Pack A — Fluency

Mathematics

Year 10 · Geometry

Pack A · Fluency

Name: _________________________________

Date: _________________ Class: ___________

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

BronzeQuestions 1–10

1.
Find the area of a rectangle with length $8$ cm and width $5$ cm.
2.
Find the area of a triangle with base $10$ cm and height $6$ cm.
3.
Find the area of a circle with radius $5$ cm. Give your answer in terms of $\pi$ .
4.
In a right-angled triangle, the legs are $3$ and $4$ cm. Find the hypotenuse.
5.
A right-angled triangle has hypotenuse $10$ cm and one leg $6$ cm. Find the other leg.
6.
Find the circumference of a circle with radius $4$ cm, in terms of $\pi$ .
7.
A ship sails due east. State its bearing.
8.
Find the volume of a cuboid measuring $4$ cm × $5$ cm × $6$ cm.
9.
In a right-angled triangle, the side opposite to angle $\theta$ has length $3$ cm and the hypotenuse is $5$ cm. Find $\sin\theta$ .
10.
Find the perimeter of a rectangle measuring $9$ cm by $4$ cm.

SilverQuestions 11–20

11.
Find the area of a trapezium with parallel sides $6$ cm and $10$ cm, height $4$ cm.
12.
A shape consists of a rectangle $8$ × $5$ with a right triangle (legs $4$ and $3$ ) attached. Find the total area.
13.
A right-angled triangle has hypotenuse $10$ cm and angle $\theta = 30°$ . Find the length of the side opposite $\theta$ (to 2 d.p.).
14.
A right-angled triangle has hypotenuse $12$ cm and angle $\theta = 40°$ . Find the side adjacent to $\theta$ (to 2 d.p.).
15.
A right-angled triangle has adjacent side $8$ cm and angle $\theta = 35°$ . Find the opposite side (to 2 d.p.).
16.
In a right-angled triangle, the opposite side is $5$ cm and the hypotenuse is $10$ cm. Find $\theta$ (to 1 d.p.).
17.
A walker travels on a bearing of $50°$ , then turns right through $90°$ . What is the new bearing?
18.
Find the volume of a cylinder with radius $4$ cm and height $10$ cm, in terms of $\pi$ .
19.
A right-angled triangle has legs $2$ and $3$ cm. Find the hypotenuse exactly.
20.
Find the area of a parallelogram with base $12$ cm and perpendicular height $7$ cm.

GoldQuestions 21–30

21.
Find the area of a triangle with sides $a = 6$ cm, $b = 8$ cm, and included angle $C = 30°$ (to 2 d.p.).
22.
A ship sails $50$ km on a bearing of $60°$ . How far east and north does it travel? (1 d.p.)
23.
A tree of height $12$ m casts a shadow of length $5$ m. Find the angle of elevation of the sun (to 1 d.p.).
24.
In a triangle, $A = 50°$ , $B = 60°$ , and the side opposite $A$ is $8$ cm. Find the side opposite $B$ (to 2 d.p.).
25.
In a triangle, two sides are $5$ cm and $7$ cm with included angle $60°$ . Find the third side (to 2 d.p.).
26.
A ship sails from A on a bearing of $60°$ to B, $10$ km away. From B, it sails on bearing $150°$ to C, $12$ km away. (a) State the angle $\angle ABC$ . (b) Use cosine rule to find $AC$ (2 d.p.).
27.
A right-angled triangle has hypotenuse $10$ cm and an angle of $30°$ . Find its perimeter (to 2 d.p.).
28.
Find the length of the space diagonal of a cuboid measuring $3$ × $4$ × $12$ cm (to 2 d.p.).
29.
A pyramid has a square base of side $8$ cm and apex directly above the centre, $6$ cm above the base. Find the slant height from the apex to a midpoint of a base edge (to 2 d.p.).
30.
A triangle has sides $5, 7, 9$ cm. Find the largest angle (to 1 d.p.).

PlatinumQuestions 31–40

31.
A ship at $A$ sails on bearing $60°$ for $30$ km to $B$ . From $B$ it sails on bearing $150°$ for $40$ km to $C$ . Find the distance $AC$ (to 2 d.p.).
32.
Find the area of a sector of a circle with radius $10$ cm and angle $60°$ (to 2 d.p.).
33.
From a point $25$ m from the base of a tower the angle of elevation to the top is $38°$ . Find the height of the tower (2 d.p.).
34.
In triangle $ABC$ , $a = 8$ cm, $b = 10$ cm, $A = 40°$ . Find angle $B$ (to 1 d.p.).
35.
A solid consists of a cylinder (radius $3$ cm, height $10$ cm) topped by a hemisphere (same radius). Find the volume in terms of $\pi$ .
36.
A triangle has sides $6, 8, 10$ cm. Use the cosine rule to find an angle, then compute the area (to 2 d.p.).
37.
Two ships leave a port. Ship 1 sails on bearing $60°$ at $20$ km/h; Ship 2 on bearing $150°$ at $15$ km/h. How far apart are they after $2$ hours (to 2 d.p.)?
38.
A ladder of length $5$ m leans against a vertical wall. The foot of the ladder is $3$ m from the wall. (a) How high up the wall does the ladder reach? (b) What angle does the ladder make with the ground (to 1 d.p.)?
39.
In triangle $ABC$ , $A =$ 50° $,$ B = $70°$ , and side $c =$ 10 $cm. Find side$ a$ (to 2 d.p.).
40.
A regular hexagon has side length $6$ cm. Find its area exactly (in surd form).