Derivation of the Area of a Rhombus

A rhombus is a parallelogram whose sides are congruent. The diagonals of rhombus are perpendicular to each other. They also bisect each other. In this post, we are going to find the general formula for finding the area of a rhombus using these properties. We are going to learn two methods.

Method 1

Consider the rhombus below.

area of a rhombus

We can divide it into two congruent triangles using diagonal d_1. Since the diagonals of a rhombus are perpendicular to each other, we can use d_1 as base and one half of d_2 as the height of the upper triangle (Why?). If we let A_T be the area of the upper triangle, then, calculating its area, we have  Continue reading

Proof Without Words: If x >0, x + 1/x >=2

A picture is worth a thousand words, even in mathematics. Most of the time, complicated algebraic statements can be shown using geometric representations. One example is the sum of

\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \cdots

which when added equals 1.

In this post, we show another geometric proof without words. Proof without words is not a proof per se but rather a representation that will help readers understand the proof.  Continue reading

Why the Area of a Rhombus is Half the Product of its Diagonals

A rhombus is a parallelogram with four congruent sides. Since it is a parallelogram, it has also all the properties of a parallelogram. One of these properties is that the diagonals bisect each other. That is, they divide each other into two equal parts.

Another property of a rhombus is that the diagonals are perpendicular. So, summarizing all the properties above, if we have rhombus ABCD, then,

\overline{AB} \cong \overline{BC} \cong \overline{CD} \cong \overline{DA}.


\overline{AC} \perp \overline{BD}. Continue reading

Explore Solid Nets Apps and Print Models

Are you looking for solid nets resources, printable worksheets, or interactive apps? Check out the five websites below. These sites contains resources about solids, their properties, and their nets. Printable nets are also available.

Annenberg Media has an interactive page on Platonic solids, prisms, and pyramids. It allows users to rotate 3d shapes and see how the solids are formed from their nets.

Illuminations has a page that lets you explore nets of Platonic solids. What is good about this interactive program is that it also allows you to create your own net.

Math Interactives  allows users to explore the relationship between the volume of the solids and their nets. Users can also check if their visualization skill by predicting the top view, front view, and side view of the solids.

solid nets

Maths is Fun – Maths is fun has a page about Platonic solids. The resource includes properties, printable nets, as well as 3d interactive animations.

SEN Teacher is a website that provides free learning materials. The website has free printable nets of polyhedra which is available in PDF. The website also allows users to create  customize nets putting images on them creating “photonets.”

Angle Sum of Polygons in Tessellated Triangles

Tessellated triangles are not only beautiful but that they are also interesting. Tessellating them will prove the angle sum of polygons particularly parallelogram, trapezoid, and hexagon. The term angle sum means the sum of the interior angles.

Let us start with the knowledge that the angle sum of a triangle is 180 degrees. Copying a triangle with angle measures x, y, and z, and rotating it 180 degrees will give us the first two figures. The tessellated copies are shown in the next figure.

tessellated triangle

Using the three figures above, we can prove the following.  Continue reading

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