The square is the simplest figure in geometry. It is from this, the rectangle and square, that they begin to study this subject. The ability to solve problems with a square will help you master more complex material. This article will tell you how to find the diagonal of a square.

Solving geometric problems is interesting because they can be solved in several ways. Each method is interesting in its own way. The diagonal of a square is no exception, which can be found in direct and indirect ways.

How to find the diagonal of a square - formula

There is quite simple formula to find the diagonal of a square. It looks like this: a√2. a is the side of the square. Remember that all sides of a square are equal. So if you know the size of one side, you know the size of the other three sides. To find out the diagonal of a square, you need to multiply its side by the root of two.

Example 1: Find the diagonal of a square if it is known that its side is 5.

Solution: Substituting the value into the above formula, it is not difficult to guess that the diagonal will be equal to 5√2.

Example 2: Find the side of a square if it is known that its diagonal is 5√2.

Solution: The diagonal is indicated by a small Latin letter d. d = a√2. Therefore, to find the side knowing the diagonal, it is necessary to divide the diagonal value by the root of two. Having done this action, we find out the side of the square, which, in this case, is equal to 5.

How to find the diagonal of a square through a right triangle

If you draw a diagonal in a square, it is easy to see that two right triangle. Remember that a right triangle has one right angle. It consists of two legs (sides at an angle of 90 degrees) and a hypotenuse (opposite 90 -degree angle sides). Square of the hypotenuse equal to the sum squares of legs. In this case, the hypotenuse is the diagonal of our square. Since the legs are the sides of the square, the formula will have next view: d² = a² + a² = 2a². It follows that d = √2a² = a√2.

Example 3: Find the diagonal of a square if its side is 3.

Solution:

1. Add the squares of the sides, we get 18.
2. We count the root of 18 and get 3√2.

Despite the fact that the last method is longer and ultimately we end up with the formula from the first example, it is necessary to know it. In essence, this method is a proof of the formula for the diagonal of a square. This is the proof that can come in an exam or an Olympiad. Learn it well as it can help you in the above mentioned events.

Online calculator

Although solving such problems is not difficult, some students may forget the formula. For such cases there is online calculator, which allows you to find the correct answer based on what is given in the problem. To use this service, follow the link.

1. Scroll down the page and you will find the subtitle “find the diagonal of a square given the side.
2. Below this subheading there will be a formula, after looking at which you will not need a calculator.
3. But still, if you are not sure, enter the value of the length of the square in the field, and then click the “calculate” button.
4. The calculator will give you the correct answer in 1 second.

Now, knowing several ways to solve a problem on this topic, you won’t have to leaf through a math book looking for the right formula, but simply use an online calculator or the examples given above.

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Definition.

Rectangle is a quadrilateral in which two opposite sides are equal and all four angles are equal.

The rectangles differ from each other only in the ratio of the long side to the short side, but all four corners are right, that is, 90 degrees.

The long side of a rectangle is called rectangle length, and the short one - width of the rectangle.

The sides of a rectangle are also its heights.

Basic properties of a rectangle

A rectangle can be a parallelogram, a square or a rhombus.

1. The opposite sides of the rectangle have the same length, that is, they are equal:

AB = CD, BC = AD

2. Opposite sides of the rectangle are parallel:

3. The adjacent sides of a rectangle are always perpendicular:

AB ┴ BC, BC ┴ CD, CD ┴ AD, AD ┴ AB

4. All four corners of the rectangle are straight:

∠ABC = ∠BCD = ∠CDA = ∠DAB = 90°

5. The sum of the angles of a rectangle is 360 degrees:

∠ABC + ∠BCD + ∠CDA + ∠DAB = 360°

6. The diagonals of a rectangle have the same length:

7. The sum of the squares of the diagonal of a rectangle is equal to the sum of the squares of the sides:

2d 2 = 2a 2 + 2b 2

8. Each diagonal of a rectangle divides the rectangle into two identical figures, namely right triangles.

9. The diagonals of the rectangle intersect and are divided in half at the intersection point:

		AO=BO=CO=DO=	d
			2

10. The point of intersection of the diagonals is called the center of the rectangle and is also the center of the circumcircle

11. The diagonal of a rectangle is the diameter of the circumcircle

12. You can always describe a circle around a rectangle, since the sum of opposite angles is equal to 180 degrees:

∠ABC = ∠CDA = 180° ∠BCD = ∠DAB = 180°

13. A circle cannot be inscribed in a rectangle whose length is not equal to its width, since the sums of the opposite sides are not equal to each other (a circle can only be inscribed in special case rectangle - square).

Sides of a rectangle

Definition.

Rectangle length call the length more long pair its sides. Rectangle width is the length of the shorter pair of its sides.

Formulas for determining the lengths of the sides of a rectangle

1. Formula for the side of a rectangle (length and width of the rectangle) through the diagonal and the other side:

a = √ d 2 - b 2

b = √ d 2 - a 2

2. Formula for the side of a rectangle (length and width of the rectangle) through the area and the other side:

b = dcos	β
	2

Diagonal of a rectangle

Definition.

Diagonal rectangle Any segment connecting two vertices of opposite corners of a rectangle is called.

Formulas for determining the length of the diagonal of a rectangle

1. Formula for the diagonal of a rectangle using two sides of the rectangle (via the Pythagorean theorem):

d = √ a 2 + b 2

2. Formula for the diagonal of a rectangle using the area and any side:

4. Formula for the diagonal of a rectangle in terms of the radius of the circumscribed circle:

d = 2R

5. Formula for the diagonal of a rectangle in terms of the diameter of the circumscribed circle:

d = D o

6. Formula for the diagonal of a rectangle using the sine of the angle adjacent to the diagonal and the length of the side opposite to this angle:

8. Formula for the diagonal of a rectangle through sine acute angle between the diagonals and the area of ​​the rectangle

d = √2S: sin β

Perimeter of a rectangle

Definition.

Perimeter of a rectangle is the sum of the lengths of all sides of a rectangle.

Formulas for determining the length of the perimeter of a rectangle

1. Formula for the perimeter of a rectangle using two sides of the rectangle:

P = 2a + 2b

P = 2(a + b)

2. Formula for the perimeter of a rectangle using area and any side:

P=	2S + 2a 2	=	2S + 2b 2
	a		b

3. Formula for the perimeter of a rectangle using the diagonal and any side:

P = 2(a + √ d 2 - a 2) = 2(b + √ d 2 - b 2)

4. Formula for the perimeter of a rectangle using the radius of the circumcircle and any side:

P = 2(a + √4R 2 - a 2) = 2(b + √4R 2 - b 2)

5. Formula for the perimeter of a rectangle using the diameter of the circumscribed circle and any side:

P = 2(a + √D o 2 - a 2) = 2(b + √D o 2 - b 2)

Area of ​​a rectangle

Definition.

Area of ​​a rectangle called the space limited by the sides of the rectangle, that is, within the perimeter of the rectangle.

Formulas for determining the area of ​​a rectangle

1. Formula for the area of ​​a rectangle using two sides:

S = a b

2. Formula for the area of ​​a rectangle using the perimeter and any side:

5. Formula for the area of ​​a rectangle using the radius of the circumscribed circle and any side:

S = a √4R 2 - a 2= b √4R 2 - b 2

6. Formula for the area of ​​a rectangle using the diameter of the circumcircle and any side:

S = a √D o 2 - a 2= b √D o 2 - b 2

Circle circumscribed around a rectangle

Definition.

A circle circumscribed around a rectangle is a circle passing through the four vertices of a rectangle, the center of which lies at the intersection of the diagonals of the rectangle.

Formulas for determining the radius of a circle circumscribed around a rectangle

1. Formula for the radius of a circle circumscribed around a rectangle through two sides:

Instructions

Determine the length of the diagonal of a rectangle with sides 3 and 4 cm.

Find the sum of the squares of the sides of the rectangle 32 + 42 = 9 + 16 = 25.

Extract from the result Square root– diagonal length 5 cm.

Video on the topic

note

The diagonals of a rectangle are equal. If the length of one is found, then the length of the second will be exactly the same.

Sources:

• how to find the length of the diagonal in a rectangle

A square is a beautiful and simple flat geometric figure. This is a rectangle with equal sides. How to find diagonal square, if the length of its side is known?

Instructions

diagonal length square equal to the length of its side multiplied by two.

Video on the topic

Helpful advice

If the accuracy of the mathematical result is not very important, then instead of the root of two, you can use its approximate value of 1.41.

Tip 6: How to find the diagonal of a parallelogram given the sides

A parallelogram is a quadrilateral whose opposite sides are parallel. The straight lines connecting its opposite angles are called diagonals. Their length depends not only on the lengths of the sides of the figure, but also on the values ​​of the angles at the vertices of this polygon, therefore, without knowing at least one of the angles, it is possible to calculate the lengths of the diagonals only in exceptional cases. These are special cases of a parallelogram - a square and a rectangle.

Instructions

If the lengths of all sides of a parallelogram are the same (a), then this figure can also be called a square. The values ​​of all its angles are 90°, and the lengths of the diagonals (L) are the same and can be calculated using the Pythagorean theorem for a right triangle. Multiply the length of the side by the root of two - the result will be the length of each of its diagonals: L=a*√2.

If it is known about a parallelogram that it is a rectangle with the specified length (a) and width (b), then in this case the lengths of the diagonals (L) will be equal. And here, too, use the Pythagorean theorem for a triangle in which the hypotenuse is the diagonal, and the legs are two adjacent sides of the quadrilateral. Calculate the required value by extracting the root of the squared and rectangle: L=√(a²+b²).

For all other cases, knowledge of the side lengths alone is sufficient only for a value that includes the lengths of both diagonals at once - the sum of their squares, by definition, is equal to twice the sum of the squares of the side lengths. If the angle between them (γ) is also known for the lengths of two adjacent sides of the parallelogram (a and b), then this will allow us to calculate the lengths of each segment connecting opposite angles. Find the length of the diagonal (L₁) lying opposite the known angle using the cosine theorem - add the squares of the lengths of adjacent sides, subtract the product of the same lengths by the cosine of the angle between them from the result, and extract the square root from the resulting value: L₁ = √(a²+b² -2*a*b*cos(γ)). To find the length of the other diagonal (L₂), you can use the property of a parallelogram given at the beginning of this step - double the sum of the squares of the lengths of the two sides, subtract the square of the already calculated diagonal from the result, and take the root from the resulting value. IN general view this formula can be written as follows: L₂ = √(a²+b²- L₁²) = √(a²+b²-(a²+b²-2*a*b*cos(γ))) = √(a²+b²-a²-b² +2*a*b*cos(γ)) = √(2*a*b*cos(γ)).

Sources:

• how to find the length of the diagonal of a parallelogram

We can call it a parallelogram whose diagonals bisect the angles at the vertices of the figure. In addition to this property of the diagonal rhombus are notable for the fact that they are axes of symmetry of a polygon, intersect only at right angles, and a single common point divides each of them into two equal segments. These properties make it easy to calculate the length of one of the diagonals if the length of the other and some other parameter of the figure are known - side size, angle at one of the vertices, area, etc.

Instructions

If, in addition to the length of one of (l), it is known about the quadrilateral in question that it is a special case rhombus- square, no calculations will have to be made. In this case, the lengths of both diagonals - simply equate the desired value (L) to the known one: L=l.

Knowing the side length rhombus(a) in addition to the length of one of the diagonals (l) will allow the length of the other (L) by the Pythagorean theorem. This is because the two halves of the intersecting diagonals form a side rhombus right triangle. Half of the diagonals in it are legs, and the side is the hypotenuse, so the equality resulting from the Pythagorean theorem can be written as follows: a² = (l/2)² + (L/2)². For use in calculations, convert it to this form: L = √(4*a²-l²).

With a known value of one of the angles (α) rhombus and the length of one of the diagonals (l) to find the value of the other (L), consider the same right triangle. The tangent of half a known angle in it is the ratio of the length of the opposite leg - half of the diagonal l - to the adjacent one - half of the diagonal L: tan(α/2) = (l/2)/(L/2) = l/L. Therefore, for the desired value, use the formula L = l/tg(α/2).

If the problem conditions give the length of the perimeter (P) rhombus and the size of its diagonal (l), the formula for calculating the length of the second (L) can be reduced to the equality used in the second step. To do this, divide the perimeter by four and replace the side length in with this expression: L = √(4*(P/4)²-l²) = √(P²/4-l²).

Under initial conditions, in addition to the length of one of the diagonals (l), the area (S) of the figure can also be given. Then to calculate the length of the second diagonal rhombus(L) use a very simple algorithm - double the area and divide the resulting value by the length of the known diagonal: L = 2*S/l.

Content:

A diagonal is a line segment that connects two opposite vertices of a rectangle. A rectangle has two equal diagonals. If the sides of a rectangle are known, the diagonal can be found using the Pythagorean theorem because the diagonal divides the rectangle into two right triangles. If the sides are not given, but other quantities are known, such as area and perimeter or aspect ratio, you can find the sides of the rectangle and then use the Pythagorean theorem to calculate the diagonal.

Steps

1 On the sides

1. 1 Write down the Pythagorean theorem. Formula: a 2 + b 2 = c 2
2. 2 Substitute the values ​​of the sides into the formula. They are given in the problem or need to be measured. The side values ​​are substituted for a 3
   • In our example:
     4 2 + 3 2 = c 2 4

     2 By area and perimeter

     1. 1 Formula: S = l w (In the figure, instead of S, the designation A is used.)
     2. 2 This value is substituted for S 3 Rewrite the formula to isolate w 4 Write down the formula to calculate the perimeter of a rectangle. Formula: P = 2 (w + l)
     3. 5 Substitute the perimeter of the rectangle into the formula. This value is substituted for P 6 Divide both sides of the equation by 2. You will get the sum of the sides of the rectangle, namely w + l 7 Substitute the expression to calculate w 8 into the formula Get rid of the fraction. To do this, multiply both sides of the equation by l 9 Set the equation equal to 0. To do this, subtract the first-order variable term from both sides of the equation.
        • In our example:
          12 l = 35 + l 2 10 Order the terms of the equation. The first term will be the second-order variable term, then the first-order variable term, and then the free term. At the same time, do not forget about the signs (“plus” and “minus”) that appear in front of the members. Note that the equation will be written as a quadratic equation.
          • In our example 0 = 35 + l 2 − 12 l 11
            • In our example, the equation is 0 = l 2 − 12 l + 35 12 Find l 13 Write down the Pythagorean theorem. Formula: a 2 + b 2 = c 2
              • Use the Pythagorean theorem because each diagonal of a rectangle divides it into two equal right triangles. Moreover, the sides of the rectangle are the legs of the triangle, and the diagonal of the rectangle is the hypotenuse of the triangle.
            • 14 These values ​​are substituted for a 15 Square the length and width, and then add the results. Remember that when you square a number, it multiplies by itself.
              • In our example:
                5 2 + 7 2 = c 2 16 Take the square root of both sides of the equation. Use a calculator to quickly find the square root. You can also use an online calculator. You will find c

                3 By area and aspect ratio

                1. 1 Write down an equation characterizing the ratio of the sides. Isolate l 2 Write down the formula to calculate the area of ​​a rectangle. Formula: S = l w (In the figure, instead of S, the designation A is used.)
                   • This method is also applicable when the perimeter of the rectangle is known, but then you need to use the formula to calculate the perimeter, not the area. Formula for calculating the perimeter of a rectangle: P = 2 (w + l)
                2. 3 Substitute the area of ​​the rectangle into the formula. This value is substituted for S 4 In the formula, substitute an expression characterizing the relationship of the parties. In the case of a rectangle, you can substitute an expression to calculate l 5 Write it down quadratic equation. To do this, open the brackets and set the equation equal to zero.
                   • In our example:
                     35 = w(w+2)6 Factor the quadratic equation. To obtain detailed instructions, read.
                     • In our example, the equation is 0 = w 2 − 12 w + 35 7 Find w 8 Substitute the found width (or length) into the equation characterizing the aspect ratio. This way you can find the other side of the rectangle.
                       • For example, if you calculate that the width of a rectangle is 5 cm and the aspect ratio is given by the equation l = w + 2 9 Write down the Pythagorean theorem. Formula: a 2 + b 2 = c 2
                         • Use the Pythagorean theorem because each diagonal of a rectangle divides it into two equal right triangles. Moreover, the sides of the rectangle are the legs of the triangle, and the diagonal of the rectangle is the hypotenuse of the triangle.
                       • 10 Substitute the length and width values ​​into the formula. These values ​​are substituted for a 11 Square the length and width, and then add the results. Remember that when you square a number, it multiplies by itself.
                         • In our example:
                           5 2 + 7 2 = c 2 12 Take the square root of both sides of the equation. Use a calculator to quickly find the square root. You can also use an online calculator. You will find c (displaystyle c), that is, the hypotenuse of the triangle, and therefore the diagonal of the rectangle.
                           • In our example:
                             74 = c 2 (displaystyle 74=c^(2))
                             74 = c 2 (displaystyle (sqrt (74))=(sqrt (c^(2))))
                             8 , 6024 = c (displaystyle 8,6024=c)
                             Thus, the diagonal of a rectangle whose length is 2 cm greater than its width and whose area is 35 cm 2 is approximately 8.6 cm.