Geometry and Trigonometry practice questions.

A facility must be placed inside a region as far as possible from all of the existing sites (the 'toxic waste dump' problem). Using a Voronoi diagram of the sites, the best candidates are

• the midpoints between pairs of adjacent sites
• the sites themselves
• the vertices of the Voronoi diagram
• the centroid of all the sites

Answer: the vertices of the Voronoi diagram

The nearest-site distance is locally maximised at Voronoi vertices, where three or more cells meet — each vertex is the centre of the largest empty circle through its nearby sites. Compare vertex distances (and edge–boundary intersections) and pick the largest.

In the travelling salesman problem, the route produced by the nearest-neighbour algorithm provides

• a lower bound for the optimal route length
• an upper bound for the optimal route length
• the exact optimal route
• the number of possible Hamiltonian cycles

Answer: an upper bound for the optimal route length

Nearest-neighbour outputs an actual valid tour, so the optimum can only be equal or shorter — an upper bound. Lower bounds come from the deleted-vertex method (remove a vertex, find an MST, add back the two shortest deleted edges).

A circle has radius 5 cm. The length of an arc subtending an angle of 1.2 radians at the centre is

• 4.2 cm
• 6 cm
• 7.5 cm
• 12 cm

Answer: 6 cm

With the angle in radians, arc length s = rθ = 5 × 1.2 = 6 cm. This simple formula is a key reason radians are used.

Find the area of a circle with radius 5 (leave your answer in terms of π).

• 10π
• 25π
• 5π
• 50π

Answer: 25π

Area = πr² = π × 5² = 25π.

A cuboid has dimensions 2 cm × 3 cm × 6 cm. Calculate the length of its longest internal diagonal, in cm.

• 7 cm
• 6.32 cm
• 11 cm
• 5 cm

Answer: 7 cm

Diagonal = √(2² + 3² + 6²) = √(4 + 9 + 36) = √49 = 7 cm.

A cone has base radius 5 cm and vertical height 12 cm. Calculate its total surface area (base plus curved surface), to 3 s.f. (Use A = π·r² + π·r·l.)

• 283 cm²
• 204 cm²
• 78.5 cm²
• 471 cm²

Answer: 283 cm²

Slant height l = √(5² + 12²) = 13. A = π×5² + π×5×13 = 25π + 65π = 90π = 282.7 ≈ 283 cm².