# Question Video: Finding the Areas of a Square and a Rhombus given Their Diagonals Mathematics

Determine the difference in area between a square having a diagonal of 10 cm and a rhombus having diagonals of 2 cm and 12 cm.

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### Video Transcript

Determine the difference in area between a square having a diagonal of 10 centimeters and a rhombus having diagonals of two centimeters and 12 centimeters.

Although it’s not essential to answering this problem, we’ll begin by sketching the two shapes. We have a square with a diagonal of length 10 centimeters and a rhombus with diagonals of lengths two centimeters and 12 centimeters. We then need to recall how to find the area of each of these shapes from the length of their diagonal. The area of a square with a diagonal of length 𝑑 units is 𝑑 squared over two. The area is half the square of the length of the diagonal. So the area of this square, which has a diagonal of 10 centimeters, is 10 squared over two. That’s 100 over two, which is 50 square centimeters.

The area of a rhombus, on the other hand, with diagonals of lengths 𝑑 one and 𝑑 two is 𝑑 one multiplied by 𝑑 two over two. It’s half the product of the lengths of its diagonals. So the area of this rhombus, which has diagonals of lengths 12 centimeters and two centimeters, is 12 multiplied by two over two, which simplifies to 12 square centimeters. To find the difference in area, we subtract the area of the smaller shape, that’s the rhombus, from the area of the larger shape, that’s the square, 50 minus 12, which gives a difference in area of 38 square centimeters.