Area

2016 Sample Question 6 / 2014 Question 7

A cross-country race is run on a triangular course.
The points A, B and C mark the corners of the course, as shown below.
The area within the triangular course 𝐴𝐵𝐶, in square metres, can be
calculated by evaluating

  ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯   ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯   ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

A. √3100 × 1200 × 1050 × 8050 B. √3100 × 2250 × 2050 × 1900 C. √6200 × 5300 × 4150 × 3950
1 1
D. ⎯⎯× 2050 × 2250 × sin(140°) E. ⎯⎯× 2050 × 2250 × sin(40°)
2 2

2016 Question 1
Consider the diagram below. The shaded area, in square centimetres, is
A. 35 B. 45 C. 60 D. 85 E. 95

2017 NHT Question 1

A dairy farm is situated on a large block of land.
The shaded area in the diagram below represents the block of land. 𝑑 = 3.2 km.

c. Calculate the area of this block of land. Round your answer to the nearest square kilometre. 1 mark

____________________________________________________________________________________________________________________

____________________________________________________________________________________________________________________

2018 Question 2
A triangle 𝑃𝑄𝑅 is shown in the diagram below.
The length of the side 𝑄𝑅 is 18 cm. The length
of the side 𝑃𝑅 is 26 cm. The angle 𝑄𝑅𝑃 is 30°.
The area of triangle 𝑃𝑄𝑅, in square
centimetres, is closest to
A. 117 B. 162 C. 171 D. 234 E. 468

2019 NHT Question 1

2019 NHT Question 1
A piece of cardboard is shown in the diagram below.
The area of the cardboard, in square centimetres, is
A. 4 B. 5 C. 21 D. 25 E. 29

2019 NHT Question 2

Michael will present a building design at the conference.
The horizontal cross-section of the building is in the shape of a
regular hexagon. Each side of the hexagon is 12 m long. Triangle
𝐴𝐵𝐶 is shown shaded below.

a. i. Show, with calculations, that the area of triangle 𝐴𝐵𝐶, rounded

to one decimal place, is 62.4 m . 1 mark

____________________________________________________________________________________________________________________

____________________________________________________________________________________________________________________

ii. Calculate the area of the hexagon. Round your answer to the nearest square metre. 1 mark

____________________________________________________________________________________________________________________

____________________________________________________________________________________________________________________

2019 Question 1
The four bases of a baseball field form four corners of a
square of side length 27.43 m, as shown in the diagram
below.

A player ran from home base to first base, then to second

base, then to third base and finally back to home base.

The minimum distance, in metres, that the player ran is

A. 27.43 B. 54.86 C. 82.29 D. 109.72 E. 164.58

2019 Question 1
2019 Question 1
The following diagram shows a cargo ship viewed from above.

The shaded region illustrates the part of the deck on which shipping containers are stored.
a. What is the area, in square metres, of the shaded region? 1 mark

____________________________________________________________________________________________________________________

____________________________________________________________________________________________________________________
Area
2014 Question 7
A cross-country race is run on a triangular course.
The points A, B and C mark the corners of the course, as shown below.

The area within the triangular course 𝐴𝐵𝐶, in square metres, can be
calculated by evaluating
  ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯   ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
A. √3100 × 1200 × 1050 × 8050 B. √3100 × 2250 × 2050 × 1900

  ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
C. √6200 × 5300 × 4150 × 3950 D. 1
⎯⎯× 2050 × 2250 × sin(140°)
1
E. ⎯⎯× 2050 × 2250 × sin(40°)
2 2
2015 Question 5
A company logo is in the shape of a regular hexagon with side length 2 cm, as
shown below. The hexagon is divided into six equilateral triangles.
Every second triangle is shaded.

The shaded area of the logo, in square centimetres, is closest to

A. 1.7 B. 2.0 C. 5.2 D. 6.0 E. 10.4

2014 Question 2
The chicken coop has two spaces, one for nesting and one for eating. The nesting and eating spaces
are separated by a wall along the line AX, as shown in the diagrams below.

𝐷𝑋 = 3.16 m, and 𝐴𝑋 = 2.31 m. ∠𝐴𝐷𝑋 = 45° and ∠𝐴𝑋𝐷 = 60°.

d. Calculate the area of the floor of the nesting space, ADX.
Write your answer in square metres, correct to one decimal place. 1 mark

____________________________________________________________________________________________________________________

____________________________________________________________________________________________________________________

____________________________________________________________________________________________________________________

2013 Question 1
Competitor X is 40 m from G and competitor Y is 52 m from G.
The angle XGY is 113°.

b. ii. Find the area of triangle XGY, correct to the nearest square
metre. 1 mark

____________________________________________________________________________________________________________________

____________________________________________________________________________________________________________________

2012 Question 4
2012 Question 4
𝑂𝐴𝐵𝐶𝐷 has three triangular sections, as shown in the
diagram below. Triangle 𝑂𝐴𝐵 is a right-angled triangle.
Length 𝑂𝐵 is 10 m and length 𝑂𝐶 is 14 m.
Angle 𝐴𝑂𝐵 = angle 𝐵𝑂𝐶 = angle 𝐶𝑂𝐷 = 30°

The length, 𝑂𝐴 is 8.66 m.

b. Determine the area of triangle 𝑂𝐴𝐵. Write your answer, in m , correct to one decimal place. 1 mark

____________________________________________________________________________________________________________________

____________________________________________________________________________________________________________________

2011 Question 9
In the diagram below, ∠𝐴𝐵𝐷 = ∠𝐴𝐶𝐵 = 𝜃°. 𝐵𝐷 = 24
cm and 𝐵𝐶 = 40 cm. The area of the triangle 𝐴𝐵𝐷 is
100 cm .

The area of triangle 𝐴𝐵𝐶, in cm , is closest to

A. 167 B. 178 C. 267 D. 278 E. 378

2011 Question 3 Figure 2 Figure 3

The lighthosue has a lightroom, shown
shaded in Figure 2 below.

The floor of the lightroom is in the shape of

a regular octagon.

The longest distance across the floor is 4

metres.

The lightroom floor and ∠𝑃𝑂𝑄 = 𝜃° are

shown in Figure 3 below.

The size of the angle 𝜃 is 45°

b. Determine the area of triangle 𝑃𝑂𝑄.

Write your answer in square metres correct to one decimal place. 1 mark

____________________________________________________________________________________________________________________

____________________________________________________________________________________________________________________

2009 Question 2
The area (in m ) of triangle 𝑋𝑌𝑍 can be found
using Heron’s formula
  ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
𝐴 = 𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐), with
𝑎 = 1.92, 𝑏 = 8.24, 𝑐 = 9.20 and 𝑠 =
A. 4.40 B. 6.45 C. 9.20 D. 9.68 E. 19.36

2010 Question 1
2010 Question 1
𝐶𝑊 is a 90 metre straight path between the canoeing activity and a water slide located at point 𝑊.
𝐺𝑊 is a straight path between the entry gate and the water slide. The angle 𝐺𝐶𝑊 is 120°.

d. i. Find the area that is enclosed by the three paths, 𝐺𝐶, 𝐶𝑊 and 𝐺𝑊.
Write your answer in square metres, correct to one decimal place. 1 mark

____________________________________________________________________________________________________________________

____________________________________________________________________________________________________________________

____________________________________________________________________________________________________________________

2009 Question 3
The ferry has a logo painted on its side. The logo is a regular
pentagon with centre 𝑂 and side length 30 cm. It is shown in the
diagram below.

Angle 𝑃𝑂𝑄 is equal to 72°. The length 𝑂𝑃 is 25.52 cm.

c. Find the area of the pentagon.

Write your answer correct to the nearest cm . 2 marks

_____________________________________________________________________________

_____________________________________________________________________________

____________________________________________________________________________________________________________________

____________________________________________________________________________________________________________________

____________________________________________________________________________________________________________________

2008 Question 3
An orienteering course is triangular in shape and
is marked by three points, 𝐴, 𝐵 and 𝐶, as shown
in the diagram below.

In this course, 𝐵 is 7.0 km from 𝐴, 𝐶 is 8.0 km

from 𝐵 and 𝐴 is 12.3 km from 𝐶. The area (in
km ) enclosed by this course is closest to
A. 21 B. 24 C. 25 D. 26 E. 28

2008 Question 8
2008 Question 8
A regular hexagon has side length 3.0 cm and height 5.2 cm as shown
in the diagram above.

The area (in cm ) of the hexagon is closest to

A. 11.7 B. 13.5 C. 15.6 D. 18.0 E. 23.4

2008 Question 2
The shed has the shape of a prism. Its front
face, 𝐴𝑂𝐵𝐶𝐷, is shaded in the diagram below.
𝐴𝐵𝐶𝐷 is a rectangle and 𝑀 is the mid point of
𝐴𝐵. The length of 𝑂𝑀 is 1.6 m.

b. Show that the area of the front face of the

shed, 𝐴𝑂𝐵𝐶𝐷, is 18 m . 1 mark

____________________________________________________________________________________________________________________

____________________________________________________________________________________________________________________

____________________________________________________________________________________________________________________

____________________________________________________________________________________________________________________

2001 Question 3
The diagram below shows the kite, 𝐴𝐵𝐶𝐷. The length of side 𝐴𝐵 is 40
centimetres and 𝐵𝐶 is 80 centimetres. The size of ∠𝐴𝐵𝐶 is 125°.

The frame of the kite is made from two supporting pieces of

lightweight wood, 𝐴𝐶 and 𝐵𝐷, which cross at right angles.
The length 𝐴𝐶 is 108 m.

b. The kite is covered with a lightweight fabric which is cut exactly to

fit the kite shape. Assuming there is no overlapping of the fabric,
show that the area of fabric needed to make the kite is 2621 cm to
the nearest square centimetre. 2 marks

_______________________________________________________________________________

_______________________________________________________________________________

____________________________________________________________________________________________________________________

____________________________________________________________________________________________________________________

____________________________________________________________________________________________________________________

2007 Question 1
2007 Question 1
On the front face of Figure 1, 𝐴𝐵𝑅𝑄, Tessa marks point 𝑊 halfway
between 𝑄 and 𝑅 as shown in Figure 2 below. She then draws line
segments 𝐴𝑊 and 𝐵𝑊 as shown.

d. What fraction of the area of the rectangle 𝐴𝐵𝑅𝑄 does the area of
the triangle 𝐴𝑊𝐵 represent? 1 mark

____________________________________________________________________________

Figure 2

2007 Question 3
Tessa is a student in a woodwork class. The class will construct
geometrical solids from a block of wood. Tessa has a piece of wood
in the shape of a rectangular prism. This prism, 𝐴𝐵𝐶𝐷𝑄𝑅𝑆𝑇, shown
in Figure 1, has base length 24 cm, base width 28 cm and height 32
cm.

Tessa’s next task is to carve the right rectangular pyramid ABCDY

shown in Figure 4 below. She marks a new point, Y, halfway
between points W and V in Figure 3. She uses point Y to construct
this pyramid.
Figure 1
c. Using 𝐴𝑌 as 37 cm, demonstrate the use of Heron’s formula to
calculate the area, in cm , of the triangular face YAB. 2 marks

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________ Figure 4

2006 Question 5
A block of land is triangular in shape. The three sides measure 36 m, 58 m and 42 m. To calculate the
area, Heron’s formula is used. The correct application of Heron’s formula for this triangle is

  ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
A. Area = 136(136 − 36)(136 − 58)(136 − 42)
  ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
B. Area = 136(136 − 18)(136 − 29)(136 − 21)
  ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
C. Area = 68(68 − 36)(68 − 58)(68 − 42)
  ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
D. Area = 68(68 − 18)(68 − 29)(68 − 21)
  ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
E. Area = 68(136 − 36)(136 − 58)(136 − 42)

2005 Question 1
The area of the triangle 𝑃𝑄𝑅 is closest to
A. 3.3 cm B. 6.3 cm C. 10.6 cm
D. 12.5 cm E. 22.7 cm
2004 Question 1
A yacht has two flat triangular sails as shown in
the diagram below.

The sail 𝐷𝐸𝐹 has side lengths 𝐷𝐸 = 2.7 metres

and 𝐷𝐹 = 8.3 metres. The angle 𝐸𝐷𝐹 is 130°.
The length 𝐸𝐹 is 10.2 m.

d. Calculate the area of the sail 𝐷𝐸𝐹.

Write your answer in square metres, correct to
one decimal place. 1 mark

____________________________________________________________________________________________________________________

____________________________________________________________________________________________________________________

2003 Question 2
A paved area is constructed in the shape of a regular octagon as shown below.
The size of the angle 𝐺𝑂𝐻 is 45°, where point 𝑂 is the centre of the octagon
b. The length 𝑂𝐺 = 𝑂𝐻 = 2.30 metres.
Calculate the area of the octagonal paved area.
Write your answer correct to the nearest square metre. 2 marks

____________________________________________________________________________________________________________________

____________________________________________________________________________________________________________________

____________________________________________________________________________________________________________________

____________________________________________________________________________________________________________________

____________________________________________________________________________________________________________________

2002 Question 2
A chocolate bar is made in the shape of a
triangular prism. Each end is an equilateral
triangle of side length 3 centimetres.

The area of a triangular end of this chocolate

bar is closest to
A. 2.3 cm B. 3.9 cm C. 4.2 cm
D. 4.5 cm E. 9.0 cm

2006 Question 1
2006 Question 1
A farmer owns a flat allotment of land in the shape of triangle 𝐴𝐵𝐶
shown below. Boundary 𝐴𝐵 is 251 metres. Boundary 𝐴𝐶 is 142
metres. Angle 𝐵𝐴𝐶 is 45°. A straight track, 𝑋𝑌, runs perpendicular to
the boundary 𝐴𝐶. Point 𝑌 is 55 m from 𝐴 along the boundary 𝐴𝐶.

The size of angle 𝐴𝑋𝑌 is 45. The length of 𝐴𝑋 77.8 m.

e. Determine the area of triangle 𝐴𝐵𝐶 correct to the nearest square

metre. 1 mark

_______________________________________________________________________________

_______________________________________________________________________________

_______________________________________________________________________________

______________________________________________________________________________

A farmer plans to build a fence, 𝑀𝑁, perpendicular to the boundary 𝐴𝐶.

The land enclosed by triangle 𝐴𝑀𝑁 will have an area of 3200 m .

g. Determine the length of the fence 𝑀𝑁. 2 marks

________________________________________________________________________________

________________________________________________________________________________

________________________________________________________________________________

________________________________________________________________________________

________________________________________________________________________________

2002 Question 1
The diagram below shows a camping
ground by the sea. The boundary 𝑃𝑄 is
1203 metres long and runs beside an
east-west road. The boundary 𝑃𝑆 is
1048 metres long and the bearing of 𝑆
from 𝑃 is 015° True. The boundary 𝑄𝑅
is 951 metres long and the angle 𝑃𝑄𝑅
is 80°. The fourth boundary of the
camping ground is along a cliff edge by
the sea.

a. Find the area of triangle 𝑃𝑄𝑅 in square metres, correct to the nearest square metre. 2 marks

____________________________________________________________________________________________________________________

____________________________________________________________________________________________________________________