Mensuration — Free IB MYP4 Mathematics (Standard) Questions

1QuestionVolume of Cones Spheres and HemispheresConcept Practice

2 marks~3 minCriterion C

A manufacturer designs a novelty ice-cream scoop where a single spherical scoop of radius

r

sits on top of a cone of radius

r

and height

2r

.

V_{\text{cone}} = \frac{1}{3}\pi r^2 h \qquad V_{\text{sphere}} = \frac{4}{3}\pi r^3

a

Show that

V_{\text{cone}} = \dfrac{2}{3}\pi r^3

. [1]

b

The manufacturer claims the sphere holds twice the ice cream of the cone. Justify whether this claim is correct. [1]

Solutions

2QuestionSurface Area of Cones and SpheresConcept Practice

2 marks~3 minCriterion A

A party hat is designed in the shape of a cone. The base has a radius of 4 cm and a vertical height of 3 cm. The manufacturer needs to know the slant height to calculate the amount of card required.

Calculate the slant height of the cone. [2]

Solutions

3QuestionUnderstanding the Density FormulaConcept Practice

2 marks~3 minCriterion A

A solid rectangular prism used in a materials science demonstration has a length of 8 cm, a width of 5 cm, and a height of 3 cm. Its mass is 360 g. A material is classified as a heavy-density solid if its density exceeds 2.5 g/cm³.

a

Calculate the density of the prism. [1]

b

Justify whether the prism qualifies as a heavy-density solid. [1]

Solutions

4QuestionChanging Units in Density ProblemsConcept Practice

2 marks~3 minCriterion B

The density of three substances is given in two unit systems.

Water:

1.00 \ \text{g/cm}^3

and

1000 \ \text{kg/m}^3

Aluminium:

2.70 \ \text{g/cm}^3

and

2700 \ \text{kg/m}^3

Lead:

11.3 \ \text{g/cm}^3

and

11300 \ \text{kg/m}^3

a

Deduce the relationship between a density value expressed in

\text{g/cm}^3

and the same density expressed in

\text{kg/m}^3

. [1]

b

A sample of iron has a density of

7.87 \ \text{g/cm}^3

. Justify whether this density is greater or less than

8000 \ \text{kg/m}^3

. [1]

Solutions

5QuestionArea of Circular Sectors and SegmentsConcept Practice

2 marks~3 minCriterion A

A circle has centre

O

and radius

r

. Sector

OAB

has central angle

\theta°

.

a

Explain why sector

OAB

represents the fraction

\dfrac{\theta}{360}

of the full circle. [1]

b

Justify why the area of sector

OAB

is given by

A = \dfrac{\theta}{360} \times \pi r^2

. [1]

Solutions

6QuestionArea of Rectangles Triangles and ParallelogramsConcept Practice

4 marks~6 minCriterion A

A landscape architect is designing a parallelogram-shaped lawn with a base of 6 m and a perpendicular height of 4 m. She explains to her client that cutting a triangular section from one end of the lawn and repositioning it at the other end produces a rectangular shape with the same area.

a

Calculate the area of the rectangular shape formed. [1]

b

Deduce a general formula for the area of any parallelogram in terms of its base

b

and perpendicular height

h

. [1]

c

A rectangular patio of area 20 m² is planned alongside the lawn. The architect claims the lawn has a greater area than the patio. Justify whether this claim is correct. [2]

Solutions

7QuestionNets and Surface Area of Cubes and CuboidsConcept Practice

3 marks~5 minCriterion A

A packaging company designs cardboard trays. Each tray has a fixed width of 4 cm and a fixed height of 3 cm, but its length

l

(in cm) varies. The graph below shows the total surface area

S

(in cm²) of a closed cuboid tray as

l

varies from 0 to 10 cm.

a

Show that the surface area of the cuboid can be written as

S = 14l + 24

. [1]

b

Explain why the graph of

S

against

l

is a straight line. [1]

c

The company requires a tray with surface area no greater than 130 cm². Justify whether a tray of length 8 cm satisfies this requirement. [1]

Solutions

8QuestionComparing Volumes of Different SolidsAssessment Practice

2 marks~3 minCriterion B

A cone and a cylinder share the same base radius

r

and height

h

.

V_{\text{cone}} = \frac{1}{3}\pi r^2 h \qquad V_{\text{cylinder}} = \pi r^2 h

Justify the relationship between

V_{\text{cone}}

and

V_{\text{cylinder}}

, using correct mathematical notation and a general statement valid for all values of

r

and

h > 0

. [2]

Solutions

9QuestionVolume of Cones Spheres and HemispheresAssessment Practice

10 marks~15 minCriterion D

An engineer is designing a spherical water tower with a conical roof. The water tower consists of a hemisphere (the lower part) and a right circular cone (the roof) placed on top of the hemisphere, sharing the same circular base. The total height of the water tower (from the base of the hemisphere to the apex of the cone) is 12 m. The total volume of the water tower must be exactly 500 m³. The engineer uses the following model: The hemisphere has radius

r

(m). The cone has base radius

r

and height

h

(m). The total height:

r + h = 12

. The total volume:

\frac{2}{3}\pi r^3 + \frac{1}{3}\pi r^2 h = 500

.

a

Using the given equations, determine the radius

r

of the hemisphere (and cone). Show all working.

b

The engineer's model assumes the tank walls have negligible thickness. Discuss one real-world limitation of this assumption and how it would affect the actual water capacity compared to the calculated volume.

c

Suppose the height measurement of 12 m has a possible error of

\pm 0.1

m. Evaluate how this error could affect the calculated radius and the resulting water capacity. In your answer, comment on the potential impact on the water supply for a community relying on this tower.

Solutions

10QuestionChanging Units in Density ProblemsAssessment Practice

4 marks~6 minCriterion C

A graph of mass (kg) against volume (m³) for a sample of aluminium produces a straight line through the origin. The gradient of the line is

2700 \ \text{kg/m}^3

.

a

State the conversion factors needed to change kg to g and m³ to cm³. [1]

b

Show that

1 \ \text{kg/m}^3 = 0.001 \ \text{g/cm}^3

, and hence calculate the density of aluminium in g/cm³. [2]

c

A student claims that because the numerical value decreased from 2700 to 2.7, the density itself has decreased. Critique this claim. [1]

Solutions

11QuestionMulti-Step Problems with Density and VolumeAssessment Practice

4 marks~6 minCriterion D

A cargo shipping company uses the formula

\text{mass} = \text{density} \times \text{volume}

to estimate the total mass of mixed cargo loaded into a standard 20-foot container with an internal volume of

33.2 \text{ m}^3

. The cargo consists of 50 steel drums and 80 wooden crates. Each steel drum has a volume of

0.20 \text{ m}^3

and a density of

7800 \text{ kg/m}^3

. Each wooden crate has a volume of

0.25 \text{ m}^3

and a density of

600 \text{ kg/m}^3

.

a

Calculate the estimated total mass of the cargo. [2]

b

The company's safety regulations state that the total cargo mass must not exceed

95\,000 \text{ kg}

. Advise the shipping company whether the cargo should be cleared for loading, identifying one reason why the actual mass could differ from your estimate. [2]

Solutions

12QuestionArc Length and Sector Area CalculationsAssessment Practice

6 marks~9 minCriterion C

A landscape architect is designing decorative stone edging for four circular garden beds. Each bed uses a sector of stone with an arc length of 10 cm per unit scale. The sectors have radii of 3 cm, 6 cm, 9 cm, and 12 cm respectively.

The perimeter of a sector is given by

P = L + 2r

, where

L

is the arc length and

r

is the radius.

a

Calculate the perimeter of each sector. Record your results as ordered pairs

(r, P)

in order of increasing radius. [2]

b

Deduce the perimeter of a sector with radius 15 cm and the same arc length. Justify your reasoning using the pattern in your results. [2]

c

The architect claims that doubling the radius of a sector always produces a more efficient design because the perimeter does not double. Assess this claim using the formula

P = L + 2r

, and interpret what this means for the architect's choice of sector size. [2]

Solutions

13QuestionArc Length and Sector Area CalculationsAssessment Practice

6 marks~9 minCriterion D

A landscape architect is designing a circular fountain with a radius of 2.5 m. A sector with a central angle of 120° will be decorated with tiles. The client's budget allows for exactly 6.5 m² of tiling. The architect's radius measurements have a tolerance of ±0.1 m, and tile grout reduces the effective tiled area by 2%.

a

Calculate the area of the 120° sector using

A = \frac{\theta}{360} \pi r^2

where

\theta

is the central angle in degrees. [2]

b

Deduce whether the sector area exceeds the budget, stating the difference. [1]

c

Advise the architect whether the design is feasible within the budget, justifying your recommendation by considering the effect of the radius tolerance on the calculated area and the impact of grout on the effective tiled area. [3]

Solutions

14QuestionArea of Circular Sectors and SegmentsAssessment Practice

5 marks~8 minCriterion B

A circle has radius

r

and centre

O

. A sector of the circle has central angle

\theta°

.

a

State the arc length of the sector and the area of the full circle. [1]

b

A sector is dissected into many thin triangular strips, each with vertex at

O

, height approximately

r

, and base along the arc. The strips are rearranged into a shape that approximates a rectangle. Show that the area of the sector is

\dfrac{\theta}{360}\pi r^2

. [2]

c

A designer claims that a sector with

\theta = 120°

and

r = 8

cm has sufficient area to cover a surface of

165\ \text{cm}^2

. Assess whether this claim is correct, justifying your answer with a calculation. [2]

Solutions

15QuestionArea of Rectangles Triangles and ParallelogramsAssessment Practice

6 marks~9 minCriterion D

A rooftop garden is being designed on a parallelogram-shaped plot. The plot has a base of

8

m, a perpendicular height of

5

m, and slanted side lengths of

7

m.

a

Decompose the parallelogram into one rectangle and two congruent right-angled triangles by dropping perpendiculars from each top vertex to the base. Use the Pythagorean theorem to find the horizontal leg

x

of each right-angled triangle. [2]

b

Calculate the area of the rectangle and the combined area of the two right-angled triangles. Show that their sum equals

b \times h

. [2]

c

The garden designer claims that using a steeper slanted side of

9

m (keeping base

8

m and perpendicular height

5

m unchanged) would increase the area available for planting. Advise the designer whether this change should be made, justifying your answer with reference to what the slanted side length determines about the shape. [2]

Solutions

16QuestionArea of Rectangles Triangles and ParallelogramsAssessment Practice

5 marks~8 minCriterion C

A parallelogram has base

b

and perpendicular height

h

. A rectangle shares the same base

b

and perpendicular height

h

.

a

State the area formula for the parallelogram using algebraic notation. [1]

b

A diagonal is drawn across the parallelogram, forming two triangles. Justify, using congruence and algebraic reasoning, that the area of each triangle equals

\dfrac{1}{2}bh

. [2]

c

An architect claims that knowing the triangle area formula is sufficient to derive the rectangle area formula, without needing to treat the rectangle separately. Defend whether this claim is correct, showing your reasoning with algebraic notation. [2]

Solutions

17QuestionArea of Trapeziums and RhombusesAssessment Practice

5 marks~8 minCriterion B

A landscape architect is designing a trapezoidal flower bed with parallel sides of length

a

(top) and

b

(bottom), where

b > a

, and perpendicular height

h

.

a

Construct a clearly labelled diagram showing the trapezium decomposed into one rectangle and two right-angled triangles. Label the parallel sides

a

and

b

, the height

h

, and the horizontal bases of the two triangles

x

(left) and

y

(right). [1]

b

Using your diagram, justify that

x + y = b - a

, write algebraic expressions for the area of each component shape, and show that their sum simplifies to

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

. [3]

c

The architect states that a trapezoidal bed with

a = 3

m,

b = 7

m, and

h = 4

m has sufficient area to plant 20 lavender bushes, each requiring at least

0.9

m² of space. Assess whether the architect's claim is correct. [1]

Solutions

18QuestionNets and Surface Area of Cubes and CuboidsAssessment Practice

4 marks~6 minCriterion D

A cereal company is designing a cuboid-shaped box with a fixed volume of

3000 \text{ cm}^3

. Three sets of dimensions (length

\times

width

\times

height) are proposed:

Set A:

10 \text{ cm} \times 10 \text{ cm} \times 30 \text{ cm}

Set B:

5 \text{ cm} \times 20 \text{ cm} \times 30 \text{ cm}

Set C:

12.5 \text{ cm} \times 12.5 \text{ cm} \times 19.2 \text{ cm}

a

Calculate the surface area of Set A and Set B. [1]

b

Calculate the surface area of Set C. [1]

c

Advise the company which set of dimensions to choose to minimise material costs, and identify two real-world factors that the surface area model does not account for, explaining why each matters. [2]

Solutions

19QuestionNets and Surface Area of Cubes and CuboidsAssessment Practice

6 marks~9 minCriterion B

A packaging company designs closed rectangular boxes with a fixed volume of

24 \text{ cm}^3

. Minimising surface area reduces material cost.

a

Calculate the surface area of each box using

S = 2(lw + lh + wh)

. [2]

Dimensions

(l, w, h)

in cm:

(1, 4, 6)

;

(2, 3, 4)

;

(1, 3, 8)

;

(2, 2, 6)

.

b

Justify which set of dimensions minimises surface area by identifying a geometric property shared by those dimensions. [1]

c

Let the dimensions be

x

,

y

,

z

with

xyz = 24

. Show that

S = 2\!\left(xy + \dfrac{24}{x} + \dfrac{24}{y}\right)

. Hence prove, using the AM–GM inequality, that surface area is minimised when all three dimensions are equal. Advise the company whether to use the

(2, 3, 4)

box or commission a new cubic design, justifying your recommendation with reference to the theoretical minimum. [3]

Solutions

20QuestionNets and Surface Area of Cubes and CuboidsAssessment Practice

6 marks~9 minCriterion C

A packaging company is designing a cardboard box (a cuboid) with length

l = 8

cm, width

w = 5

cm, and height

h = 3

cm.

a

Construct a fully labelled net of the cuboid, showing the dimensions of all six faces. [2]

b

Show that the surface area of the cuboid is

158 \text{ cm}^2

, using the formula

SA = 2(lw + lh + wh)

. [2]

c

A trainee designer claims the surface area is

150 \text{ cm}^2

and orders cardboard accordingly. Advise the packaging manager whether this order should be accepted, identifying the most likely source of the designer's error. [2]

Solutions

Mensuration — Free MYP4 Mathematics (Standard) Practice Questions