Geometry – Measurement & Calculation — Free IB MYP3 Mathematics Questions

1QuestionSolving Surface Area Problems in ContextConcept Practice

2 marks~3 minCriterion A

The net of a cuboid has side lengths of 3 cm, 4 cm, and 5 cm.

a

Calculate the area of each different rectangular face of the cuboid. [1]

b

Calculate the total surface area of the cuboid. [1]

Solutions

2QuestionIntroduction to Volume of CylindersConcept Practice

2 marks~3 minCriterion A

A cylinder has a radius of 4 cm and a height of 10 cm.

a

Write the formula for the volume of a cylinder. [1]

b

Calculate the volume of the cylinder. Give your answer in terms of

\pi

. [1]

Solutions

3QuestionArea and Perimeter of Compound ShapesConcept Practice

2 marks~3 minCriterion A

The diagram shows a compound shape made from a rectangle and a semicircle. The rectangle has a length of 10 cm and a width of 6 cm. The semicircle is attached to one of the 6 cm sides.

a

Calculate the radius of the semicircle. [1]

b

Explain how many sides of the rectangle form part of the perimeter of the compound shape. [1]

Solutions

4QuestionCreating and Interpreting Scale DiagramsConcept Practice

2 marks~3 minCriterion A

A scale diagram of a park uses a scale where

1 \text{ cm}

represents

5 \text{ m}

.

On the diagram, the distance between the fountain and the bench measures

3 \text{ cm}

.

(a) Calculate the actual distance, in metres, between the fountain and the bench. [2]

Solutions

5QuestionConverting Area UnitsConcept Practice

3 marks~5 minCriterion B

The diagram shows a square with a side length of

1\ \text{m}

, divided into a grid of smaller squares each measuring

10\ \text{cm} \times 10\ \text{cm}

.

a

Calculate the total number of smaller squares in the grid. [1]

b

Calculate the area of the large square in square centimetres (

\text{cm}^2

). [1]

c

Explain what your answers to (a) and (b) show about the conversion factor between square metres and square centimetres. [1]

Solutions

6QuestionConverting Area UnitsConcept Practice

2 marks~3 minCriterion A

A rectangle has a length of 4 cm and a width of 3 cm. Each small square on the grid measures 1 cm by 1 cm.

a

Calculate the area of the rectangle in

\text{cm}^2

. [1]

b

Calculate the area of the rectangle in

\text{mm}^2

. Show your conversion clearly. [1]

Solutions

7QuestionArea of Trapeziums and Compound ShapesConcept Practice

2 marks~3 minCriterion B

The diagram shows a trapezium. The two parallel sides are labelled

a

and

b

. The perpendicular distance between them is labelled

h

.

a

Identify which measurements are the bases and which is the height of the trapezium. Use correct mathematical notation in your response. [1]

b

Explain why the area of a trapezium is calculated using the formula

A = \dfrac{1}{2}(a+b)h

. [1]

Solutions

8QuestionArea of Rectangles Triangles and ParallelogramsConcept Practice

2 marks~3 minCriterion A

The diagram shows a right triangle and a parallelogram drawn on a centimetre grid.

a

Calculate the area of the triangle. [1]

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}

b

Calculate the area of the parallelogram. [1]

\text{Area} = \text{base} \times \text{height}

Solutions

9QuestionCalculating Perimeter of Squares and RectanglesConcept Practice

2 marks~3 minCriterion C

A gardener plans to place fencing around a rectangular vegetable plot.

a

State one real-world situation, other than a garden, where calculating the perimeter of a rectangle is useful. [1]

b

Explain why the formula

P = 2(l + w)

works for a rectangle, and identify one assumption this formula makes about the shape. [1]

Solutions

10QuestionUnderstanding Surface Area ConceptuallyAssessment Practice

5 marks~8 minCriterion C

An L-shaped prism is formed by joining Prism A (dimensions

3 \text{ cm} \times 4 \text{ cm} \times 5 \text{ cm}

) and Prism B (dimensions

3 \text{ cm} \times 2 \text{ cm} \times 5 \text{ cm}

) along a

3 \text{ cm} \times 5 \text{ cm}

face. The surface area of Prism A alone is

94 \text{ cm}^2

and of Prism B alone is

62 \text{ cm}^2

.

a

Calculate the area of the shared face. [1]

b

Explain why the total surface area of the L-shaped prism is not

94 + 62 = 156 \text{ cm}^2

. [2]

c

Draw a net of the L-shaped prism, label the hidden faces, and calculate the correct surface area. [2]

Solutions

11QuestionSolving Surface Area Problems in ContextAssessment Practice

6 marks~9 minCriterion D

A packaging company designs a rectangular box with length 20 cm, width 15 cm, and height 10 cm to hold a fragile cylindrical object with radius 6 cm and height 9 cm. The surface area of the box is found using:

S = 2(lw + lh + wh)

a

Calculate the total surface area of the box. [2]

b

Explain why the box dimensions are sufficient to contain the cylinder. [1]

c

Explain two reasons why the surface area formula alone may underestimate the amount of cardboard needed to manufacture this box. [3]

Solutions

12QuestionUnderstanding Surface Area ConceptuallyAssessment Practice

3 marks~5 minCriterion B

The graph shows how the surface area of a cube changes as its side length increases from 1 cm to 5 cm.

Side length (cm)	1	2	3	4	5
Surface area (cm²)	6	24	54	96	150

a

State the formula for the surface area of a cube with side length

s

. Use it to calculate the surface area when

s = 4

cm. [1]

b

Explain why the surface area does not increase by the same amount each time the side length increases by 1 cm. [1]

c

The side length doubles from 2 cm to 4 cm. Analyse what happens to the surface area and explain why this result is not a surprise given the formula. [1]

Solutions

13QuestionSolving Volume Problems with UnitsAssessment Practice

5 marks~8 minCriterion C

A student claims that a cylindrical water tank with a radius of

0.3

m and a height of

1.2

m holds more water than a rectangular prism tank with a length of

1.5

m, a width of

0.4

m, and a height of

0.6

m, because the cylinder has a large radius.

a

Calculate the volume of each tank. Show all working and include units. [3]

b

Explain which tank holds more water, using your results from part (a). [1]

c

Analyse the flaw in the student's reasoning. [1]

Solutions

14QuestionIntroduction to Volume of CylindersAssessment Practice

2 marks~3 minCriterion D

A farmer needs to store water for irrigation and is considering two cylindrical tanks. Tank A has a radius of 1.5 m and a height of 4 m. Tank B has a radius of 2 m and a height of 2.25 m. The volume of a cylinder is given by

V = \pi r^2 h

. Use

\pi = 3.14

.

a

Calculate the volume of Tank A and Tank B. [1]

b

The farmer's shed has a height limit of 2.5 m. Explain which tank is more practical for the farmer, and state one limitation of using volume alone to decide which tank to purchase. [1]

Solutions

15QuestionVolume of a Rectangular PrismAssessment Practice

3 marks~5 minCriterion B

A rectangular prism has a fixed length of

l = 5

cm and a fixed width of

w = 4

cm. Its volume

V

(cm³) is plotted against its height

h

(cm) in the graph below.

a

Apply the formula

V = l \times w \times h

to write an equation for

V

in terms of

h

only. [1]

b

Explain why the graph of

V

against

h

is a straight line through the origin. [1]

c

The graph shows a point at

h = 3

cm. Interpret what the corresponding value of

V

tells you about the prism at that height. [1]

Solutions

16QuestionArea and Perimeter of Compound ShapesAssessment Practice

6 marks~9 minCriterion D

A gardener plans two overlapping rectangular flower beds and needs to find the total area to calculate the volume of soil required at a uniform depth of

0.3 \, \text{m}

.

Bed A:

4 \, \text{m} \times 3 \, \text{m}

Bed B:

5 \, \text{m} \times 2 \, \text{m}

Overlapping region:

2 \, \text{m} \times 1 \, \text{m}

a

Calculate the total area of the two flower beds combined. [2]

b

Calculate the total volume of soil required. [1]

c

The gardener's model assumes perfectly rectangular beds and a uniform soil depth of

0.3 \, \text{m}

. Explain two reasons why this model may not accurately predict the actual volume of soil needed in a real garden. [3]

Solutions

17QuestionDrawing and Interpreting Nets of 3D ShapesAssessment Practice

6 marks~9 minCriterion B

A right prism has a regular pentagonal base with side length 4 cm and height 10 cm.

The area of a regular polygon with

n

sides of length

s

is

A = \dfrac{ns^2}{4\tan(\pi/n)}

.

a

Calculate the lateral surface area of the pentagonal prism. [2]

b

Calculate the area of one pentagonal base. Show your working. [2]

c

A student claims: "The more sides the base has, the greater the total surface area." Using your results and the total surface areas of a triangular prism (

\approx 133.9\ \text{cm}^2

) and a hexagonal prism (

\approx 323.1\ \text{cm}^2

), all with side length 4 cm and height 10 cm, analyse whether this claim is correct. [2]

Solutions

18QuestionBreaking Down Compound Shapes into Simple PartsAssessment Practice

3 marks~5 minCriterion C

The table below shows the side length and area of a square.

Side length $s$ (cm)	0	1	2	3	4	5
Area $A$ (cm²)	0	1	4	9	16	25

a

State the formula that connects the side length

s

and the area

A

of a square. [1]

b

Explain why the graph of this relationship is a curve rather than a straight line. [1]

c

A student claims: "If the side length doubles, the area also doubles." Analyse this claim using values from the table. [1]

Solutions

19QuestionCreating and Interpreting Scale DiagramsAssessment Practice

4 marks~6 minCriterion D

A hiker plans a route using a map with a scale of

1:50\,000

. The straight-line trail measures

24\ \text{cm}

on the map. A fold in the map causes a measurement error of

3\ \text{mm}

.

a

Calculate the actual distance, in kilometres, represented by the

3\ \text{mm}

error. [2]

b

Explain how this measurement error and the fact that a map is a flat representation of curved terrain affect the reliability of using the map to plan the hiking route. [2]

Solutions

20QuestionMeasuring Bearings with a ProtractorAssessment Practice

5 marks~8 minCriterion C

A surveyor stands at point

A

and measures the bearing of point

B

as

056°

. The interior angle

\angle BAC

of the triangular field is

47°

, with

C

lying clockwise from

B

as seen from

A

.

a

State what a three-figure bearing measures. [1]

b

Calculate the bearing of

C

from

A

. [2]

c

Justify whether a bearing of

009°

for

C

from

A

would be consistent with the given information. [2]

Solutions

21QuestionMeasuring Bearings with a ProtractorAssessment Practice

8 marks~12 minCriterion B

A ship sails in a straight line past a lighthouse

L

. The table shows the bearing of

L

measured from the ship at four equally spaced positions.

Position	$P_1$	$P_2$	$P_3$	$P_4$
Bearing of $L$	$340°$	$350°$	$010°$	$020°$

a

Describe the pattern in the bearings as the ship moves from

P_1

to

P_4

. [2]

b

Apply the pattern to predict the bearing of

L

from position

P_5

, the next equally spaced position along the same line. Explain your reasoning. [3]

c

Using the scale

1 \text{ cm} = 10 \text{ m}

, the spacing between consecutive positions is

2 \text{ cm}

. A scaled diagram is provided showing

P_1

to

P_5

and the lighthouse

L

. Measure the bearing of

L

from

P_5

using a protractor, and analyse whether your measurement supports your prediction from part (b). [3]

Solutions

22QuestionCommon Mistakes in Unit ConversionAssessment Practice

6 marks~9 minCriterion C

A homeowner measures a rectangular room: length

5\ \text{m}

, width

4\ \text{m}

. The flooring supplier charges per square centimetre. The homeowner calculates the area as

20\ \text{m}^2

, then converts it as

20 \times 100 = 2000\ \text{cm}^2

and orders flooring for

2000\ \text{cm}^2

.

a

Identify the error in the homeowner's unit conversion. [1]

b

Calculate the correct area of the room in

\text{cm}^2

. [2]

c

The supplier quotes a price based on the homeowner's order of

2000\ \text{cm}^2

. Analyse how the conversion error affects the total cost, and explain one assumption about the room that could affect the accuracy of the area. [3]

Solutions

23QuestionConverting Between Metric Length UnitsAssessment Practice

6 marks~9 minCriterion D

An architect creates a scale model of a building using a scale of

1 : 200

, where

1\ \text{cm}

on the model represents

2\ \text{m}

in reality. A wall on the model measures

12.5\ \text{cm}

. Construction workers measure to the nearest centimetre.

a

Calculate the real length of the wall in metres. [2]

b

Explain why rounding the real length to the nearest whole metre would cause problems for construction workers. [2]

c

The architect considers rounding the real length to the nearest centimetre instead. Analyse whether this is a more suitable level of precision for construction, given the limitations of the scale model. [2]

Solutions

24QuestionArea of Rectangles Triangles and ParallelogramsAssessment Practice

4 marks~6 minCriterion D

A rectangular garden measures 12 m by 8 m. A triangular flower bed, with base 4 m and height 3 m, is cut from one corner of the garden, leaving a grass area.

a

Calculate the area of the rectangular garden. [1]

b

Calculate the area of the triangular flower bed. [1]

c

The gardener claims the remaining grass covers more than 90% of the original garden. Using your results from (a) and (b), determine the remaining grass area and explain whether the gardener's claim is correct. [2]

Solutions

Geometry – Measurement & Calculation — Free MYP3 Mathematics Practice Questions