What is the height of a rhombus according to the formula? The height of a rhombus is equal to half of its side.
The geometric figure rhombus is a variation of a parallelogram with equal sides. Its height is the part of the line passing through the vertex of the figure and forming an angle of 90° when intersecting with the opposite side. A special case of a rhombus is a square. Knowledge of the properties of a rhombus, as well as the correct graphical interpretation of the problem conditions, allows you to correctly determine the height of the figure using one of the acceptable methods.
Finding the height of a rhombus based on the area of the figure
There is a rhombus in front of you. As you know, to find its area it is necessary to multiply the side value 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 – numerical value corresponding to the length of the height of the rhombus.
This ratio allows us to determine the height of the figure as: H = S/k(S is the area of the rhombus, known from the conditions of the problem or calculated earlier, for example, as half the product of the diagonals of the figure).
Finding the altitude 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 a circle inscribed in a rhombus,
- H – the desired height of the figure.
From this relationship 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 using the angles of the figure
In front of you is a rhombus MNKP, the side of which is MN = NK = KP = PM = m. Two straight lines are drawn through vertex M, each of which forms a perpendicular with the opposite side (NK and KP) - a height. Let's denote them as MH and MH1, respectively. Consider triangle MNH. It is rectangular, which means, knowing ∠N and the definition of trigonometric functions, you can determine its side-height of the rhombus: sinN = MH/MN ⇒ MH = MN * sinN, where:
- sinN – sine of the angle at the vertex of an equilateral parallelogram (rhombus),
- MN (m) – the size of the side of a given rhombus.
Because Since the angles of a rhombus lying opposite each other are equal to each other, then the value of the second perpendicular dropped from the vertex M is also determined as the product of MN by sinN.
H = m * sinN– the height of a figure such 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 vertex.
By determining the length of one height of a rhombus, you obtain 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 to each other.
Knowing the diagonals, it is easy to find the height of the rhombus. In that The Pythagorean theorem will help us. And although it concerns right triangles, they also exist in a rhombus - they are formed by the intersection of two diagonals d1 and d2:
Let's imagine that diagonal 1 is 30 centimeters, and diagonal 2 is 40 cm.
So, our actions:
We calculate the size of the side using the Pythagorean theorem. Side BC is the hypotenuse (because it lies opposite the obtuse angle) of triangle BXD (X is the intersection of diagonals d1 and d2). This means that the size of this side squared is equal to the sum of the squares of sides BX and XC. We also know their size (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, multiply d1 by d2 and divide the result by 2. It turns out: 30 multiplied by 40 (= 1200) and divided by 2 - it comes out to 600 cm2. - this 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 a rhombus): 1200 divided by 25 - it comes out to 48 centimeters. This is the final answer.
How to find the height of a rhombus if the area and perimeter are known (what is the 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 = Height times Side Length).
Let's assume that the perimeter is 124 cm and the area is 155 cm square.
It plays into our hands that all sides of a rhombus are the same, so its perimeter is 4 times the length of one leg.
- Let's find the length of the side of the rhombus using the known perimeter. To do this, divide the perimeter value (124) by 4, and get a value of 31 centimeters - the length of the leg.
- We calculate the height using the area formula.We divide the area (155 cm square) 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. Let's imagine that the size of the side 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 (side in a square multiplied by the sine of the angle). To find out the sine of any degree, use my answer. The sine of 90 degrees is equal to 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) multiplied by sine 90 g. (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 side is 10 cm and diagonal is 12 cm.
Our actions:
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 the diagonal will be equal to the difference between the square of the leg (10 squared) and the square of half 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 total length is 16 centimeters.
We calculate the area of a rhombus using two diagonals.We multiply the length of the first diagonal (12 cm) by the length of the second (16 cm) and divide it by 2 - we get 96 cm square. (this is the area of the rhombus).
We calculate the height, knowing the side size and area.To do this, divide 96 by 10 - it comes out 9.6 centimeters is the final answer.
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 lowered from the corner to one of the sides. In a quadrilateral, heights can be lowered from each corner to two sides. Let's look at how to find the heights of a rhombus and how they relate to each other.
How to find the height of a rhombus
Quadrilaterals are figures whose angles can change while the lengths of the sides remain the same. 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 angles or the height. For example, if the angles of a rhombus are 90°, the result is a square. In this case, the height coincides with the side. Let's look at how to find the height of a rhombus at angles other than right angles.
Determine the value of two heights of a 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's lower the height AH to side BC, and lower the other height AH1 from the same corner to side DC.
- Then 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 = AD (all sides of the rhombus are all equal), then the height AH = AH1. Similarly, one can prove that two heights dropped from any angle are equal to each other.
How do the remaining 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∟С = sin∟A = sin∟B.
Consequently, all heights omitted from any corner of a rhombus are equal to each other, and the side, angle and height are related to each other by a rigid relation: h = a × sin∟A, where a is the length of any side, ∟A is any angle of the rhombus.