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.

Standardized School Assessments

These are the big ones that most students across the provinces will run into. They almost always include problems where you have to break a shape down into rectangles and triangles to find the total area.

• Ontario EQAO (Grade 9 Math): This is a staple. The “Geometry and Measurement” strand specifically tests the ability to solve problems involving the perimeter and area of composite two-dimensional shapes.
• BC Graduation Numeracy Assessment (Grade 10): Instead of just straight math, this one is “contextualized.” A student might be asked to calculate the amount of flooring needed for a weirdly shaped room or the area of a garden path.
• Alberta Provincial Achievement Tests (PATs – Grades 6 & 9): Geometry is a significant chunk here. By Grade 9, students are expected to handle composite 2-D shapes and even surface areas of composite 3-D objects.

Trade and Technical Entrance Exams

If you’re working with someone heading into the trades, they can’t escape geometry. Calculating material for complex footprints is a “real-world” skill these tests love.

• Pre-Admission Technical Math (Colleges like Mohawk, BCIT, etc.): Many technical programs require a math placement test. These almost always feature “composite area” problems because they’re essential for construction and engineering tech.
• Canada Trades, Ontario Skilled Trades, Alberta Trades
• Ironworker or Carpenter Pre-Apprenticeship Exams: These are very practical. Expect questions like: “What is the total area of this L-shaped concrete slab?”

Law Enforcement & Military

“Big Prep” often forgets that spatial reasoning and basic geometry are part of the “Numerical” or “Spatial” sections of these exams.

• CFAT (Canadian Armed Forces Aptitude Test): While it leans heavily on word problems and patterns, the “Problem Solving” section frequently includes basic geometry (area/perimeter) of shapes that aren’t just simple squares.
• Ontario Security Guard Exam  and BC Security Guard: Surprisingly, basic geometry pops up here too, often in the context of map reading or identifying areas on a floor plan.
• CritiCall (Emergency Dispatch): Some versions include “Map Reading” modules where calculating distances or identifying relative areas of zones is required.

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