THREE DIMENSIONAL GEOMETRY (3D)

1. The figure below shows a square ABCD point V is vertically above middle of the base ABCD. AB =
10cm and VC=13cm.

(a) The length of diagonal AC to two decimal places. (1marks)

(b) The height of the pyramid to two decimal places. (2marks)

(c) The acute angle between VB and base ABCD. (2marks)

(d) The acute angle between BVA and ABCD. (2marks)

(e) The angle between AVB and DVC. (3marks)

 2. In the figure below, VABCD is a right pyramid on a rectangular base. Point O is vertically below
the vertex V. AB=12cm, BC=5cm and VA=VB=VC=VD= 18cm.

Calculate

(a) the height VO (3marks)

(b) the angle

(i) between VC and the plane ABCD. (2marks)

(ii) the planes VAB and ABCD. (2marks)

(iii) the planes VAD and VBC. (3marks)

3. The figure below represents a triangular prism. The faces ABCD, ADEF and CBFE are rectangles.
AB=8CM, BC= 14cm, BF = AF = 7cm while M is the midpoint of FE. Calculate

a) The angle between faces BCEF and ABCD. (3marks)

b) The angle between faces ABE and ABCD. (3marks)

c) The angle between faces ABM and ABCD. (3marks)

d) The angle between faces CMD and ME. (3marks)

4. The figure below shows a right pyramid with a square base ABCD. VC = 20cm, AB = BC = 10.

X and Y are the mid-point of AB and BC respectively. Calculate

(a) the vertical height VO to 2d.p (3marks)

(b) the angle between VD and ABCD (3marks)

(c) the angle which plane VXY makes with the base (5marks)
5. The diagram below shows a regular tetrahedrone ABCD of side 10cm.

a) Find the angle between AB and the base (3 marks)

b) Find the angle between plane ACB and plane BCD (3 marks)
6. The figure below shows a wedge in which ABC and PQR are right angled triangles which are
congruent.

Calculate

a) The length CP. (2 marks)

b) The angle between CP and plane BCRQ. (2 marks)

c) Angles between places ACRP and ABQP. (3 marks)

d) Given that the mass of the wedge is 2.4kg, calculate the density of the material used to make the
wedge. (3 marks)
7. The figure below shows a right pyramid mounted onto a cuboid AB = BC = 15√2cm, CG = 8 cm and
VG=15√2cm

Calculate:

a) The length of AC. (1 mark)

b) The angle between the line AG and the plane ABCD. (3marks)

c) The vertical height of point V from the plane, ABCD. (3marks)

d) The angle between the planes EFV and ABCD. (3marks)

e) The angle between the line VB ad the plane ABCD. (3marks)

f) The angle between the plane VBC and the plane ABCD. (3marks)

g) The angle between the plane VEH and the plane BCHE (3marks)
8. In the figure below, AB = 16m, AE = 12m angle FBC = 25° and rectangle DCFE is perpendicular to
the rectangle base ABCD.

a) The length AF (2marks)

b) The length BC (2marks)

c) The angle between the line EF and CA (3marks)

d) The angle between the line AF and plane ABCD (3marks)

e) The angle between the line EC and AD (3marks)

9. The figure below shows a solid frustum of a pyramid with a rectangular top of side 6cm by 4 cm and
a rectangular base of side 10cm by 8 cm. The slant edge of the frustum is 8cm.

a) Calculate the height of the frustum (3marks)

b) Calculate the volume of the solid frustum. (3marks)

c) Calculate the angle between the line FC and the plane FGHE (2marks)

d) Calculate the angle between the planes BCHG and the base EFGH. (2marks)
10. The roof of a building is as shown in the figure below with a rectangular base ABC. AB=20m and
AD = 8m. The ridge EF = 10m and is centrally placed. The faces ADE and BFC are equilateral
triangles.

Calculate

(i) The height of E above the base ABCD (2 marks)

(ii) The angle between the planes ABCD and ADFE (3 marks)

(iii) The angle between the planes AED and ABCD (2 marks)

(v) The acute angle between lines DB and EF (3 marks)

11. The figure below shows a cuboid ABCDEFGH with AB 8cm, BC6cm and CH Sem.

Calculate to 1 d.p

(a) The length BE (2marks)

(b) The angle between BE and plane ABCD (1mark)

(c) The angle between lines FH and BC. (2marks)

(d) The acute angle between lines FH and BD (1marks)

(e) The angle between place AGHD and planes ABCD (2marks)

(f) Point K and L divides EH and FG respectively in the ratio 1: 3. Determine the angle between planes
ADKL and BCKL (2marks)
12. The figure below is a frustum of a rectangular pyramid with AB = 12cm, EF = 8cm, BC=9cm and
height of 6cm

8cm

6cm

12cm

(a) Calculate the full height of the pyramid.

9cm

(2marks
(b) Find the angle that the plane ABFE makes with the base ABCD.

(2marks)

(c) Find the angle that AG makes with the base ABCD.

(3marks)

(d) Find the angle that AC makes with line EF.

(1mark)

(e) Find the angle that plane BCGF makes with the base ABCD.

(2marks)

13. The figure above shows the structure of a building under construction. HA = IB = JC = ED = 12m
and BC = AD = IJ = HE 16m; and AB = DC=HIEJ = 10m and HG = IG=FJ = FE = 7m and GF = 9m.

7m

9m

7m

10m

7m

7m

10m

16m

12m

B
Calculate:
a) the angle face GHI makes with base ABCD.

(3 marks)

(b) vertical height of ridge GF above base ABCD.

(3 marks)

(c) angle face GFJI makes with ABCD.

(3 marks)

(d) M is mid point of AB. What is the projection of MF to the base ABCD?

(1 mark)

14. The figure below is a square based pyramid ABCDV with AD = DC = 6cm, and height VO = 10 cm.
a) State the projection of VA on the base ABCD.

(1 mark)

b) Find

i) The length of VA

(3 marks)

ii) The angle between VA and ABCD

(2 marks)

iii) The angle between the planes VDC and ABCD

(2 marks)

iv) Volume of the pyramid

(2 marks)
. The figure below shows a right pyramid with a square block atitis base. The sides of the base are 40cm
and the height of the base is 30cm.M is the midpoint of WX such that VM=29cm.

29 gm

40 cm

40 cm

30 cm

Calculate:

(3 marks)

(a) The vertical height of vertex V from the plane ABCD.

(b) The angle between plane of VWX and VyZ.

(2 marks)

(c) The projection of the line yV to the plane wxyz.

(2 marks)

(d) The angle between the planes WXCD and ABCD.

(3 marks
16. The diagram below represents a wooden prism. ABCD is a rectangle. Points E and F are directly
below C and B respectively. M is the mid-point of CD. AB = 8 cm, BC = 10 cm and CE = 4.5 cm.

10 cm

(a) Calculate the size of angle CDE

(2 marks)

(b) Calculate the

(i) Length of AC

(2 marks)

(ii) Angle AC makes with the plane ADEF

(2 marks)

(c) Find the:

(i) Length of MB

(2 marks)
ii) Angle CBM

(2 marks)

17. The triangle below shows a triangular prism. AB=20m, BC=10m. AE=ED=BF=FC=8cm.

8m

10m

(a) Find the length

20m

(1mark)

(1) AC

(ii) AF

(2marks)

(b) (0) Calculate the angle between line AF and the base ABCD

(3marks)

(ii) Find the angle between plane ADF and the base ABCD

(2marks)

(c) Find the volume of the prism

(2marks)
Exercise 6.5

1. Two points distance bee earth's surface have the coordinates A (35°S, 17°E) and B (35°S, 28°W)
respectively. Calculate the distance between A and B in kilometers giving your answer to the nearest
whole number. (3marks) Take TT 22/, and radius of the earth = 6370km.

2. An airplane leaves point A (60°S, 10°W) and travels due east for a distance of 960 nautical miles
to point B. determine the position of B and the time difference between points A and B.

(3 marks)

3. If the local time of town A (52°N, 0°) is 12.00noon. Find the local time of town B(1° S, 37° E)

(3marks)

4. A point K (30° N, 15° E) is on the surface of the Earth. Point R is 900 nautical miles East of K.
Find

(a) The longitude difference between K and R to 4 significant figures.

(2 mark)

(b) The position of R.

(1 mark)
A globe representing the earth has a radius of 0.2m. Points P (60°N, 140°E) and Q (60°N, 120°W)
are marked on the globe. If O is the centre of the latitude 60°N, find the area of the minor sector
OPQ.

(3 marks)

11.

Ex

1.

6. Points A and B lies on the same circle of latitude PoN if A and B are on longitude 41°W and 3°E
respectively and the distance between them is 1370nm. Calculate the latitude P.

(2marks)

7. Points A and B lies on the same circle of latitude PºN if A and B are on longitude 41°W and 3°E
respectively and the distance between them is 1370nm. Calculate the latitude P.

(2marks)

8. Calculate the shortest distance between X(40°N,80°W) and Y (40°N, 100°E) in kilometers taking
π = Radius = 6371km. (Give your answer to the nearest whole number) 22 7 and

(3marks)

9. The latitude and longitude of two stations P and Q are (47°N, 25°W) and (47°N, 70°W)
respectively. Calculate the distance in nautical miles between P and Q along the latitude 47°N

D. A jet flies from (34°N, 12°E) to (34°E, 24°E) in 1½ hrs. Find its average speed in knots.

(3marks
11. Two points A(70°, 15°E) and Blie on the same circle of latitude on the earth's surface. Given that
the shortest distance between the two points along the circle of latitude is 2133.6km. Giving
coordinates to the nearest degree, find the location of B.(Take 22 and radius of earth = 6370km) 7
(3marks)

Exercise 6.6

1. A plane leaves an airport P (10°S, 62°E) and flies due north at 800km/h. (a) Find its position after
2 hours

(3 marks)

(b) The plane tums and flies at the same speed due west. It reaches longitude Q, 12°W. (1) Find the
distance it has traveled in nautical miles.

(3 marks)

(ii) Find the time it has taken (Take = 2, the radius of the earth to be 6370km and 1 nautical mile to
be 1.853km)

(2 marks)

(c) If the local time at P was 1300 hours when it reached Q, find the local time at Q when it landed at
Q

(2 marks)
The positions of two towns on the earth's surface are A (40°S, 45°W) and B (40°S, 135°E)

(a)alcula difference in distance between towns A and B along the parallel of latitude and along the
great circle (in nm)

(4 Marks)

(b) Two planes X and Y left town A at 8:00 a.m. flying at 758 knots each towards town B. If plane X
flies along the parallel of latitude and plane Y along the great circle; then determine the position of
one of the planes when other lands at town B.

(4 Marks)

(c) Find the local time at town B when the second plane lands

(2 Marks)

3. The position of two towns X and Y are given to the nearest degree as X (45° N, 1100 W) and Y
(45° N, 700 Ε). (Take π = 3.142, R = 6370km).Find:

(a) The distance between the two towns along the parallel of latitude in km.

(3marks)

(b) The distance between the towns along a parallel of latitude in nautical miles.

(3marks
c) A plane flew from X to Y taking the shortest distance possible. It took the plane 15hrs to move
from X and Y. Calculate its speed in Knots.

(4marks)

4. The position of two towns X and Y are given to the nearest degree as: X(45°N, 10°E) and
Y(450N, 70°E). Find:- (a) The difference in longitude.

(1mark)

(b) The distance between the two towns in: i) Kilometers (take the radius of the earth as 6371km.

(3marks)

ii) Nautical miles (take 1 nautical mile to be 1.85km)

(3marks)

(c) The local time at x when the local time at y is 2.00p.m.

(3marks)

5. P and Q are two points on the same parallel of latitude 66°25°, whose longitudes differ by 120°.
Calculate (a) the radius of the parallel of latitude where P and Q lie R (6370km).

(2marks
(b) the distance of P and Q measured along the parallel of latitude.

(2marks)

(c) (i) the length of the straight line joining PQ.

(2marks)

(ii) the distance PQ along the latitude in nautical mile.

(2marks)

(d) If an aircraft took 30min to fly P to Q. Calculate its speed in knots.

(2marks)

6. A jambo jet flies from P(30° N, 700 W) to Q due South of P on latitude 45° S. (Take π = 3.142, R
= 6370km) a) Find the distance covered by the jambo jet in

i) Kilometres

(3 marks)

ii) Nautical miles

(2 marks)

b) If the speed of the plane is 500km/hr, find the time taken from P to Q.

(2 marks)

c) Find the local time at R (20°S, 20°E) when the local time in P is 11.20pm

(3 marks)
Three points P and Q are found on the earth's surface. The position of P is (52ºS, 66°W) and Q
(52°S, 114°E). Using the earth's radius = 6370km.

(a) Find the difference in longitude between points P and Q

(1mark)

(b) Calculate the shortest distance between points P and Q along: - (i) the latitude in km(to one whole
number)

(2marks)

(ii) the longitude in km (to one whole number)

(3marks)

(c) A plane travelling at 800km/h leaves point P at 10.00 a.m. and flies through the south pole to
point Q. Find the local time the plane arrives at point Q to the nearest minute.

(4 marks)

8. The positions of two towns A and B on the Earth's surface are (36°N, 49°E) and (36°N, 131°W)
respectively. a) Find the difference in longitude between towns A and B

(2 Marks)

b) Given that the radius of Earth is 6370km, calculate the shortest distance between towns A and B

(3 Marks)
Another town C is 840km due east of town B and on the same latitude as towns A and B. Find the
longitude of town C.

(3 Marks)

d) What is the time difference between A and B and towns A and C to the nearest minute?

(2 Marks)

10.

9. Two towns A and B lie on the same parallel of latitudes 60°N. If the longitudes of A and B are
42°W and 29°E respectively.

(a) Find the distance between A and B in nautical miles along the parallel of latitude.

(2 marks)

(b) Find the local time at A if at B is 1.00pm.

(2 marks)

(c) Find the distance between A and B in km. (Take R = 6370 km and t = 2)

(2 marks)

(d) If C is another town due South of A and 10010km away from A, Find the co-ordinate of C.

(4 marks)
The position of two towns are A (30°S, 20°W) and B (30°S, 80°E) find (a) the difference in
longitude between the two towns.

(1 mark)

(b) (i) the distance between A and B along parallel of latitude in km (R = 6370 km and ㅠ = 2).

(3 marks)

(ii) in nm.

(2 marks)

(c) Find local time in town B when it is 1.45pm in town A.

(4 marks)

11. The position of two towns A and B on earth surface are (36°N, 49° E) and (36° N, 131° W)
respectively. Take R = 6370

a) Find the difference in longitude between the town A and B.

(1 mark)

b) Calculate the distance between A and B along the latitude in

i) nautical miles

(2 marks)

ii) kilometres

(2 marks)

c) i) Another town C is 840km East of town B and on the same latitude as town A and B. Find the
position of town C.

(3 marks
ii) If the local time in B is 7.30 a.m, find the local time in C.

(2 marks)

12. The positions of three ports in the Indian Ocean are P(40°N, 30°W) Q(40° N, 20°E) and R(36°S,
30°W) respectively.

a) Find the distance in nautical miles to the nearest nm between:

(i) Ports P and Q

(3 Marks)

(ii) Ports P and R

(2 Marks)

b) A ship left port P on Tuesday 1430 hours and sailed to port Q at 20 knots. Calculate:

(i) The local time at port Q when the ship left port P

(2 Marks)

(ii) The day and time the ship arrived at port Q

(3 Marks)

13. Two points A and B are found on the earth's surface. The position of A is (52°S, 66°W) and B
(52°S, 114°E). Use Earth's radius as 6 370 km.

(a) Find the longitude difference between A and B.

(1 mark)
Calculate the shortest distance between A and B along: (i) the latitude in kilometres to the nearest
whole number.

(2 marks)

(ii) the longitude in kilometres to the nearest whole number.

(3 marks)

(c) A plane travelling at 800 km/h leaves point A at 10.00 a.m. and flies through South Pole to point
B. Find the local time the plane arrives at point B to the nearest minutes.

(4 marks)

14. An aircraft flies from a point A (1°15'S, 37°E) to a point B directly North of A. The arc AB
subtends an angle of 48.9º at the centre of the earth. From B the aero plane flies due west to a point C
on longitude 23°W. Take radius of the earth as 6370km.

(a) (i) State the location of B

(2 marks)

(ii) Find the distance in km traveled by the aero plane between B and C

(3 marks)

(b) (i) The aeroplane left B at 1.00am local time. Find the local time at C?

(2 marks)

(ii) If it maintained an average speed of 840km/h between B and C, at what local time did it arrive at
C?

(3 marks
. Points P(30%, 20%), Q(30%, 40E) R(60N, FE) and SHIN, W) are four points on the surface of the
earth. R is due North of Q and S is due West of R and due North of P.

(a) State the values of a, b and c.

(3marks)

(b) Given that all distances are measured along parallels of latitudes or along meridians, and in
nautical miles, find the distance of R from P using two alternative routes via Q and S.

(4marks)

(c) Two pilots start flying from P to R one along the route PQR at 400 knots and the other along PSR
at 300 knots which one reaches R earlier and by how long?

(3marks)

16. A plane leaves an airport P (10°S, 60°E) and flies due north at 800km/hr. By taking radius of the
earth to be 6370 km and 1 nautical mile to be 1.853km,

(a) Find its position after 2hrs

(b) The plane turns and flies at the same speed due West to reach Q longitude 12°W.Find the
distance it has traveled due in nautical miles.
(c) Find the time it has taken

(d) If the local time at P was 1300hrs when it reached Q. Find the local time at Q when it landed at Q

17. Points R and S are two points on the surface on a latitude 48°S. The two points lie on longitudes
30°W and 150°E respectively. By taking the earth's radius to be 6370km, calculate:

(a) The distance from R to S along a parallel of latitude.

(b) An aeroplane flies at an average speed of 280km/h from R to S along a great circle through the
South Pole. Calculate the total time taken.

(c) The local time of R when the local time of S is 2.15 pm.

(d) Another point Q is 600nm North of R. Find the location of Q

P and Q are two points on a geographical globe of diameter 50 cm. They both lie on a parallel
latitude 50° North Phas longitude 90° West and Q has longitude 90° East. A string AB has one end at
point P and another at point Q when it is stretched over the North pole. Taking t = 3.142;

(1) Calculate the length of the string.

(ii) If instead the string is laid along the parallel of latitude 50°N with A at point P, calculate the
longitude of point B.

(ii) State the position of B if the string is stretched along a great circle of P towards the South Pole if
point Ais static at P.

19. P, Q and R are points on the surface of the earth such that P (60°N, 20°W), Q (60°S, 20°W) and
R(60°N, 80°E) find:

a) The shortest distance between P and Q on the surface of the earth in kilometres and nautical
miles(nm)

b) The length of latitude 60°N and hence the length of the minor arc PR in kilometres

c) The distance from P to the North Pole.

. A jet flies Frond the X(50 deg * S, 20 deg * E) directly to Y(50 deg * S, 28 deg * W) and then due
South for 1200mm to Z (a) (i)

(ii) Calculate the distance XY along a parallel of latitude 50°S in km

(b) (i) Given that the average speed of the jet is 400 knots, calculate the time taken to reach Z from X
to the nearest 0.1 hour

(ii) Find the time of arrival at Z given that the plane left X at 7.40a.m. Take pi = 2/7 and R = 6370km
)

21. A jet on a rescue mission left town A(35 deg * S, 15 deg * E) to town B(45 deg * N, 15 deg * E)
and then to town C(45 deg * N,45 deg * W) If the radius of the earth is 6370km. Find

(a) the distance in nautical miles from A to C via B correct to 4 s.f

(b) the distance in kilometers from A to B to the nearest km

(c) the jet flew at 840km/h from A to C. If the jet left town A at 8.15am, find the local time at town C
when it arrived there