the other two sides are: 10 cm and 6 cm, so the perimeter of the parallelogram = 10 + 6 + 10 + 6 = 32 cm
Various examples of parallelograms
- First example: a parallelogram with an area of 24 square cm², and a base length of 4 cm, find its height.
Solution: By applying the law of the area of a parallelogram, the area = base x height = 24 = 4 x height, so height = 6 cm.
Perimeter of a Cuboid
A cuboid is a three-dimensional solid; Since the definition of the perimeter is the line or thread that wraps around the two-dimensional shape such as square, rectangle, circle, triangle, and parallelogram; We conclude from this that it is not possible to calculate the perimeter of a cuboid at all, and the calculation of the perimeter can be replaced by calculating the lateral area, that is, calculating the area of each face of the cuboid separately, and the total area of it can also be calculated by adding the side areas to each other algebraically, and it is a unit In both cases, the area is the squared units of length—a square meter, or a square cm, and so on.
The lateral area of a cuboid can also be calculated as:
- Lateral area = perimeter of the base x height
- Base circumference = base length + base width
- Total area = lateral area + sum of the areas of the two bases
- The sum of the areas of the two bases = the area of the first base + the area of the second base, if any
- Area of the first base = length x width
It should be noted that some rectangular parallelepipeds have one base, so this must be taken into account when applying the law.
Area of a rectangular prism
A cuboid has six faces, and its area can be calculated by finding the sum of the areas of these faces. and height, as follows:
- Total area of a cuboid = (2 x length x width) + (2 x width x height) + (2 x length x height)
It should be noted here that the number was multiplied by 2; Because every two opposite sides of a cuboid are congruent; That is, they have the same area, and the area is measured in square units
Lateral area of a cuboid
The lateral area of a cuboid is defined as the sum of the areas of all faces except the upper and lower sides, and the following formula can be used to find the lateral area
- The lateral area of the cuboid = the area of the four sides of the cuboid = 2 x (width x height) + 2 x (height x length), and by taking out the height as a common factor, and ordering the terms, the lateral area is equal to: lateral area of the cuboid = 2 x height x (length + width),
Examples of calculating the area of a cuboid
First example: What is the total area of a cuboid box with length 6 cm, width 5 cm, and height 4 cm?
- Solution: Area of the cuboid = 2 x (length x width + width x height + height x length) = 2 x (6 x 5 + 5 x 4 + 4 x 6) = 2 x (30 + 20 + 24) = 2 x 74 = 148 cm 2.
Second example: What is the total area of a cuboid whose length is 20 cm, width is 12 cm, and height is 9 cm?
- Solution: Area of the cuboid = 2 x (length x width + width x height + length x height) = 2 x ((20 x 12) + (12 x 9) + (20 x 9) = 2 x (240 + 108 + 180) = 2 x 528 = 1056 cm 2.
Third example: What is the lateral area of a cuboid if its length is 12 cm, its width is 13 cm, and its height is 15 cm?
- Solution: The lateral area of a cuboid = 2 x height x (length + width) = 2 x 15 x (12 + 13) = 750 cm 2.
Triangle Laws
Triangle area formula.
The term area is given to the space confined within the boundaries of a body or a flat or two-dimensional shape, and the unit of area measurement is the square unit of side length, and the unit of measurement m 2 is the standard unit of area measurement, and the area of a triangle can be measured using the following area law
- Area of a triangle = 0.5 x base x height
The previous law is used to calculate the area of all types of triangles. If there is a triangle with a base length of 20 m and a height of 12 m, then according to the aforementioned law, the area of the triangle is = 0.5 x 20 x 12 = 120 m 2, and the law of the area of the triangle can be reached using the law of the area of the rectangle, which states The area of the rectangle is equal to the base multiplied by the height, and the diagonal of the rectangle - which is the distance between the two opposite angles in the rectangle - separates the rectangle into two triangles of equal area, the area of each of them is equal to half the area of the rectangle, meaning that the area of the triangle = 5.0 x base x Height
The area of a triangle can be found using specific data. If the measure of the length of two adjacent sides of the triangle is known, in addition to the measure of the included angle between them, it is possible to calculate the area of the triangle using the following formula:
- The area of the triangle = 0.5 x the first adjacent x the second adjacent x sin(x)
- Sin(x) = sine of the angle x between the first side and the second side.
It is also possible to calculate the area of a triangle if the measure of one of its sides is available in addition to the measures of both angles adjacent to this side, and the height of the triangle is not available, through the following law:
- Area of a triangle = side 2 * sin(x) * sin(y)/(2 * (sin(x+y))
- Area of a triangle = 0.5 * height * base
As for an equilateral triangle, the general formula for the area law of a triangle is as follows
- Area of an equilateral triangle = a²*((3^½)/4)
Various examples of calculating the area of a triangle
- An acute-angled triangle has a base length of 13 inches and a height of 5 inches. Find its area. Solution: Using the area law of a triangle:
Area = 0.5 * Base * Height Area = 0.5 * 13 * 5 = 32.5 inches 2.
- The second example: a right-angled triangle, with a base length of 7 cm, and a height of 8 cm, find its area. Solution: Using the area law of a triangle,
Area = 0.5 * base * height, area = 0.5 * 7 * 8 = 28 cm 2.
Perimeter of the triangle
The perimeter can be defined as the total length of the borders of the geometric figure that surround it from the outside. The perimeter is measured by a number of length units such as the meter (m), (cm), and (mm), and the perimeter of any triangle is equal to the sum of the lengths of its sides, and therefore:
- Perimeter of an equilateral triangle = 3 x a, where a is the length of one of the sides of the triangle.
- Perimeter of an isosceles triangle = 2 x a + b, where a is the length of one of the equal sides, and b is the length of the base of the triangle.
- Perimeter of a scalene triangle = a + b + c, where a, b, and c are the lengths of the three sides of the triangle.
- Perimeter of a right triangle = a+b+c = a+b+(a²+b²)√;
The previous law was obtained using the Pythagorean theorem, which can be used to find the perimeter of a right triangle in cases where one of its sides is not known, as follows:
The Pythagorean theorem states that the sum of the squares of the lengths of the two sides of a right angle is equal to the square of the length of the hypotenuse, that is: C² = A² + B², so the hypotenuse (C) = (A² + B²)√, and substituting in the law of the perimeter of a right triangle, the perimeter = A + B +(a²+b²)√.
Examples of calculating the perimeter of a triangle
- The first example: A triangular garden with the longest sides 90 m, 70 m, and 40 m, is intended to be surrounded by a fence. What is the length of the fence that is needed to enclose it?
Solution: The length of the fence = the perimeter of the triangle, and therefore the perimeter of the triangle = the sum of the lengths of its sides = 90 + 70 + 40 = 200 m.
- Second example: What is the perimeter of a triangle whose three sides are 5 cm, 4 cm, and 2 cm
Solution: Perimeter of a triangle = sum of the lengths of its sides = 5 + 4 + 2 = 11 cm.
- Third example: What is the perimeter of an equilateral triangle whose length of one of its three sides (a) is 10 cm
Solution: Perimeter of an equilateral triangle = 3 x a = 3 x 10 = 30 cm.
Rhombus Rules
rhombus circumference law
A rhombus is a flat shape that has four equal sides, and four angles whose measurements are not required to be 90 degrees. The perimeter of a rhombus is defined as the total distance that surrounds the outer shape. In general, the perimeter of a rhombus is given by the following relationships:
- Calculate the perimeter of a rhombus using the side length:
Perimeter of a rhombus = 4 x side length.
- Calculate the circumference of a rhombus using the length of the two diagonals:
Perimeter of the rhombus = 2× ((first diameter)²+(second diameter)²)√.
Examples of calculating the perimeter of a rhombus
- First example: What is the perimeter of a rhombus whose side length is 5 cm.
Solution: Apply the formula for the perimeter of a rhombus = 4 x side length = 4 x 5 = 20 cm.
- Second example: If the perimeter of the rhombus is 260 cm, find the length of its side.
Solution: By applying the law: perimeter of rhombus = 4 x side length, it results that side length = circumference of rhombus ÷ 4 = 4 / 260 = 65 cm.
Rhombus area law
The area of a rhombus, like the area of other geometric shapes, represents the inner region within its boundaries.
- Calculating the area in terms of height and length of one of the sides: The area of a rhombus can be calculated in terms of its height and the length of one of its sides using the following formula:
The area of the rhombus = height x length of the side, where the height of the rhombus is the perpendicular line segment connecting the two opposite sides of each other. Because all the sides of the rhombus are equal in length.
- Calculating the area in terms of the lengths of the two diagonals: the area of the rhombus can be calculated in terms of the lengths of the diagonals; Where the diagonals of a rhombus can be defined as the two straight segments connecting each pair of opposite angles, using the following law:
The area of the rhombus = ((first diameter x second diameter) ÷ 2),
- Calculating the area in terms of the length of a side and the measure of one of its angles: Through this method, the area of a rhombus can be calculated if the length of the side and the measure of one of its angles are known, and the law is:
The area of a rhombus = the square of the side length of the rhombus x the sine of one of the angles of the rhombus, .
Various examples of calculating the rhombus area
- First example: Calculate the area of a rhombus wooden plank if you know the length of one of its sides is 2 m, and the measure of one of its angles is 60 degrees.
The solution: by applying the rhombus area law in terms of the length of a side and the measure of one of its angles = (l)² x sin angle, and substituting the value of the side length and angle measure with the law. The area of the plank = (2 m)² x sac (60°) = 4 m² x s’ 60° = 4 m² x 0.866, so the area of the plank = 3.46 m²
- Second example: Calculate the area of a given rhombus if you know that its height is 6 cm, and the length of one of its sides is 2 cm.
The solution: By applying the law of the area of a rhombus in terms of height and its side length: area = height x side length, and substituting the value of the height and side length into the law, so that the area of the rhombus = 6 cm x 2 cm, so the area of the rhombus = 12 cm².
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