Angles, circles and 3D shapes – Year 6 maths worksheet answers to download as PDF

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1. Answer to Question 1

   1. Angles in a triangle add up to 180°. \(x=180-50-60=\) 70°.
   2. The right angle is 90°. \(180-90-35=\) 55°.
   3. The two base angles are equal. \(180-40=140\), and \(140\div2=\) 70° each.
   4. \(180\div3=\) 60°.

2. Answer to Question 2

   1. \(x=360-90-85-110=\) 75°.
   2. Opposite angles are equal, so another angle is 65°. The other two angles are equal and add up to \(360-65-65=230\), so each is 115°. The angles are 65°, 115°, 65°, 115°.
   3. \(360-100-100-60=\) 100°.
   4. Each angle is 90°, and \(4\times90=360\).

3. Answer to Question 3

   1. Angles on a straight line add up to 180°, so \(x=180-125=\) 55°. Check: 125 + 55 = 180.
   2. Again the angles on the line total 180°: \(40+75=115\), so \(y=180-115=\) 65°.
   3. Angles around a point make a full turn of 360°: \(110+130=240\), so \(z=360-240=\) 120°.
   4. \(90+80+100=270\), so \(w=360-270=\) 90°, a right angle.

4. Answer to Question 4

   1. The angle opposite 48° is vertically opposite, so \(c=\) 48°. Angles on a straight line add up to 180°, so \(b=180-48=\) 132°. \(d\) is opposite \(b\), so \(d=\) 132°.
   2. The 48° angle and \(b\) are next to each other on a straight line, so they add up to 180°. Then \(b=180-48=132°\).
   3. Vertically opposite angles are equal, so \(3x=120\) and \(x=\) 40.

5. Answer to Question 5

   1. \(d=2\times6=\) 12 cm.
   2. \(r=15\div2=\) 7.5 cm.
   3. \(r=9.4\div2=\) 4.7 cm.
   4. The circle is 7 cm wide, which is the diameter. The radius is \(7\div2=\) 3.5 cm, so set the compasses to 3.5 cm.
   5. The line from the centre to the edge is the radius. The line across the circle through the centre is the diameter.

6. Answer to Question 6

   1. Triangular prism: 5 faces, 9 edges, 6 vertices. Square-based pyramid: 5 faces, 8 edges, 5 vertices. Cylinder: 2 flat faces and 1 curved face, 2 curved edges and 0 vertices.
   2. Cube: 8 + 6 − 12 = 2. Triangular prism: 6 + 5 − 9 = 2. Square-based pyramid: 5 + 5 − 8 = 2. All of them give 2.

7. Answer to Question 7

   1. \(180-100=80\) and \(80\div2=\) 40° each.
   2. \(180-70=110\) and \(110\div2=\) 55° each.
   3. Both base angles are 55°. \(180-55-55=\) 70°.
   4. Two angles of 100° add up to 200°, which is more than the 180° total for a triangle.

8. Answer to Question 8

   1. The angles at the centre make a full turn of 360°, shared equally by 8 slices: 360 ÷ 8 = 45° each.
   2. Three slices of 45°: 3 × 45 = 135°, an obtuse angle.
   3. Shared between 6 slices: 360 ÷ 6 = 60° each.
   4. 6 − 4 = 2 slices are left, so 2 × 60 = 120°. Check: 4 × 60 + 120 = 360.

9. Answer to Question 9

   1. 360 ÷ 12 = 30°.
   2. 3 numbers apart: 3 × 30 = 90° (a right angle).
   3. 4 numbers apart: 4 × 30 = 120°.
   4. 6 numbers apart: 6 × 30 = 180°, which is a straight line.
   5. 150 ÷ 30 = 5, so the hands are 5 numbers apart: 5 o'clock (the minute hand points at 12 and the hour hand at 5).

10. Answer to Question 10

    1. A quarter turn clockwise from north is east.
    2. 135° is between east (90°) and south (180°), so it faces south-east. The angle to south is 180 − 135 = 45°.
    3. Total turn = 135 + 225 = 360°, a complete turn, so it faces north again.
    4. East to west is a half turn: 180°.

11. Answer to Question 11

    1. The angles of any quadrilateral add up to 360°, so \(120+60+x+2x=360\), which simplifies to \(180+3x=360\).
    2. Subtract 180 from both sides: \(3x=180\). Divide by 3: \(x=\) 60.
    3. \(x=60°\) and \(2x=2\times60=120°\). The four angles are 120°, 60°, 60° and 120°.
    4. \(120+60+60+120=\) 360°, so the answer is correct.

12. Answer to Question 12

    1. \(66\div2=\) 33 cm.
    2. 360 ÷ 36 = 10°.
    3. 90 ÷ 10 = 9 spokes.
    4. The model diameter is 33 cm, so the radius is 33 ÷ 2 = 16.5 cm.

13. Answer to Question 13

    1. The net has 6 squares, one for each face of the cube.
    2. A triangular prism has 3 rectangular faces and 2 triangular faces, so the net has 3 rectangles and 2 triangles.
    3. A cuboid has 6 faces. With a square base, 2 faces are squares (top and bottom) and 4 are rectangles that are not squares.
    4. A row of 6 squares would have 4 squares that wrap round, but the 5th and 6th squares would overlap the 1st and 2nd. So some faces would overlap and some faces would be missing: it does not fold into a cube.

14. Answer to Question 14

    1. The radius is half the diameter: \(4\div2=\) 2 m.
    2. 1 m = 100 cm, so 4 m = 4 × 100 = 400 cm.
    3. Each 1 cm on the drawing stands for 50 cm, so divide the real length by 50: 400 ÷ 50 = 8 cm.
    4. The compasses are opened to the radius, which is half the drawn diameter: 8 ÷ 2 = 4 cm.

15. Answer to Question 15

    1. \(x+x+(x+30)=180\), so \(3x+30=180\).
    2. \(3x=150\) and \(x=\) 50.
    3. The angles are 50°, 50° and 80°. 50 + 50 + 80 = 180.
    4. \(x+x+2x=180\) gives \(4x=180\) and \(x=45\). The angles are 45°, 45° and 90°.
    5. The first triangle has a third angle of 80° and the second has 90°. The second is larger by 10°.

16. Answer to Question 16

    1. 360 ÷ 3 = 120°.
    2. 90° out of 360° is \(\frac{90}{360}=\) \(\frac{1}{4}\) of a full turn.
    3. It must turn 120°, then another blade points straight up.
    4. 6 seconds is 360°, so 90° takes 6 ÷ 4 = 1.5 seconds. In one second it turns 360 ÷ 6 = 60°.
    5. 2 minutes = 120 seconds. 120 ÷ 6 = 20 complete turns.

17. Answer to Question 17

    1. Angles in a triangle add up to 180°. Angle C \(=180-50-60=\) 70°.
    2. Angles on a straight line add up to 180°. \(e=180-60=\) 120°.
    3. The two interior angles that are not next to \(e\) are 50° and 70°. \(50+70=120\), which equals \(e\). So the exterior angle equals the sum of the other two angles.
    4. Interior angle at B \(=180-40-75=65\)°. The exterior angle is \(180-65=\) 115°. (Check: 40 + 75 = 115.)

18. Answer to Question 18

    1. \(\frac{120}{360}=\) \(\frac{1}{3}\) of the circle.
    2. 360 ÷ 120 = 3 sprinklers.
    3. Two sprinklers cover 2 × 120 = 240°. The uncovered angle is 360 − 240 = 120°.
    4. \(\frac{270}{360}=\) \(\frac{3}{4}\) of the circle. The angle that is not watered is 360 − 270 = 90°.
    5. The radius is 4 m, so the diameter is 2 × 4 = 8 m.

19. Answer to Question 19

    1. Two triangular faces joined by rectangles give 5 faces, 9 edges and 6 vertices. It is a triangular prism, like a Toblerone box.
    2. Six square faces and 12 edges describe a cube.
    3. Four triangles meeting at a point on a square base: 5 faces and 8 edges. It is a square-based pyramid.
    4. Two flat circular faces joined by one curved face, with no corners: a cylinder.
    5. For the triangular prism: 6 + 5 − 9 = 2, so Euler's rule works.

20. Answer to Question 20

    1. \(x+2x+3x+4x=360\), so \(10x=360\) and \(x=36\). The angles are 36°, 72°, 108° and 144°.
    2. 2 + 3 + 4 = 9 parts, and 180 ÷ 9 = 20°. The angles are 40°, 60° and 80°.
    3. The largest angle of the quadrilateral is 144° and of the triangle is 80°. The quadrilateral angle is bigger by 144 − 80 = 64°.
    4. 36° + 144° = 180° and 72° + 108° = 180°. So the pairs are 36° and 144°, and 72° and 108°.