Pack B — Fluency

Mathematics

Year 9 · Architecture

Pack B · Fluency

Name: _________________________________

Date: _________________ Class: ___________

Answer all questions. Show your working. Questions are grouped by challenge level.

BronzeQuestions 1–30

1.
A right-angled triangle has the two shorter sides of length 6 cm and 8 cm. Find the hypotenuse.
2.
Simplify $\sqrt{{n}}$ fully.
3.
Simplify $\sqrt{{n}}$ fully.
4.
A triangle has sides 7, 24, 25 cm. Use Pythagoras' converse to decide whether it is right-angled.
5.
Simplify $\sqrt{{a}} \times \sqrt{{b}}$ .
6.
Identify which numbers are rational vs irrational: $5, \sqrt{2}, \pi, 0.\overline{3}, \sqrt{4}$ .
7.
Estimate $\sqrt{50}$ to 1 d.p.
8.
Find the hypotenuse of a right-angled triangle with legs 9 and 12 cm.
9.
Add: $\sqrt{50} + \sqrt{18}$ (give simplified form).
10.
A right-angled triangle has hypotenuse 13 cm and one shorter side 5 cm. Find the other side.
11.
A cuboid has length 10 cm, width 6 cm, height 3 cm. Find its volume.
12.
A cube has side 7 cm. Find its total surface area.
13.
A cylinder has radius 5 cm and height 8 cm. Find the volume in terms of π.
14.
A triangle base 6 cm, perpendicular height 8 cm. Find the area.
15.
A trapezium has parallel sides $a$ and $b$ , height $h$ . Find $A$ when $a = 4$ , $b = 10$ , $h = 5$ .
16.
A square pyramid has base side 6 cm and height 9 cm. Find the volume.
17.
A cone has radius 6 cm, height 12 cm. Find the volume in terms of π.
18.
A sphere has radius 6 cm. Find the volume in terms of π.
19.
A cuboid with dimensions 6 × 4 × 3 cm. Find the surface area.
20.
A cylinder has radius 4 cm and height 10 cm. Find its total surface area in terms of π (including top and bottom).
21.
In a right-angled triangle, the angle $\theta$ has opposite side 8 cm and hypotenuse 17 cm. Write down $\sin \theta$ as a fraction in its simplest form.
22.
Identify the opposite, adjacent, and hypotenuse for angle 30° in a right-angled triangle with hypotenuse 10 cm.
23.
Find sin 60° in surd form.
24.
A right triangle has opposite 5 and adjacent 12. Find tan θ.
25.
In a right triangle, $\sin \theta = 0.6$ and hypotenuse = 10. Find the opposite side.
26.
Use $\sin 30° = 0.5$ to find the opposite side when angle = 30° and hyp = 8 cm.
27.
Find $\tan 45°$ exactly.
28.
A right triangle has hypotenuse 10 and an angle 30°. Find the side opposite the 30° angle.
29.
Find tan 30° exactly.
30.
A right triangle has hyp 13 and one leg 12. Find the third side, and then $\sin \theta$ for the angle opposite the 5-side.

SilverQuestions 31–60

31.
Pythagoras with surds: legs $2\sqrt{3}$ and 3. Find the hypotenuse.
32.
Find the distance between $(1, 2)$ and $(4, 6)$ . Give answer as a surd in simplest form.
33.
Simplify $\sqrt{8} + \sqrt{32} - \sqrt{18}$ .
34.
Find the hypotenuse of a right-angled triangle with legs 5 and 10 cm. Give answer as a simplified surd.
35.
Simplify $\sqrt{75} \div \sqrt{3}$ .
36.
A right-angled triangle has hypotenuse $\sqrt{50}$ and one leg $\sqrt{18}$ . Find the other leg.
37.
Find the distance between $(-2, 1)$ and $(3, 13)$ .
38.
Find the perimeter of a right-angled triangle with legs 6 and 8 cm.
39.
Square the surd: $(2 + \sqrt{3})^2$ . Expand.
40.
Rationalise the denominator: $\dfrac{6}{\sqrt{3}}$ .
41.
A triangular prism has triangular cross-section with legs 5 and 12 (right-angled), and length 20 cm. Find the volume.
42.
A square pyramid has base side 8 cm and slant height 10 cm. Find the surface area.
43.
A sphere has radius 6 cm. Find its surface area in terms of π.
44.
Make $h$ the subject of $V = \pi r^2 h$ .
45.
A cone has volume $48\pi$ cm³ and radius 4 cm. Find its height.
46.
Find the volume of a cuboid pyramid (rectangular base) with base 6 × 4 and height 12.
47.
A cylinder has volume $200\pi$ cm³ and height 10 cm. Find its radius.
48.
A hemisphere has radius 5 cm. Find (a) volume, (b) curved surface area.
49.
A rectangular tank holds 1200 L of water. Length 2 m, width 1.5 m. Find the depth.
50.
Convert: a tank holds 250 L. Find its volume in (a) m³ and (b) cm³.
51.
In a right triangle, the side opposite $\theta$ is 7 cm and hyp = 15 cm. Find $\theta$ to 1 d.p.
52.
A right triangle has hyp 12 cm and angle 50°. Find the opposite side to 2 d.p.
53.
A right triangle has adjacent 5 cm and angle 45°. Find the opposite side to 2 d.p.
54.
A right triangle has adjacent 6 and opposite 8. Find (a) the hypotenuse, (b) the angle θ to 1 d.p.
55.
Find the angle of elevation: a tree casts a 12 m shadow when the tree is 7.5 m tall.
56.
A right triangle has hyp 10 and angle 30°. Find the adjacent (to 2 d.p.).
57.
A ladder of length 5 m leans against a wall, making a 60° angle with the ground. Find the height up the wall.
58.
A right triangle has opposite 8 and adjacent 15. Find (a) the angle θ to 1 d.p., (b) the hypotenuse.
59.
A 25 m vertical building. From the top, the angle of depression to a car is 20°. Find the horizontal distance.
60.
Find the angle in a right triangle with hyp 13 and opposite 5.

GoldQuestions 61–90

61.
A right-angled triangle has shorter sides 5 and 10. Hypotenuse as a simplified surd.
62.
Find the length of the space diagonal of a cuboid with sides 8, 6, 4. Use 3D Pythagoras.
63.
Pythagoras in 3D: cuboid 12 × 9 × 6. Find the space diagonal.
64.
A square has diagonal 10 cm. Find its side length and area.
65.
A ladder of length 10 m leans against a wall. Foot 3 m from the base. Find the height reached.
66.
Simplify $(3 + \sqrt{2})(2 - \sqrt{2})$ .
67.
The area of a square is 12 cm². Find the side as a simplified surd.
68.
Find the length of the diagonal of a rectangle 6 cm by 8 cm.
69.
Show that the triangle with sides 5, 5, $5\sqrt{2}$ is right-angled.
70.
A ladder of length 8 m makes a 60° angle with the ground. Find the height up the wall (using trig).
71.
A triangular prism has a right-angled triangular cross-section with legs 6 cm and 8 cm, and prism length 15 cm. Find its volume.
72.
A cuboid has square base side $x$ cm and height 3 cm, SA 80 cm². Find $x$ .
73.
Rearrange $V = (1/3)\pi r^2 h$ , then find $h$ when $V = 100$ cm³, $r = 5$ .
74.
A solid metal sphere of radius 6 cm is melted and recast as a cylinder of radius 4 cm. Find the height.
75.
A cylinder open at the top has radius 3 cm and height 10 cm. Find the surface area (including base, no top).
76.
A cone has radius 6 cm and slant height 10 cm. Find the (a) curved surface area, (b) total SA.
77.
A cone has volume $48\pi$ cm³ and slant height 5 cm. Find the radius and height. (Hint: use Pythagoras for slant.)
78.
A composite shape: cylinder topped by a hemisphere. Cylinder r = 4, h = 10. Find total volume in terms of π.
79.
A cuboid has volume 360 cm³. Two sides are 6 and 5. Find the third.
80.
A swimming pool is a cuboid 25 m by 10 m by 2 m. Find (a) volume in m³, (b) capacity in litres.
81.
A vertical tree casts a shadow of 15 m on horizontal ground. The angle of elevation from the tip of the shadow to the top of the tree is 48°. Find the tree height to 2 d.p.
82.
A cuboid has length 8 cm, width 6 cm, height 4 cm. Find the angle between the space diagonal and the base.
83.
A right triangle has angle 30° and hyp 8 cm. Use exact values to find the opposite and adjacent.
84.
An aircraft's angle of approach to a runway is 3°. How high should it be when 7 km away horizontally? Give in m.
85.
A right triangle has hyp 10 and one acute angle 35°. (a) Find the opposite and adjacent. (b) Find the area.
86.
Find $x$ in: angle in a right triangle is $(x + 10)°$ and the opposite/adjacent ratio is 1.
87.
A right triangle has one acute angle of 25° and hypotenuse 20 cm. Find the perimeter to 2 d.p.
88.
Two right-angled triangles share a common leg. The first has opposite 3 and adjacent 4. The second has opposite 5 and the same adjacent 4. Find the difference in their hypotenuses.
89.
A right triangle has hyp 25 cm and an angle 60° at one acute vertex. Find the lengths of the two legs.
90.
A boat sails 10 km on bearing 030°. (a) How far north? (b) How far east?

PlatinumQuestions 91–120

91.
A cuboid has length 8 cm, width 6 cm, height 4 cm. Find the angle between the space diagonal and the base, to 1 d.p.
92.
Make $a$ the subject of $c = \sqrt{a^2 + b^2}$ .
93.
A cliff of height $h$ m stands on level ground. From point A 28 m from the base, the line of sight to the top is 52 m. From B further away, sight to top is 60 m. Find $h$ and B's distance.
94.
Apply Pythagoras to find a diagonal of a square pyramid: base side 8, apex height 12 above the centre. Find the slant distance from a base vertex to the apex.
95.
Rationalise: $\dfrac{1}{1 + \sqrt{2}}$ .
96.
An isosceles triangle has equal sides 13 cm and base 10 cm. Find (a) the perpendicular height, (b) the area.
97.
Find the area of an equilateral triangle of side $s = 10$ cm.
98.
A right-angled triangle has legs $a$ and $b$ with $a + b = 14$ and $a^2 + b^2 = 100$ . Find $a$ and $b$ .
99.
Two ladders lean against opposite walls of an alley. One reaches 6 m up the left wall; the other 8 m up the right wall. The alley is 5 m wide. They cross at some height $h$ . Find $h$ (use similar triangles).
100.
A triangle has sides $\sqrt{8}$ , $\sqrt{18}$ , $\sqrt{32}$ . Show it is right-angled.
101.
A cone has radius 6 cm and height 12 cm. The top is sliced off parallel to the base, producing a smaller cone of radius 3 cm. Find the volume of the remaining frustum (in terms of π).
102.
A spherical balloon has volume $36\pi$ cm³. (a) Find the radius. (b) The balloon's radius doubles. By what factor does the volume increase?
103.
Make $r$ the subject of $V = (4/3)\pi r^3$ .
104.
A skyscraper modelled as a cuboid is 154 m tall, 87 m wide, 30 m deep. (a) Find the glass area (4 vertical sides). (b) Find the space diagonal.
105.
A cylinder has radius $r$ and height $h$ where $h = 2r$ . Volume = $54\pi$ cm³. Find $r$ .
106.
A solid is formed by adding a cone to a cylinder. The cone and cylinder have the same radius 4 cm. The cylinder is 10 cm tall and the cone is 6 cm tall. Find the total volume.
107.
A hemispherical bowl has volume $\dfrac{2}{3}\pi r^3 = 250$ ml. Find $r$ (give 1 d.p.).
108.
Two cylinders are similar with linear scale factor 3. The smaller has volume 16 cm³. Find the larger volume.
109.
A pyramid has rectangular base 6 × 8 m and apex height 5 m above the centre of the base. Find the slant height to the midpoint of a long side.
110.
A cylinder of radius $r$ and height $h$ has surface area $A = 2\pi r^2 + 2\pi r h$ . Make $h$ the subject.
111.
PQRS is a square base 8 cm; pyramid apex V is 12 cm above the centre. M = midpoint of QR. Find (a) GM (G = centre), (b) VM, (c) angle of face VQR with the base.
112.
In a right triangle, the angle at A is 30° and the hypotenuse is 12 cm. Use exact values to find the legs.
113.
A cliff of height $h$ stands on level ground. From point A, 28 m from the base, line of sight to the top is 52 m. From B further, sight is 60 m. Find $h$ and the distance from B to the base.
114.
A flagpole is observed from two points 50 m apart. From the closer point, elevation = 35°. From the further, 22°. Find the height of the flagpole.
115.
A ladder of length 6 m leans against a wall. The angle with the ground is initially 70°; it slips to 60°. By how much does the top descend?
116.
Find $\sin 75°$ exactly using the identity $\sin(45° + 30°) = \sin 45 \cos 30 + \cos 45 \sin 30$ .
117.
A triangular prism has a right-angled triangular cross-section with legs 6 and 8 cm. The hypotenuse is the base of the prism, lying on a table. Find the angle the prism's slanted face makes with the table.
118.
A square pyramid has slant edge from base vertex to apex = 13 cm and base diagonal half-length = 5 cm. Find the apex height above the centre.
119.
A skier descends a 100 m slope inclined at 15° to horizontal. Find (a) horizontal distance covered, (b) vertical drop.
120.
A 3D Pythagoras-with-trig problem: a square pyramid has base side 10 cm and slant edge 13 cm. Find the angle between a slant edge and the base.