IB · Mathematics: Analysis & Approaches SL & HL · Geometry & trigonometry

3D geometry: IB Mathematics: Analysis & Approaches questions with answers

Free 3D geometry questions for IB Mathematics: Analysis & Approaches (SL & HL), part of Geometry & trigonometry. Each question has the answer and the marking points an examiner would look for.

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3D geometry practice questions

Question 1Structured · 6 marksA* challenge

A sector of a circle with radius 15 cm and angle 4 radians is folded so that its two straight edges meet, forming the curved surface of a cone.

(a)

Find the base radius of the cone.

[3]
(b)

Find the volume of the cone.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)9.55 cm
arc length = 15 × 4 = 60.0 (A1)
arc length = circumference of base M1
r = 9.55 cm A1
3
(b)1105 cm3
h = √152 − r2 = 11.6 (M1)(A1)
1105 cm3 A1
3
Question 2Structured · 5 marksA* challenge

A cuboid ABCDEFGH has AB = 8 cm, BC = 3 cm and AE = 4 cm.

(a)

Find the length of the diagonal AG.

[2]
(b)

Find the angle between AG and the base ABCD.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)9.43 cm
3D Pythagoras (M1)
9.43 cm A1
2
(b)25.1°
AC = 8.54 (A1)
correct trig ratio (M1)
25.1° A1
3
Question 3Structured · 7 marks

A right circular cone has base radius 4 cm and height 12 cm.

(a)

Find the volume of the cone.

[2]
(b)

Find the total surface area of the cone.

[3]
(c)

Find the angle between the slant edge and the base.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)201 cm3
13πr2h (M1)
201 cm3 A1
2
(b)209 cm2
slant height 12.6 (A1)
πrl + πr2 (M1)
209 cm2 A1
3
(c)71.6°
tan θ = 124 (M1)
71.6° A1
2
Question 4Structured · 7 marks

A right pyramid VABCD has a square base ABCD with sides of length 8 cm. The vertex V is 9 cm vertically above the centre M of the base.

(a)

Find the length of the edge VA.

[3]
(b)

Find the angle between the edge VA and the base ABCD.

[2]
(c)

Find the volume of the pyramid.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)10.6 cm
AM = 12√82 + 82 = 5.66 (M1)(A1)
VA = √92 + AM2 = 10.6 cm A1
3
(b)57.8°
tan VÂM = 95.66 (M1)
57.8° A1
2
(c)192 cm3
13 × 82 × 9 (M1)
192 cm3 A1
2
Question 5Structured · 6 marksA* challenge

A sector of a circle with radius 10 cm and angle 4 radians is folded so that its two straight edges meet, forming the curved surface of a cone.

(a)

Find the base radius of the cone.

[3]
(b)

Find the volume of the cone.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)6.37 cm
arc length = 10 × 4 = 40.0 (A1)
arc length = circumference of base M1
r = 6.37 cm A1
3
(b)327 cm3
h = √102 − r2 = 7.71 (M1)(A1)
327 cm3 A1
3
Question 6Structured · 5 marksA* challenge

A cuboid ABCDEFGH has AB = 4 cm, BC = 3 cm and AE = 4 cm.

(a)

Find the length of the diagonal AG.

[2]
(b)

Find the angle between AG and the base ABCD.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)6.40 cm
3D Pythagoras (M1)
6.40 cm A1
2
(b)38.7°
AC = 5.00 (A1)
correct trig ratio (M1)
38.7° A1
3

More Geometry & trigonometry questions

Sine & cosine rulesRadians, arcs & sectorsTrig identities & exact valuesTrig equations & graphsCompound angles & reciprocal trigVectors, lines & planes

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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