Depth · one concept from first principles · step 2 of 4

Triangle height depends on chosen base

What is the area of trapezoid ABCD on the right, in $\text{cm}^2$ ?

Figure description: In trapezoid ABCD, A is the top-left vertex, D the top-right, B the bottom-left, and C the bottom-right. The angles at B and at C are right angles, so side AB and side DC are parallel to each other and both perpendicular to side BC. Side AB is $10\ \text{cm}$ long and side DC is $16\ \text{cm}$ long. The diagonal AC is drawn; its length is $20\ \text{cm}$ , and the perpendicular segment dropped from vertex B to the diagonal AC (the height to that diagonal) is $6\ \text{cm}$ .

Geometry

Your answer

How to solve

Strategy Identify Subproblems — The diagonal AC cuts the trapezoid into triangle ABC and triangle ACD, so the total area is the sum of two simpler triangle areas (subproblems). Triangle ABC is the bridge: measuring it with base AC and height 6 gives its area, and the SAME area measured with base AB lets us recover the missing width BC. That width is exactly the height for triangle ACD. The whole idea is that one triangle's area stays the same no matter which side you call the base, so picking the convenient base each time unlocks the answer.

1STEP 1

Split the trapezoid with the diagonal

Diagonal AC splits the trapezoid into triangles ABC and ACD, so its area is those two added.

Area_ABCD = Area_ABC + Area_ACD

2STEP 2

Area of triangle ABC using base AC

Taking AC = 20 cm as the base with height 6 cm gives triangle ABC an area of 60.

Area_ABC = 1/2 × 20 × 6 = 60 cm²

3STEP 3

Use the same triangle to find the width BC

Measured off base AB = 10 cm instead, that same 60 gives height BC = 12 cm.

1/2 × 10 × BC = 60 → BC = 12 cm

4STEP 4

Area of triangle ACD using base DC

Base DC = 16 cm with that height 12 cm gives triangle ACD an area of 96.

Area_ACD = 1/2 × 16 × 12 = 96 cm²

5STEP 5

Add the two triangle areas

The trapezoid's area is the sum of the two triangles.

60 + 96 = 156 cm²

Answer

156 cm²

The answer is an area, so cm² units are correct. A quick check with the trapezoid formula confirms it: average of the parallel sides times the height between them, (10 + 16) / 2 * 12 = 13 * 12 = 156 cm², matching the triangle-by-triangle total.

Takeaway

A triangle's area stays the same whichever side you call the base, so pick the base that makes the height easy to use.

Split the trapezoid with the diagonal
Area of triangle ABC using base AC
Use the same triangle to find the width BC
Area of triangle ACD using base DC
Add the two triangle areas

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