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GCSE Maths · Higher · Chapter 22: Triangles · Lesson 1

Further 2D and 3D problems

Grades 6–8About 50 minutesCalculator allowed37 marks
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NameClassDate
A

Do Now

Grades 4–6

The first question is what today's lesson needs. The others bring back earlier topics.

  1. A1
    Needed today

    A cuboid is 2 cm by 6 cm by 9 cm. Find the length of its longest diagonal.

    [1 mark]
    Answer
  2. A2
    From chapter 19

    A bag has 5 red and 5 blue. Two are taken without replacement. Find P(both red).

    [1 mark]
    Answer
  3. A3
    From chapter 7

    Two triangles have a right angle, the hypotenuse and one other side equal. Which condition shows they are congruent?

    [1 mark]
    Answer
  4. A4
    From chapter 12

    Similar shapes have matching sides 6 cm and 18 cm. Find the scale factor.

    [1 mark]
    Answer
B

Example, then your turn

Grades 6–7

Your teacher works through each example with you. Then try the one beside it.

Example 1

Find . Give your answer to 1 d.p.

ABCD13 cm19 cm58°
Not drawn accurately
Hint The perpendicular makes two right-angled triangles.
  1. B1

    Find . Give your answer to 1 d.p.

    [4 marks]
    ABCD15 cm24 cm47°
    Not drawn accurately
    Answer
Example 2

Find the angle between and the base , to 1 d.p.

ABCDEFGH28 cm21 cm14 cm
Not drawn accurately
Hint The angle is between and .
  1. B2

    Find the angle between and the base , to 1 d.p.

    [3 marks]
    ABCDEFGH15 cm8 cm11 cm
    Not drawn accurately
    Answer
C

Practice

Grades 6–7
  1. C1

    is a diameter. Find to 3 s.f.

    [2 marks]
    OABC15 cm28°
    Not drawn accurately
    Answer
  2. C2

    is a side of a regular pentagon in the circle. Find to 3 s.f.

    [3 marks]
    OAB9 cm
    Not drawn accurately
    Answer
  1. C3

    From the angles of elevation of and are and . Find to 1 d.p.

    [3 marks]
    PFTW50 m
    Not drawn accurately
    Answer
  2. C4

    Find the area of triangle to 3 s.f.

    [3 marks]
    ABC12 cm19 cm64°
    Not drawn accurately
    Answer
D

Exam-style questions

Grades 7–8
  1. D1

    is a pyramid on a rectangular base with cm and cm. is cm directly above , the centre of the base. is the midpoint of .

    ABCDVOM21 cm20 cm
    Not drawn accurately
    (a)

    Work out the angle between the edge and the base, to 1 d.p.

    [3 marks]
    Answer
    (b)

    Work out the angle between the face and the base, to 1 d.p.

    [2 marks]
    Answer
    Total for D1: 5 marks
  1. D2

    A wedge has a horizontal rectangular base and a vertical face . Tom says, "The angle between and the base is ."

    ABCDEF40 cm15 cm9 cm
    Not drawn accurately
    (a)

    Explain why Tom is wrong, and work out the correct angle to 1 d.p.

    [3 marks]
    Answer
    (b)

    Work out the angle between and the edge , to 1 d.p.

    [3 marks]
    Answer
    Total for D2: 6 marks
E

Extension

Grade 8 and beyond

No route is given. The hints are at the end of the sheet. Use one at a time.

  1. E1

    The top of the pyramid is cm above the centre of the square base. Each sloping edge makes with the base. Find to 3 s.f.

    [4 marks]
    ABCDVOx cm
    Not drawn accurately
    Hint 1 · What to try

    Use triangle : you know and the angle at .

    Hint 2 · The first line

    Hint 3 · The full method

    is half the diagonal of the square, so the diagonal is ; then use Pythagoras on the square.

    Answer
F

How did it go?

Colour one circle on each line.
Not yetNearlyYes
I can draw a right angle inside a harder shape.
I can find the angle between a line and a plane.
I can work in a cuboid, a pyramid and a wedge.

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