Basic Geometry – How to find the Area of Complex Shapes

Complex figures can be divided into several smaller shapes where the perimeter or area formula is known, then added.

To determine the area of any composite figure, simply ADD the areas of each component basic figure.  Be sure to write your final answer with square units.

Determine the area of the given shape. 

The original shape can be redrawn as a rectangle and a triangle.  Rectangles have opposite sides that are congruent (exactly the same).

Area _Composite = Area _Triangle + Area _Rectangle

Area _Triangle = (1/2)(Base)(Height) = (1/2)(3m)(1.5m) = 2.25 m²

Area _Rectangle = (Base)(Height) = (3m)(1.5m) = 4.5 m²

Area _Composite = (2.25m²) + (4.5m²) = 6.75 m²

To determine the surface area of any composite solid, simply add the surface areas of each component basic solid.  You must also subtract the area of any internal face.  Be sure to write your final answer with square units.

Ex. Determine the surface area of the given shape.  Leave the final answer in terms of pi.

The original shape can be redrawn as a cylinder and a cone.  We will have to subtract the area of the circle where the figures meet from each surface area equation because they are “inside” the solid.

SurfaceArea _Composite = S.Area _Cone + S.Area _Cylinder

S.Area _Cone = (Base Area)+(1/2)(Perimeter)(Height) = (1/2)(dπ)(h) = (1/2)(6π)(2) = 6π ft²

S.Area _Cylinder = 2(Base Area)+(Perimeter)(Height) = (πr²)+(dπ)(h) = (π3²)+(6π)(5) = 39π ft²

S.Area _Composite = (6π ft²) + (39π ft²) = 45π ft²