Unit 5
Geometry
About this unit
Calculate areas and perimeters of composite shapes, volumes of solids, surface areas, and angles in polygons. Develop spatial intuition through paper-folding and 3D reasoning.
What types of questions will you face?
- 1Find the area of a rectangle given its perimeter and a length-to-breadth ratio — set up one equation in one variable using P = 2(l + b), solve, then multiply to get area
- 2Count unit cubes with exactly 0, 1, 2, or 3 painted faces after a large cube is assembled, all outer faces painted, and then broken apart — use the interior-dimension formula (n−2)^3 for zero painted faces
- 3Find a missing angle inside a composite polygon figure (parallelogram, rhombus, or a square combined with an equilateral triangle) using angle sum rules and known properties
- 4Calculate the area or perimeter of a composite 2D shape formed by combining or subtracting rectangles, triangles, semicircles, or trapeziums
- 5Find the volume of water poured between tanks of different dimensions, or calculate the height of water after adding an object — using Volume = base area × height
Skills you will build
- Setting up one-variable equations from perimeter and ratio clues: let b = breadth, l = 2b, substitute into P = 2(l + b), solve
- Applying the interior-dimension formula for painted cubes: strip one layer from each end of each dimension to get (n−2) × (n−2) × (n−2) unpainted interior cubes
- Using angle properties of parallelograms (opposite angles equal, co-interior angles sum to 180°) and equilateral triangles (all angles 60°) to find unknown angles in composite figures
- Decomposing irregular shapes into rectangles and triangles, calculating each piece separately, then adding or subtracting as needed
- Applying Volume = base area × height and its rearrangement height = volume ÷ base area to solve tank problems, including those with fractional fill levels
By the end of this unit, you will be able to
- Find the dimensions and area of any rectangle from its perimeter and a dimension ratio, using one-variable algebra
- Determine how many unit cubes in a painted n×n×n cube have exactly 0, 1, 2, or 3 painted faces
- Calculate unknown angles in composite polygon figures using properties of parallelograms, rhombuses, squares, and equilateral triangles
- Find the area or perimeter of any composite shape that combines or subtracts standard shapes
- Solve multi-step volume and capacity problems involving tanks, water transfer, and submerged objects
Difficulty profile
Questions in this unit range from Very Easy to Difficult. Simple perimeter and angle questions are Very Easy to Easy. Composite area, cube-painting, and volume-transfer problems are Medium. Multi-step area-relationship and surface-area problems can reach Difficult.
Exam tip: Geometry
For any rectangle problem giving perimeter and a ratio: use one variable only — never introduce two unknowns when one ratio links them. For painted-cube problems: always apply the (n−2)^3 formula for zero painted faces rather than drawing and counting — it works for any size cube and takes under 10 seconds.
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