# Form 4 Mathematics: Three dimensional geometry questions and answers

Form 4 Mathematics: Three dimensional geometry questions and answers

Lessons (**26**) * SHARE*

- 1.
The figure below shows a right pyramid VABCD whose rectangular base is 18cm by 12cm. The altitude is 24cm. The plane PQRS
and ABCD are parallel and 6cm apart.
Calculate:
(a) The angle between the planes ABCD and VAB
(b) The area of the rectangle PQRS

10m 3s - 2.
The figure below shows a shape of a roof with a horizontal rectangular base ABCD. The ridge EF is also horizontal. The measurements of the roof are
AB=8m, BC=5m,EF=4.5m and EA=ED=FC=3.5m.
Calculate:
(a) The height of the ridge EF above the base ABCD
(b) The angle between the face AED and the base ABCD.

14m 5s - 3.
A right pyramid VABCD has a square base ABCD of side 4. The slant edge VA, VB, VC and VD are #2 sqrt 5# m long
(a) Calculate:
(i) The height of the pyramid
(ii) The angle between the plane VAB and the base ABCD
(b) C` and D` are midpoint of VC and VD respectively. Calculate the angle between the planes ABCD and A’B’C’D’.

15m 26s - 4.
The figure below shows a triangular prism with dimensions as shown.
Calculate:
(a) The angle between the faces FBCE and ABCD
(b) The volume of the prism
(c) The angle between the planes DCF and ABCD

15m 56s - 5.
In the figure below ABCDEFGH is a frustum of a right pyramid. The altitude of the frustum is 2cm.
Calculate
(a) The altitude of the pyramid
(b) The volume of the frustum
(c) The angle between the base of the frustum and the face ABGF.

10m 4s - 6.
The base of a right pyramid is a square ABCD of side 29 cm. the slant edges VA,VB, VC and VD are each of length 39 cm.
(a) Sketch an label the pyramid
(b) Find the angle between a slanting edge and the base.

6m 30s - 7.
A pyramid of height 10cm stands on a square base ABCD of side 6 cm
(a) Draw a sketch of the pyramid
(b) Calculate the perpendicular distance from the vertex to the side AB

4m 23s - 8.
The triangular prism shown below has sides AB= DC = EF = 12 cm. The ends are equilateral triangle of sides 10cm. The point N is the midpoint FC
(a) Find the length of
(i) BN
(ii) EN
(b) Find the angle between the line EB and the plane CDEF

7m 29s - 9.
An equilateral triangle ABC lies in a horizontal plane. A vertical flag AH stand at A. If AB = 2 AH find the angle between the places ABC and HBC.

7m 30s - 10.
The diagram below shows a right pyramid VABCD with V as the vertex. The base of the pyramid is rectangle ABCD, with AB = 4 cm and BC= 3 cm. The
height of the pyramid is 6cm.
(a) Calculate the
(i) Length of the projection of VA on the base
(ii) Angle between the face VAB and the base
(b) P is the mid- point of VC and Q is the mid – point of VD.Find the angle between the planes VAB and the plane ABP

13m 58s - 11.
A pyramid VABCD has a rectangular horizontal base ABCD with AB= 12 cm and BC = 9cm. The vertex V is vertically above A and VA = 6cm. Calculate the volume of the pyramid.

1m 15s - 12.
An electric pylon is 30m high. A point S on top of the pylon is vertically above another point R on the ground. Points A and B are on the same horizontal
ground as R. point A is due south of the pylon and the angle of the elevation of S from A is #26^0#. Point B is due west of the pylon and the angle of elevation of S from B is #32^0#. Calculate:
(a) Distance from A to B
(b) Bearing of B from A

16m 18s - 13.
The figure below represents a right prism whose triangular faces are isosceles. The base and height of each triangular face are 12cm and 8cm respectively. The length of the prism is 20cm.
Calculate the:
a) Angle CE
b) Angle between
i) The line CE and the plane BCDF
ii) The plane EBC and the base BCDF

9m 58s - 14.
Three points O, A and B are on the same horizontal ground. Point A is 80 metres to the north of O. Point B is located 70 metres on a bearing of #060^0#
from A. A vertical mast stands at point B. The angle of elevation of the top of the mast from o is #20^0#.
Calculate:
(a) The distance of B from O.
(b) The height of the mast in metres

8m 12s - 15.
The diagram below represents a cuboid ABCDEFGH in which FG= 4.5 cm, GH= 8cm and HC = 6 cm
Calculate:
(a) The length of FC
(b) (i) the size of the angle between the lines FC and FH
(ii) The size of the angle between the lines AB and FH
(c) The size of the angle between the planes ABHE and the plane FGHE

10m 47s - 16.
The figure below represents a triangular prism. The faces ABCD, ADEF and CBFE are rectangles. AB=8cm, BC=14cm, BF=7cm and AF=7cm.
Calculate the angle between faces BCEF and ABCD.

3m 52s - 17.
The figure below shows a right pyramid mounted onto a cuboid. AB=BC= 15#sqrt2# CG= and VG = 17#sqrt2#
Calculate:
a) The length of AC
b) The angle between the line AG and the plane ABCD
c) The vertical height of point V from the plane ABCD
d) The angle between the planes EFV and ABCD

9m 45s - 18.
The figure below represents a rectangular based pyramid VABCD. AB = 12 cm and AD =16 cm. Point O is vertically below V and VA = 26 cm.
Calculate:
(a) The height, VO, of the pyramid
(b) The angle between the edge VA and the plane ABCD
(c) The angle between the planes VAB and ABCD.

8m 16s - 19.
In the figure below, VABCD is a right pyramid on a rectangular base. Point O is vertically below the vertex V AB = 24cm, BC= 10cm and CV = 26cm.
Calculate the angle between the edge CV and the base ABCD.

4m 30s - 20.
The figure ABCDEF below represents a roof of a house. AB = DC = 12m, BC = AD = 6m, AE = BF = CF = DE = 5m and EF=8m.
Calculate, correct to 2 decimal places, the perpendicular distance of EF from the plane ABCD.
Calculate the angle between:
(i) The planes ADE and ABCD
(ii) The line AE and the plane ABCD, correct to 1 decimal place
(iii) the planes ABF E and DCFE, correct to l decimal place.

16m 54s - 21.
The figure below shows a right pyramid VABCDE. The base ABCDE is a regular pentagon. AO = 15cm and VO = 36cm.
Calculate:
(a) The area of the base correct to 2 decimal places
(b) The length AV
(c) The surface area of the pyramid correct to 2 decimal places
(d) The volume of the pyramid correct to 4 significant figures.

14m 8s - 22.
The figure below represents a cuboid PQRSTUVW.
Calculate the angle between lines PW and plane PQRS, correct to 2 decimal places.

3m 40s - 23.
The figure below represents a cuboid ABCDEFGH in which AB = 16 cm, BC = 12 cm and CF = 6 cm.
(a) Name the projection of the line BE on the plane ABCD.
Calculate correct to 1 decimal place:
(i) The size of the angle between AD and BF
(ii) The angle between line BE and the plane ABCD
(iii) The angle between planes HBCE and BCFG.
(c) Point N is the midpoint of EF. Calculate the length BN, correct to

17m 7s - 24.
The figure below represents a wedge ABCDEF. EF 10 cm, angle FBE 45° and the angle between the planes ABFE and ABCD is 20°.
Calculate length BC, correct to l decimal place.

3m 27s - 25.
The figure below is a model of a watch tower with a square base of side 10 cm. Height PU is 15 cm and slanting edges UV = TV = SV = RV = 13 cm.
Giving the answer correct to two decimal places,
calculate:
(a) length MP
(b) the angle between MU and plane MNPQ
(c) Length of VO
(d) The angle between planes VST and RSTU

9m 2s - 26.
The figure below is a right pyramid VEFGHI with a square base of 8cm and a slant edge of 20cm Points A B C and D lie on the
slant edges or the pyramid such that VA = VB = VC = VD = I0 cm and plane ABCD is parallel to the base EFGH.
(a) Find the lenght of AB.
(b) Calculate to 2 decimal places
(i) The lengh of AC
(ii) The perpendicular height of the pyramid VABCD
(c) The pyramid VABCD was cut off

8m 3s

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