Area of Trapezium Formula for Different Type of Trapezium

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Area of Trapezium Formula for Different Type of TrapeziumNow, our book asks Will this formula hold for a trapezium that leans so far over that its top corner isn't even above the bottom base? We use two ways to approach this, let’s look into both of them in detail Approach 1: Rectangle and Triangles STEP 1 OF 6 The Slanted Trapezium Consider trapezium ABCD where the altitude drops outside the base DC. Let top base , and bottom base . Height is . Previous Next Step STEP 2 OF 6 Establishing the Rectangle Construct altitudes AF and BE. The interior shape ABEF is a rectangle. Area ABEF . Previous Next StepSTEP 3 OF 6 Finding Area of ABED Area(ABED) = Area(ABEF) - Area(AAFD) Let . Then: Previous Next Step STEP 4 OF 6 Adding the Right Triangle Now add the final piece, . Area(ABCD) = Area(ABED) + Area( Let . Then: Previous Next StepSTEP 5 OF 6 Algebraic Substitution Observe that total base . In our figure: . Since : &implies; Area Previous Next Step STEP 6 OF 6 Final Derivation Substitute with (b -a : Area Area Area Area Previous RestartApproach 2: Parallelogram and TriangleSTEP 1 OF 6 Starting with a Trapezium Consider the same slanted trapezium. We can dissect the shape differently into a Parallelogram and a Triangle. Previous Next Step STEP 2 OF 6 Drawing Parallel BG Draw a line . Since and , the shape ABGD is a parallelogram with base a. Previous Next StepSTEP 3 OF 6 Drop Perpendicular from B To find the area of both the parallelogram and the triangle, let's drop a perpendicular from to point on the base. This vertical line is the shared height . Previous Next Step STEP 4 OF 6 Identifying the Bases In the parallelogram ABGD , the base is a. In the remaining triangle , the base is . Previous Next StepSTEP 5 OF 6 Area of the Two Pieces Area(Parallelogram) base × height = ah Area(Triangle) base × height Previous Next Step STEP 6 OF 6 Summing and Simplifying Total Area Expand: Combine terms: Area Previous RestartSTEP 5 OF 6 Area of the Two Pieces Area(Parallelogram) base × height = ah Area(Triangle) base × height Previous Next Step STEP 6 OF 6 Summing and Simplifying Total Area Expand: Combine terms: Area Previous Restart