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How to find the area of a composite rectangle and triangles

The rule: split the figure into non-overlapping shapes

Draw or identify boundaries that divide the figure into familiar shapes. Find each area separately, then add the areas because the pieces do not overlap. For a rectangle, $A = w h$ . For a triangle, $A = \frac{1}{2} bh$ , where the height is perpendicular to the chosen base. Shared edges are boundaries, not extra area, so they are not counted separately.

Problem

A composite figure consists of a central rectangle $3$ units wide and $6$ units high. A triangle is attached to the top and another to the bottom. Each triangle has base $3$ units and perpendicular height $2$ units. Find the area of the rectangle, each triangle, and the entire figure.

Split the figure along the two dashed shared edges, then add the rectangle and triangle areas.

Answer

24

square units.

Checked by adding the independently computed areas 18, 3 and 3.

Step-by-step solution

1. Area of the rectangle

The rectangle is $3$ units wide and $6$ units high:

A_{rectangle} = 3 \cdot 6 = 18

2. Area of the top triangle

Its base is $3$ units and its perpendicular height is $2$ units:

A_{top} = \frac{1}{2} \cdot 3 \cdot 2 = 3

3. Area of the bottom triangle

It has the same base and height, so its area is also

A_{bottom} = \frac{1}{2} \cdot 3 \cdot 2 = 3

4. Add the non-overlapping pieces

A_{total} = 18 + 3 + 3 = 24

Therefore, the area of the composite figure is $24$ square units.

A useful shortcut

The two congruent triangles together have area $3 + 3 = 6$ square units. You may add that directly to the rectangle: $18 + 6 = 24$ . The shortcut works because the triangles do not overlap the rectangle or each other.

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