Your acquaintance with rhombus properties
Have you ever gazed at a diamond-shaped object and thought how to calculate its area? When talking about a diamond shape, mathematicians normally mean a rhombus. In fact, rhombus properties differ from those of other four-sided objects.
Formally, a rhombus can be defined as a quadrilateral, featuring a two-dimensional shape with four sides and the same number of corners. All of its sides share the same length. Its opposite sides are parallel to each other. Taking into account the fact that all four sides of this shape have the same length, it’s not a problem at all to figure out the perimeter of this figure. As for the perimeter, it’s the distance around the edge of this geometry figure. In order to calculate the rhombus’ perimeter, you require multiplying the length of one of its sides by four. For instance, a rhombus with a side of 8 inches long will have the perimeter of 32 inches.
The rhombus and other quadrilaterals
From other quadrilaterals, the rhombus can be distinguished by a few characteristics. Well, a square boasts sides of the same length as well as parallel opposite sides. As for a square’s four corners, they’re 90 degree angles. As we already know the rhombus’ corners can’t be 90 degrees. Then, a parallelogram also features two sets of parallel opposite sides, though its sides shouldn’t be necessarily the same length.
The area of a rhombus
You can easily calculate the rhombus’ area by simply utilizing the length of its diagonals. As we already know, diagonals are used to measuring the distance between the two opposite corners of this geometry figure. The area of this figure is equal to the length of its two diagonals multiplied by each of them, and then divided by one. In order to calculate the rhombus’ area you simply require multiplying length at times b, then dividing by two.
A = 1/2 x a x b
Let’s assume the rhombus’ two diagonals account for 10 inches and 7 inches long. Respectively, in order to calculate the area, we should multiply ten by seven and then divide this stuff by two. We’ll get 35 square inches.
The Rhombus’ angles
It’s a common occurrence that a rhombus boasts up to four corners also dubbed vertices. We should stress that the angles of its opposite corners are the same. If one corner of this figure features a 45- degree angle, then its opposite corner will also have a 45-degree angle. Secondly, the angles of any two adjacent corners of this shape add up to 180 degrees. Accordingly, if one corner of this figure features a 45-degree angle, then each of two corners next to it will be 135 degrees.
Well, you see that the two adjacent angles to the 45-degree one are opposite each other. The total of the four angles of this shape is 360 degrees. Needless to remind that every angle need to be larger than 9 degrees and less than 180.
A bit of history
Let’s get back to Euclidean geometry. This discipline states that a rhombus is a simple quadrilateral, whose 4 sides share the same length. An alternative name for this figure is equilateral quadrilateral. The name suggests that all of its sides have the equal length. By the way, the rhombus is often dubbed a diamond.
By the way, every rhombus appears to be a parallelogram. If a rhombus features right angles, that’s a square.
The world “rhombus” originally descended from the Greek word ῥόμβος, which stands for turning round and round. The given word was actively utilized by Archimedes and Euclid. To be exact, they called it solid rhombus, pointing out to two right circular cones, which share a common base.
Key rhombus properties
A simple quadrilateral can be defined as a rhombus if it features any of the following:
- If its diagonals are perpendicular and also bisect each other.
- If it boasts four sides of equal length.
- If every diagonal bisects the two opposite interior angles.
- Any parallelogram with a diagonal bisecting an interior angle.
- Any parallelogram with two consecutive sides equal in length.
- If the parallelogram’s diagonals are perpendicular.
Every rhombus comes with two diagonals, which connect pairs of opposite vertices as well as two pairs of parallel sides. Making use of congruent triangles, you can easily prove that this geometry figure appears to be fully symmetric across every of these diagonals.
Any rhombus satisfies the following requirements:
- The two diagonals of this geometry figure are perpendicular.
- Its opposite angles are equal.
- Its angles are bisected by diagonals.
We’ve just told above that a rhombus is also a parallelogram. As follows from this a rhombus naturally comes with all the properties typical to a parallelogram. For instance, its adjacent angles are supplementary, its opposite sides are parallel, any line running through the midpoint bisects the area and so on.
However, we should clearly realize that not every parallelogram can be regarded as a rhombus, although any parallelogram featuring perpendicular diagonals appears to be a rhombus. Secondly, we may associate a rhombus with a kite. Respectively, any quadrilateral, which is both a parallelogram and a kite appears to be a rhombus.
What could be said else about a rhombus? At least we can say that it’s a tangential quadrilateral. Apart from that, we can add that the dual polygon of a rhombus appears to be a rectangle.
Furthermore, we shouldn’t overlook the following properties:
- This figure’s opposite angles are equal, while as for a rectangle, its opposite sides are equal.
- A rhombus boasts equal sides, while equality for a rectangle refers to its angles.
- A rhombus boasts an inscribed circle. A rectangle comes with a circumcircle.
- The figure created by joining the midpoints of the rhombus’ sides appears to be a rectangle.
- A rhombus boasts an axis of symmetry through every pair of opposite vertex angles. As for a rectangle, it features an axis of symmetry running through every pair of its opposite sides.
- The diagonals of a rhombus manages to intersect at the same angles. In case of a rectangle, its diagonals share the same length.
Other rhombus properties
- Identical rhombi is capable of tiling the 2D plane in three different ways, including the rhombille tiling for instance.
- One of the five 2D lattice types appears to be the rhombic lattice. It’s often dubbed a centered rectangular lattice.
- Several polyhedra feature rhombic faces, including the trapezo-rhombic dodecahedron and the rhombic dodecahedron.
- Three-dimensional analogues of a rhombus come with a bicone and a bipyramid.
Several intriguing faces of a polyhedron
The rhombic dodecahedron can be defined as a convex polyhedron, coming with twelve congruent rhombi as it faces.
A rhombohedron is just a three-dimensional figure, just like a cube, though its six faces appear to be rhombi instead of being squares.
A stellation of the rhombic triacontahedron is the rhombic hexecontahedron. Additionally, that’s a nonconvex with up to sixty golden rhombic faces of icosahedral symmetry.
The rhombic triancontahedron could be depicted as a convex polyhedron with thirty golden rhombi.
The trapezo-rhombic dodecahedron is simply a convex polyhedron, coming with six trapezoidal and rhombic faces.
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