Area of Rhombus

Transcript

Area of RhombusA Rhombus is a parallelogram with 4 sides equal Diagonals bisect each other at Right angles It’s area is Let’s see where this formula comes from We use Dissection here Finding Area of Rhombus using Dissection Watch the shapes physically break apart, flip, and reform into a rectangle. STEP 1 OF 6 Properties of a Rhombus Since ABCD is a rhombus, all its outer sides have equal length (single ticks). Furthermore, its diagonals (AC and BD) are perpendicular bisectors of each other. They intersect at at point O , perfectly cutting each other in half. Previous Next Step Finding Area of Rhombus using Dissection Watch the shapes physically break apart, flip, and reform into a rectangle. STEP 2 OF 6 Dissecting into Two Isosceles The horizontal diagonal BD perfectly splits the rhombus into two identical isosceles triangles: Top: ABD (Peach) Bottom: (Green) Previous Next Step Finding Area of Rhombus using Dissection Watch the shapes physically break apart, flip, and reform into a rectangle. STEP 3 OF 6 Four Identical Right Triangles The vertical diagonal AC further splits these pieces. We now have exactly four identical rightangled triangles separated out: , and . Previous Next StepFinding Area of Rhombus using Dissection Watch the shapes physically break apart, flip, and reform into a rectangle. STEP 4 OF 6 Flip and Rearrange! Watch closely! We flip the right-hand triangles ( and ) completely upside down. By sliding them over, their hypotenuses lock perfectly with the left-hand triangles, forming two separate solid rectangles. Previous Next Step Finding Area of Rhombus using Dissection Watch the shapes physically break apart, flip, and reform into a rectangle. STEP 5 OF 6 Combining to Form WXYZ Finally, the two smaller rectangles stack seamlessly together to form a single, large rectangle: WXYZ. This new rectangle has the exact same area as our original rhombus ABCD , because no pieces were added or removed. Previous Next StepFinding Area of Rhombus using Dissection Watch the shapes physically break apart, flip, and reform into a rectangle. STEP 6 OF 6 Deriving the Area Formula Let's analyze the dimensions of the new rectangle WXYZ: Its Height (XW) is the length of the full diagonal AC. Its Width (WZ) is exactly half the length of the diagonal BD. Area of rhombus Area of rectangle WXYZ Therefore, Area of a rhombus product of diagonals. Thus, we can write