Video: Finding the Areas of Rhombuses Using Heron’s Formula

The perimeter of the given rhombus is 292 cm and the length of 𝐴𝐶 is 116 cm. Use Heron’s formula to calculate the area of the rhombus, giving the answer to three decimal places.

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Video Transcript

The perimeter of the given rhombus is 292 centimeters and the length of 𝐴𝐶 is 116 centimeters. Use Heron’s formula to calculate the area of the rhombus, giving the answer to three decimal places.

Well, the first thing we need to do is remind ourselves of Heron’s formula. So, what we’ve got is that if we have a triangle 𝐴𝐵𝐶 and the area is equal to the square root of 𝑠 multiplied by 𝑠 minus 𝑎 multiplied by 𝑠 minus 𝑏 multiplied by 𝑠 minus 𝑐, where 𝑠 is the semiperimeter which we can find by adding together each of the sides of the triangle, so 𝑎 plus 𝑏 plus 𝑐, and then dividing it by two. So, this is useful. Well, we take a look at our diagram and what we’ve got is a rhombus. Well, what we can do is if we take a look at the diagonal across our rhombus is see that our rhombus can be divided into two identical triangles. However, we still only have one side length of our triangle. But what we can do is we can use the perimeter of the rhombus to help us work out what the other side lengths are. And that’s because in a rhombus each of the sides are the same length.

So therefore, if we call each of the side lengths of our rhombus 𝑥, we can say that 𝑥 is gonna be equal to 292 divided by four. Well, this is equal to 73. So therefore, we could say that each of the side lengths of our rhombus is 73 centimeters. And more importantly, if we look at the orange triangle, we’ve now got a triangle where we know all three sides, 116, 73, and 73. Well, then, the first thing we want to do is work out what the semiperimeter is. And we do that by adding together our side lengths, so 116 plus 73 plus 73, and then dividing it by two. And this gives us a semiperimeter of 131 centimeters.

Okay, great. So now what we can do is substitute this and our side lengths into our formula, so our Heron’s formula. And when we do that, we get that the area is equal to the square root of 131 multiplied by 131 minus 116 multiplied by 131 minus 73 multiplied by 131 minus 73, which is equal to 2571.0425 et cetera. We might think, “Wow, great! We found the area. Can we finish at this point?” But no. And why is that? Well, if we take a look back at our rhombus, we can see that, in fact, it’s two identical triangles. So, therefore, we need to multiply this by two. And when we do that, we get 5142.08518 et cetera. And then if we look back at the question for the degree of accuracy we want the answer left in, we can see we want it to three decimal places. So, therefore, after rounding, we can say that the area of the rhombus is 5142.085 centimeters squared, and we set that to three decimal places.

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