3D geometry: IB Mathematics: Applications & Interpretation questions with answers
Free 3D geometry questions for IB Mathematics: Applications & Interpretation (SL & HL), part of Geometry & trigonometry. Each question has the answer and the marking points an examiner would look for.
A room is a cuboid 8 m long, 4 m wide and 3 m high. A string is stretched from a bottom corner to the diagonally opposite top corner.
(a)
Find the length of the string.
[2]
(b)
Find the angle between the string and the floor.
[2]
(c)
Find the angle between the string and the 8 m by 3 m wall containing its starting point.
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
9.43 m
√l2 + w2 + h2(M1) 9.43 m A1
2
(b)
18.5°
sin θ = 39.43(M1) 18.5° A1
2
(c)
25.1°
projection onto the wall (M1) 25.1° A1
2
Question 2Structured · 14 marks
A tent is a triangular prism. Its front is an isosceles triangle with base 3 m and sloping sides 1.8 m, and the tent is 3 m long. The floor is not made of fabric.
(a)
Find the height of the tent.
[2]
(b)
Find the angle between a sloping side and the floor.
[2]
(c)
Find the volume of the tent.
[2]
(d)
Find the area of fabric needed.
[3]
(e)
A pole runs from a bottom front corner of the tent to the top back corner of the ridge. Find its length and the angle it makes with the floor.
[5]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
0.995 m
Pythagoras (M1) 0.995 m A1
2
(b)
33.6°
cos θ = 1.51.8(M1) 33.6° A1
2
(c)
4.48 m3
area of triangle × length (M1) 4.48 m3A1
2
(d)
13.8 m2
two rectangles and two triangles (M1) correct expression (A1) 13.8 m2A1
3
(e)
3.50 m; 16.5°
horizontal distance √(w/2)2 + l2(M1) 3.35 (A1) 3.50 m A1 tan θ = height ÷ horizontal distance (M1) 16.5° A1
5
Question 3Structured · 6 marks
A grain silo is made of a cylinder of radius 2 m and height 6 m, with a hemisphere on top. It stands on the ground.
(a)
Find the volume of the silo.
[3]
(b)
The outside of the silo, except the base, is painted at a cost of $7.50 per m2. Find the cost of painting.
A storage tank is a cylinder of radius 5 m and height 12 m with a hemisphere on top.
(a)
Find the volume of the tank.
[3]
(b)
Paint costs $4.50 per m2. Find the cost of painting the curved surface and the hemisphere.
[3]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
1204 m3
cylinder (M1) hemisphere (M1) 1204 A1
3
(b)
$2403.32
2πrh (M1) 2πr2(M1) $2403.32 A1
3
Question 5Structured · 6 marksA* challenge
A room is a cuboid 5 m long, 4 m wide and 2.5 m high. A string is stretched from a bottom corner to the diagonally opposite top corner.
(a)
Find the length of the string.
[2]
(b)
Find the angle between the string and the floor.
[2]
(c)
Find the angle between the string and the 5 m by 2.5 m wall containing its starting point.
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
6.87 m
√l2 + w2 + h2(M1) 6.87 m A1
2
(b)
21.3°
sin θ = 2.56.87(M1) 21.3° A1
2
(c)
35.6°
projection onto the wall (M1) 35.6° A1
2
Question 6Structured · 14 marks
A tent is a triangular prism. Its front is an isosceles triangle with base 2.4 m and sloping sides 1.8 m, and the tent is 4 m long. The floor is not made of fabric.
(a)
Find the height of the tent.
[2]
(b)
Find the angle between a sloping side and the floor.
[2]
(c)
Find the volume of the tent.
[2]
(d)
Find the area of fabric needed.
[3]
(e)
A pole runs from a bottom front corner of the tent to the top back corner of the ridge. Find its length and the angle it makes with the floor.
[5]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
1.34 m
Pythagoras (M1) 1.34 m A1
2
(b)
48.2°
cos θ = 1.21.8(M1) 48.2° A1
2
(c)
6.44 m3
area of triangle × length (M1) 6.44 m3A1
2
(d)
17.6 m2
two rectangles and two triangles (M1) correct expression (A1) 17.6 m2A1
3
(e)
4.39 m; 17.8°
horizontal distance √(w/2)2 + l2(M1) 4.18 (A1) 4.39 m A1 tan θ = height ÷ horizontal distance (M1) 17.8° A1
Riyaz Shaikh has mentored IGCSE, IB and A Level students for 15+ years, one-on-one, online and in Mumbai. The hardest questions on this site are the ones he uses to find a student's weak points.
Original IB-style practice questions written for mentorriyaz.in. These are not IB past paper questions, and this site is not affiliated with or endorsed by the International Baccalaureate Organization. Updated 2026-11-04. Paper generator · mentorriyaz.in