Length and Area in 2-D Figures — Practice
Every area formula is a rectangle in disguise. Know where each one comes from and you will never mix them up.
What these questions 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
Find the area of a rectangle 7 cm by 9 cm.
Worked answer
63
7 × 9 = 63 cm2
Problem 2
Find the area of a triangle with base 10 and height 6.
Worked answer
30
12(10)(6) = 30
Problem 3
Find the area of a parallelogram with base 12 and height 5.
Worked answer
60
12 × 5 = 60
Problem 4
Find the circumference of a circle with radius 3. Use π ≈ 3.14.
Worked answer
18.84
2 × 3.14 × 3 = 18.84
Problem 5
Find the area of a circle with radius 5. Use π ≈ 3.14.
Worked answer
78.5
52 = 25
3.14 × 25 = 78.5
Problem 6
A trapezoid has parallel sides 7 m and 13 m, and height 6 m. Find its area.
Worked answer
60
7 + 132 = 10
10 × 6 = 60 m2
Problem 7
A triangle has area 54 cm2 and base 12 cm. Find its height.
Worked answer
9
54 = 12(12)h = 6h
h = 9 cm
Problem 8
Find the area of a circle with diameter 14. Use π ≈ 227.
Worked answer
154
r = 7
227 × 49 = 22 × 7 = 154
Problem 9
A square of side 10 cm has a circle of diameter 10 cm cut out. Find the remaining area. Use π ≈ 3.14.
Worked answer
21.5
Square: 100 cm2
Circle: 3.14 × 25 = 78.5 cm2
100 - 78.5 = 21.5 cm2
Problem 10
A rectangle has perimeter 30 and length 9. Find its width.
Worked answer
6
18 + 2w = 30
2w = 12
w = 6
Problem 11
A 10 × 6 rectangle has a 3 × 4 rectangle cut from one corner. Find the remaining area.
Worked answer
48
10 × 6 = 60
3 × 4 = 12
60 - 12 = 48
Problem 12
A circle has circumference 31.4. Find its radius. Use π ≈ 3.14.
Worked answer
5
31.4 = 6.28r
r = 31.4 ÷ 6.28 = 5
Problem 13
A shape is a 10 × 6 rectangle with a semicircle of diameter 6 on one end. Find its total area. Use π ≈ 3.14.
Worked answer
74.13
Rectangle: 10 × 6 = 60
Semicircle: 12 × 3.14 × 32 = 14.13
60 + 14.13 = 74.13
Problem 14
Among rectangles with whole-number sides and perimeter 24, which has the greatest area?
- 1 × 11
- 3 × 9
- 4 × 8
- 6 × 6
Worked answer
6 × 6
Areas: 11, 27, 32, 36
The square wins. For a fixed perimeter, the closer to square, the larger the area.
Problem 15
A circular path has outer radius 10 and inner radius 7. Find the area of the ring. Use π ≈ 3.14.
Worked answer
160.14
Outer: 3.14 × 100 = 314
Inner: 3.14 × 49 = 153.86
314 - 153.86 = 160.14
Note π(102 - 72) = 51π, not π(3)2.
Problem 16
A triangle has its base along the full 12-unit side of a 12 × 8 rectangle, with its third vertex on the opposite side. Find its area.
Worked answer
48
Base = 12, height = 8
12(12)(8) = 48
Exactly half the rectangle — and it does not matter where along the side the apex sits.