Clever Area of Trapezoid

Equal Area Substitution

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🍫 Warm-Up: The Twin Chocolate Bar

Detective Riddle

Suppose two identical chocolate bars overlap in the middle. We slice away the Shared Bite (②):

Alex eats Orange Leftover (①).
Bob eats Blue Leftover (③)!

Who gets more chocolate?

Option A Alex gets more (Orange ①)

Option B Bob gets more (Blue ③)

Option C Exactly Equal! (① = ③)

                  🌟 The Golden Principle:
                
                  Same Whole − Same Overlap = Equal Leftover!
                
                  Orange Leftover ① = Blue Leftover ③

📐 Calculating Area of a Trapezoid

🧰 Key Dimensions

Parallel Bases Top Base (a) and Bottom Base (b) never intersect

Perpendicular Height Height (h) is the distance between the bases

                Trapezoid Area Formula
              
                Area = 
                
                  (Top Base + Bottom Base) × Height
                  2
                
                = 
                
                  (a + b) × h
                  2
                
                💡 Why divide by 2? 
 Two identical trapezoids join together to form a rectangle of 
width (a + b) and height h! So one trapezoid is exactly half.

🔍 Spotting the Trap

📋 Problem Statement

The figure is made up of
two identical right trapezoids, which overlap with each other.

Find the area of the shaded part!

                ⚠️ Trap Alert: Irregular 6-Sided Shape!
              ❌ We do NOT know its inner side lengths!
❌ No simple formula exists for this shape!
🤔 Can we trade it?

✨ Equal Area Substitution

🏠 The 3 Rooms in Our House:

Room ① Target Shaded Room ② Shared Overlap Room ③ Friendly Trapezoid

Upper Twin = Room ① + Room ②

Lower Twin = Room ③ + Room ②

                Since Upper Twin = Lower Twin:
              
                ① + ② = ③ + ②
              
                Target Shaded Area ① = Blue Trapezoid ③!

📏 Measure Blue Trapezoid ③ & Solve!

Step 1: Hidden Top Base

Inside Top Base = 20 − 5 = 15

The upper twin base is 20. Subtract the 5 overhang to find the inner base!

                  Step 2: Area Calculation
                
                    Area =
                    
                      (15 + 20) × 8
                      2
                    
                    = 35 ×
                    
                      8
                      2
                    
                    =

                    ?

                      35 × 4 =
                      140
                    
                    💡 Shortcut: Divide 8 ÷ 2 = 4 first! Then 35 × 2 = 70, 70 × 2 = 140!
                  
                    Target Shaded Area ① = 140!

⚡ Challenge: Master the Twin!

Step 1: Equal Area Swap

Shaded Area ① = Bottom Trapezoid ③

Step 2: Hidden Top Base

Inside Top Base = 12 − 4 = 8

                  Step 3: Rapid Mental Math
                
                    Area =
                    
                      (8 + 12) × 6
                      2
                    
                    = 20 ×
                    
                      6
                      2
                    
                    =

                    ?

                      20 × 3 =
                      60

🏆 Master of Equal Area Substitution

1

Spot the Twins

Whenever two identical shapes overlap, their total areas are always 100% equal (Upper = Lower).

2

Subtract Shared Overlap

① + ② = ③ + ② ⟹ ① = ③.
Trade the weird 6-sided L-shape for the friendly bottom right trapezoid!

3

Hunt Shifting Lengths

Subtract the overhang to uncover hidden inner bases:
20 − 5 = 15!

              🏅 Mission Complete!
            
              You've mastered Equal Area Substitution