Area of a rhombus and a circle
Find the area of a rhombus with four triangles and with its diagonals, see why a rhombus is half the rectangle round it, and work out the area of a circle from its sectors and with the formula pi × radius × radius.
- Germany: grade 6
- Iran: grade 5
- Ukraine: grade 6
- Russia: grade 8
Guide
The diagonals of a rhombus are perpendicular and cut each other in half. They divide the rhombus into 4 equal right-angled triangles. If the diagonals are 8 and 6, the legs of each triangle are 4 and 3.
If we draw a rectangle round the rhombus whose sides equal the two diagonals, the rectangle is made of the rhombus and 4 triangles equal to the 4 inside it. So the area of the rhombus is half the area of the rectangle.
So the area of a rhombus equals the product of the two diagonals divided by 2. A common mistake is to forget the ÷ 2 or to add the diagonals.
If we cut a circle into equal sectors and lay them side by side, the shape is close to a parallelogram whose base is half the circumference and whose height is the radius. So area of a circle = half the circumference × radius.
Because the circumference is 2×3.14×r, half of it is 3.14×r. So the area of a circle is pi × radius × radius, with 3.14 for pi. If you have the diameter, halve it first.