How to find the height of a rhombus knowing the diagonals. What is the height of the rhombus according to the formula? How do the other heights of the rhombus relate to each other

A rhombus is a quadrilateral in which all sides are equal and opposite sides are parallel. This condition simplifies the formulas for determining the height - the perpendicular dropped from the corner to one of the sides. In a quadrilateral, from each corner, you can lower the heights to two sides. Consider how to find the heights of a rhombus, how they relate to each other.

How to find the height of a rhombus

Quadrilaterals are such figures in which angles can change with constant side lengths. Therefore, unlike a triangle, it is not enough to know the lengths of the sides of a quadrilateral; it is also necessary to indicate the dimensions of the corners or the height. For example, if the angles of a rhombus are 90°, then the result is a square. In this case, the height is the same as the side. Consider how to find the height of a rhombus at angles other than straight lines.

Determine the value of the two heights of the rhombus, lowered from one corner

We have a rhombus ABCD with AB//CD, BC//AD, AB = BC = CD = DA = a. The height h is the perpendicular dropped from the corner to the opposite side. Let us lower the height AH to the side BC, and lower the other height AH1 from the same angle to the side DC.

• Then the height AH = AB × sin∟B;
• Height AH1 = AD × sin∟D.

One of the properties of a rhombus is the equality of opposite angles, i.e. ∟B = ∟D. Since AB \u003d AD (all sides of the rhombus are all equal), then the height AH \u003d AH1. Similarly, one can prove that two heights dropped from any angle are equal.

How do the other heights of the rhombus relate to each other

Since opposite sides are parallel, the sum of the angles adjacent to one side is 180°. Therefore, the sines of all four angles are equal to each other:

• sin∟D = sin(180° - ∟D) = sin∟C = sin∟A = sin∟B.

Therefore, all the heights omitted from any angle of the rhombus are equal, and the side, angle and height are interconnected by a rigid relationship: h = a × sin∟A, where a is the length of any side, ∟A is any angle of the rhombus.

Knowing the diagonals, finding the height of a rhombus is easy. In that The Pythagorean theorem will help us. And although it touches right-angled triangles, they are also in the rhombus - they are formed by the intersection of two diagonals d1 and d2:

Imagine that diagonal 1 is 30 centimeters and diagonal 2 is 40 cm.

So our actions are:

We calculate the size of the side according to the Pythagorean theorem. Side BC is the hypotenuse (because it lies opposite an obtuse angle) of triangle BXD (X is the intersection of diagonals d1 and d2). So the size of this side squared is equal to the sum of the squares of the sides BX and XC. Their size is also known to us (the diagonals of the rhombus are divided in half by the intersection) - these are 20 and 15 centimeters. It turns out that the length of side BC is equal to the root of 20 squared and 15 squared. The sum of the squares of the diagonals is 625, and if we extract this number from the root, we get the size of the leg equal to 25 centimeters.

We calculate the area of ​​a rhombus using two diagonals.To do this, we multiply d1 by d2 and divide the result by 2. It turns out: 30 times 40 (= 1200) and divided by 2 - it turns out 600 cm square. is the area of ​​the rhombus.

Now we calculate the height, knowing the length of the side and the area of ​​the rhombus. To do this, you need to divide the area by the length of the leg (this is the formula for calculating the height of the rhombus): 1200 divided by 25 - it turns out 48 centimeters. This is the final answer.

How to find the height of a rhombus if the area and perimeter are known (what formula)?

Check out all the formulas for calculating the area of ​​a rhombus:

To find out the height, we need the very first formula (Area \u003d Height times the Length of the side).

Let's assume that the perimeter is 124 cm and the area is 155 cm2.

It plays into our hands that the rhombus has all the same sides, because its perimeter is 4 times the length of one leg.

1. Find the length of the side of the rhombus through the known perimeter. To do this, we divide the value of the perimeter (124) by 4, and we get the value 31 centimeters - the length of the leg.
2. We calculate the height using the area formula.We divide the area (155 cm2) by the size of the leg (31 cm) and get 5 centimeters - this is the size of the height of this geometric figure.

How to find the height of a rhombus if the side and angle are known?

The task seems difficult, but it is not. Imagine that the size of the leg of a rhombus is equal to the root of three, and the angle is 90 degrees.

To calculate the size of the height, we use the formula for the area of ​​​​a rhombus (multiply the squared side by the sine of the angle). To find out the sine of any degree, use in my answer. The sine of 90 degrees equals 1, so finding the height will be very easy. It turns out that the area is equal to the square of the length of the side (3) times the sine of 90 gr. (1), which ultimately gives the answer - 3 cm square.

And then we divide the resulting area by the size of the leg: 3 divided by the root of 3, and we get the height of the rhombus -√3.

How to calculate the height of a rhombus if the side and diagonal are known?

In this problem, you need to use a right triangle, which is formed by the intersection of the diagonals.

Let's assume that a side is 10 cm and a diagonal is 12 cm.

Our actions:

We find the size of half of the second diagonal using the Pythagorean theorem. The hypotenuse in our case is a side, therefore the value of half of the diagonal will be equal to the difference between the square of the leg (10 squared) and the square of half of the known diagonal (6 squared). It turns out that you need to subtract 36 from 100 - we have 64 centimeters. We extract the root of this number and get the length of half of the second diagonal - 8 cm. A the total length is 16 centimeters.

We calculate the area of ​​the rhombus using two diagonals.We multiply the length of the first diagonal (12 cm) by the length of the second (16 cm) and divide this by 2 - we get 96 cm square. (this is the area of ​​the rhombus).

We calculate the height, knowing the size of the side and the area.To do this, divide 96 by 10 - it turns out 9.6 centimeters is the final answer.

The geometric figure of a rhombus is a variation of a parallelogram with equal sides. Its height is the part of the straight line passing through the top of the figure and forming an angle of 90° when it intersects with the opposite side. A special case of a rhombus is a square. Knowledge of the properties of the rhombus, as well as the correct graphical interpretation of the problem statement, allows you to correctly determine the height of the figure using one of the valid methods.

Finding the height of a rhombus based on figure area data

In front of you is a rhombus. As you know, to find its area, it is necessary to multiply the size of the side by the numerical value of the height, i.e. S = k * H, where

• k - value that determines the length of the side of the figure,
• H is a numeric value corresponding to the length of the height of the rhombus.

This ratio allows you to determine the height of the figure as: H = S/k(S is the area of ​​the rhombus, known from the condition of the problem or calculated earlier, for example, as half the product of the diagonals of the figure).

Finding the height of a rhombus through an inscribed circle

Regardless of the length of the sides and the size of the angles of a rhombus, a circle can be inscribed in it. The center of this geometric figure will coincide with the intersection point of the diagonals of an equilateral parallelogram. Information about the radius of such a circle will help determine the height of the rhombus, because r = H/2, where:

• r is the radius of the circle inscribed in the rhombus,
• H is the desired height of the figure.

From this relation it follows that the height of an isosceles parallelogram corresponds to twice the radius of the circle inscribed in this parallelogram - H = 2r.

Finding the height of a rhombus through the angles of the figure

Before you is a rhombus MNKP, the side of which is MN = NK = KP = PM = m. Two straight lines are drawn through the vertex M, each of which forms with the opposite side (NK and KP) a perpendicular - the height. Let's denote them as MH and MH1, respectively. Consider triangle MNH. It is rectangular, which means that knowing ∠N and the definition of trigonometric functions, you can also determine its side-height of a rhombus: sinN = MH/MN ⇒ MH = MN * sinN, where:

• sinN - sine of the angle at the top of an equilateral parallelogram (rhombus),
• MN (m) is the size of the side of the given rhombus.

Because the angles of the rhombus lying opposite each other are equal, then the value of the second perpendicular dropped from the vertex M is also defined as the product of MN by sinN.

H=m*sinN- the height of such a figure as a rhombus can be determined by multiplying the numerical value of the length of its side by the sine of the angle at its apex.

By determining the length of one height of the rhombus, you get information about the size of the remaining three perpendiculars of the figure. This conclusion follows from the fact that all heights of a rhombus are equal.