Length and Area in 2-D Figures — Challenge
Every area formula is a rectangle in disguise. Know where each one comes from and you will never mix them up.
What these problems cover
These questions use 13 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Perimeter — Distance all the way around the outside.
- Area — Space inside, measured in square units.
- Circumference — The perimeter of a circle: C = 2π r.
- π — The circumference of any circle divided by its diameter — about 3.14159.
- Vertex — A flat surface; a line where two faces meet; a corner point.
- Base — The number being multiplied. In 53 the base is 5.
- Radius — Centre to edge. Half the diameter.
- Diameter — All the way across through the centre. d = 2r.
- Parallel — Same slope, different intercept.
- Perpendicular — Slopes multiply to -1 — negative reciprocals.
- Parallelogram — A quadrilateral with both pairs of opposite sides parallel.
- Trapezoid — A quadrilateral with exactly one pair of parallel sides.
- Height — The perpendicular distance from the base to the opposite corner.
Problem 1
What is the area of a rectangle 7 centimetres by 9 centimetres, in square centimetres?
- 16
- 32
- 56
- 63
- 72
Worked answer
63
The area of a rectangle is its length times its width.
7· 9 = 63 square centimetres. Adding the sides gives the semi-perimeter, which is 16.
Problem 2
A triangle has a base of 10 centimetres and a height of 6 centimetres. What is its area in square centimetres?
- 16
- 30
- 60
- 120
- 15
Worked answer
30
The area of a triangle is half the base times the height.
12· 10· 6 = 30 square centimetres.
Forgetting the half gives 60, which is the area of the rectangle the triangle fills half of.
Problem 3
A trapezium has parallel sides of 6 and 10 centimetres and a height of 4 centimetres. What is its area in square centimetres?
- 24
- 28
- 32
- 40
- 64
Worked answer
32
The area of a trapezium is the average of the parallel sides times the height.
The average of 6 and 10 is 8, so the area is 8· 4 = 32 square centimetres.
Problem 4
A rectangle 12 centimetres by 8 centimetres has a rectangular hole 3 centimetres by 4 centimetres cut from it. What area remains, in square centimetres?
- 72
- 80
- 84
- 88
- 96
Worked answer
84
The whole rectangle has area 12· 8 = 96 square centimetres.
The hole has area 3· 4 = 12 square centimetres.
What remains is 96-12 = 84 square centimetres.
Problem 5
What is the area of a square with side 11 centimetres, in square centimetres?
- 22
- 44
- 110
- 121
- 144
Worked answer
121
A square's area is the side multiplied by itself.
11· 11 = 121 square centimetres. The perimeter would be 44.
Problem 6
A parallelogram has a base of 9 centimetres and a perpendicular height of 5 centimetres. What is its area in square centimetres?
- 14
- 28
- 45
- 54
- 90
Worked answer
45
The area of a parallelogram is the base times the perpendicular height, with no halving.
9· 5 = 45 square centimetres.
Problem 7
A triangle has an area of 24 square centimetres and a base of 8 centimetres. What is its height in centimetres?
- 3
- 4
- 6
- 8
- 12
Worked answer
6
Start from the area formula: 24 = 12· 8· h = 4h.
Divide by 4: h = 6 centimetres.
Dividing 24 by 8 and stopping gives 3, which forgets the half.
Problem 8
A garden measures 5 metres by 3 metres. A path 1 metre wide runs all the way around the outside of it. What is the area of the path in square metres?
- 16
- 18
- 20
- 24
- 32
Worked answer
20
The path adds 1 metre on each side, so the outer rectangle is 5+2 = 7 metres by 3+2 = 5 metres.
The outer area is 7· 5 = 35 and the garden is 5· 3 = 15 square metres.
The path is the difference: 35-15 = 20 square metres.
Problem 9
What is the area of a rectangle 4 metres by 15 metres, in square metres?
- 19
- 38
- 56
- 60
- 64
Worked answer
60
The area of a rectangle is the product of its two side lengths.
4· 15 = 60 square metres.
The perimeter is a different measurement: 2(4+15) = 38 metres, which is why 38 appears among the options.
Problem 10
A rhombus has diagonals of 8 and 6 centimetres. What is its area in square centimetres?
- 14
- 24
- 28
- 48
- 12
Worked answer
24
The area of a rhombus is half the product of its diagonals.
12· 8· 6 = 24 square centimetres.
Multiplying the diagonals without halving gives 48, the area of the rectangle that encloses it.