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.
Sample Questions
When a square is split into identical rectangles, each small perimeter tells you the side lengths — work backwards from one rectangle to the whole square.
Sub-divided square perimeter items appear in the very-easy band of Selective MR — quick marks if you halve the rectangle perimeter to get one side of the square.
The examiner checks whether you can relate a small rectangle’s perimeter to its length and width inside a square, then scale up to the square’s perimeter.
A square is cut into four congruent rectangles. Each rectangle’s perimeter is given. You find the perimeter of the original square.
Best approach: Let the square side be s. Each rectangle is s × s/2 (or s/2 × s). Write 2(l + w) = given rectangle perimeter, solve for s, then perimeter of square = 4s.
Question
A square is divided into four identical rectangles. Each rectangle has a perimeter of 120 cm. What is the perimeter of the square?
- A160 cm
- B192 cm
- C200 cm
- D180 cm
- E240 cm
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Painted-cube questions with zero red faces are always the interior core: subtract one layer from each dimension, then multiply.
Unpainted unit-cube counts appear in the medium band of Selective MR — the (n − 2) formula works for cubes and cuboids once you identify interior dimensions.
The examiner tests whether you know only fully enclosed unit cubes escape the paint, and whether you can apply (L − 2)(W − 2)(H − 2) after cutting a cuboid into 1 cm cubes.
A cuboid is painted on all faces, then cut into 1 cm unit cubes. You count how many small cubes have no painted face.
Best approach: Interior length = L − 2, same for width and height (when each is at least 2). Multiply the three interior dimensions for the count of unpainted cubes.
Question
A cuboid is 5 centimetres long, 4 centimetres wide and 3 centimetres high. Paint its six faces red and then cut it into small cubes with edge length 1 centimetre. How many small cubes have no face painted red?
- A6
- B8
- C22
- D24
- E12
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For each 3D shape, count its edges. Then check: does 48 divide evenly by that number? If yes, the edge length is a whole number and that shape is possible. If there’s a remainder, it’s impossible.
3D shape edge-count divisibility questions appear in the medium band of Selective MR. Students who don’t know edge counts from memory can use Euler’s formula (V − E + F = 2) as a check, but memorising edge counts for common shapes is faster.
The examiner tests whether students (1) know edge counts for a triangular prism (9), square pyramid (8), and triangular pyramid/tetrahedron (6), (2) check divisibility of 48 by each count, and (3) correctly identify which shapes are possible rather than guessing from appearance.
A 3D object has equal-length edges summing to a given total. Several candidate shapes are listed. Students determine which shapes could be Prem’s object based on whether the edge count divides the total evenly.
Best approach: Memorise: triangular prism = 9 edges, square pyramid = 8 edges, triangular pyramid = 6 edges. Check 48 ÷ 9 (= 5.33… ✘), 48 ÷ 8 (= 6 ✔), 48 ÷ 6 (= 8 ✔). Report the shapes that pass.
Question
Prem has a 3D object.
The edges are all the same length. Each edge is a whole number of centimetres long.
The total length of the edges is 48 cm.
Which of the following is/are possibilities for Prem's object?
(The three candidate shapes are: triangular prism, square pyramid, triangular pyramid)
- Anone of them
- Btriangular prism only
- Csquare pyramid and triangular pyramid only
- Dtriangular prism and triangular pyramid only
- Etriangular prism, square pyramid and triangular pyramid
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A hexagonal pyramid has 7 faces (1 hexagon + 6 triangles), 12 edges (6 base + 6 lateral), and 7 vertices (6 base + 1 apex). Sum = 7 + 12 + 7 = 26. Euler check: F + V − E = 7 + 7 − 12 = 2 ✔
Faces, edges and vertices of pyramids appear in the medium band of Selective MR. The most common error is using 6 faces instead of 7 (forgetting the hexagonal base counts as a face).
The examiner checks whether students can correctly count faces (base + triangular sides), edges (base edges + lateral edges), and vertices (base corners + apex) for an n-gonal pyramid, and then sum all three.
Students are given a named pyramid (with an n-sided base) and must determine the number of faces, edges, and vertices separately, then add them.
Best approach: For an n-gonal pyramid: Faces = n + 1, Edges = 2n, Vertices = n + 1. For n = 6: F = 7, E = 12, V = 7. Sum = 7 + 12 + 7 = 26.
Question
What is the sum of the number of faces, the number of edges and the number of vertices of a pyramid with a hexagonal base?
- A18
- B20
- C24
- D26
- E38
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