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Home » Area of Geometric Shapes » Area of a Rhombus with a 60° angle

Area of a Rhombus with a 60° angle

Last updated: Jan 4, 2020 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

If we know the length of a side of a rhombus with a 60° angle, we can find its area in a number of different ways. We will use different properties of parallelograms, diamonds or equilateral triangles. We'll describe several of these methods here.

Problem

A rhombus with side length 10 units has a 60° angle. Find its area.

rhombus with a 60° angle

Strategy

Let's start with the most straightforward method for solving this problem. A rhombus is a special type of a parallelogram. We know that the area of any parallelogram is given by the formula A=l·h, where l is the length of one side of the parallelogram and h the height to that side from the opposite side. So let's draw that height:

rhombus with height

In a parallelogram, the opposite angles are congruent, so m∠ABE =60°. This makes  ΔABE a 30-60-90 triangle.  We know the hypotenuse of this 30-60-90 triangle - it is 10- since all the sides of the rhombus are equal. The long leg of a 30-60-90 triangle, AE, equals half the hypotenuse times √3, so it is 5√3.

AE is the height of the rhombus, and the area of the rhombus is l·h, or 10·5√3= 50√3

Now let's try it a different way:

In a rhombus, the diagonals bisect the angles, and are perpendicular to each other:

rhombus with diagonals

This creates 4 right triangles. All are congruent using the Side-Angle-Side postulate: They all have the same length of hypotenuse, all have a common side, and all have a 30° angle.

Each one of these triangles is a 30-60-90 triangle, with a hypotenuse of 10. Their height is 5√3 and their base is 5. So the area of each triangle is (5·5√3)/2. The area of the rhombus is made up of 4 such triangles, so its area is 4·(5·5√3)/2, or 50√3.

One more way to solve this:

Finally, we can think of this rhombus as being composed of 2 equilateral triangles:

rhombus with equilaterals

One of the angles is 60°, both sides are equal, so the triangle is isosceles, and so each one of the base angles is (180°-60°)/2=60°.

We have already derived a general formula for the area of an equilateral triangle using the length of its side: AreaΔequilateral=(s2*√3)/4

So each of these triangles has an area of (102*√3)/4 or 25√3, and we have two such triangles, making the area of the rhombus 50√3.

« Area of Similar Triangles
Area of an Irregular Shape »

About the Author

Ido Sarig is a high-tech executive with a BSc degree in Computer Engineering. His goal is to help you develop a better way to approach and solve geometry problems. You can contact him at [email protected]

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About

Welcome to Geometry Help! I'm Ido Sarig, a high-tech executive with a BSc degree in Computer Engineering and an MBA degree in Management of Technology. I'm here to tell you that geometry doesn't have to be so hard! My goal with this website is to help you develop a better way to approach and solve geometry problems, even if spatial awareness is not your strongest quality. Read More…

Geometry Topics

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    • Isosceles Triangles
    • Pythagorean Theorem
    • Right Triangles
    • Similar Triangles
    • Triangle Inequalities

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