Studyledge

Area, Surface Area and Volume: O-Level / IGCSE Maths (0580)

Syllabus C5.2, E5.2, C5.4, E5.4 · Strand 5 Mensuration

Questions
8
Total marks
45
Tier mix
3 Core · 5 Extended

This topic brings together two closely linked ideas from the Mensuration strand: measuring the two-dimensional size of flat shapes and the three-dimensional size of solids. On the plane you work with perimeter and area of rectangles, triangles, parallelograms and trapezia, then extend to the circle, where the circumference is C=2πrC = 2\pi r and the area is A=πr2A = \pi r^2. From there you handle parts of a circle: an arc length and sector area are found by taking the fraction θ360\dfrac{\theta}{360} of the whole, giving arc =θ360 (2πr)= \dfrac{\theta}{360}\,(2\pi r) and sector area =θ360 (πr2)= \dfrac{\theta}{360}\,(\pi r^2).

The solids half of the topic covers prisms, cylinders, spheres, cones and pyramids. A useful unifying method is that any prism or cylinder has volume V=(cross-sectional area)×(length)V = (\text{cross-sectional area}) \times (\text{length}), so a cylinder gives V=πr2hV = \pi r^2 h. The standard formulas you apply include the sphere V=43πr3V = \tfrac{4}{3}\pi r^3 with surface area 4πr24\pi r^2, the cone V=13πr2hV = \tfrac{1}{3}\pi r^2 h with curved surface area πrl\pi r l (where ll is the slant height), and the pyramid V=13×base area×heightV = \tfrac{1}{3} \times \text{base area} \times \text{height}. Surface area is built up face by face: for a closed cylinder, for instance, you add the curved part 2πrh2\pi r h to the two circular ends 2πr22\pi r^2. Careful, consistent units and clear distinction between perpendicular height and slant height are the habits that make these questions reliable.

The key methods are choosing the right formula for the shape, substituting accurately, and keeping units consistent (for example converting between cm2\text{cm}^2 and m2\text{m}^2, or cm3\text{cm}^3 and litres). Many exam-style questions combine shapes (a composite area, or a solid such as a cylinder topped with a hemisphere), so you split the figure into parts, evaluate each, then add or subtract. The worked examples below are original, written from the syllabus objective, and each is followed by a full worked solution.

Question 1

Structured Core 6 marks

A landscape designer is planning a flower bed for a town square. The flower bed is made by joining a semicircle to one of the shorter sides of a rectangle, so the two shapes form a single flat bed.

The rectangle measures 10 m10\text{ m} by 6 m6\text{ m}. The straight side that the rectangle and the semicircle share is 6 m6\text{ m} long, so the semicircle has a diameter of 6 m6\text{ m} and bulges outwards from that end of the rectangle.

Take π=3.142\pi = 3.142.

(a) Work out the total area of the flower bed. Give your answer in m2\text{m}^2. [3]

(b) The designer will fit a low metal edging around the complete outside of the flower bed. Work out the total length of edging needed. [3]

View worked solution →

Question 2

Structured Extended 8 marks

A solid ornament is made by fixing a cone on top of a cylinder, so that the flat circular face of the cone exactly covers the top circular face of the cylinder.

The cylinder has radius 6 cm6\text{ cm} and height 15 cm15\text{ cm}. The cone has the same radius, 6 cm6\text{ cm}, and a vertical (perpendicular) height of 8 cm8\text{ cm}.

The ornament stands upright with the cone sitting point-upwards on top of the cylinder, so the single solid looks like a sharpened pencil.

(a) Calculate the total volume of the ornament. Give your answer in terms of π\pi and also correct to the nearest cubic centimetre. [4]

(b) The entire outer surface of the ornament is painted. Calculate the total area that is painted. Give your answer in terms of π\pi and also correct to 3 significant figures. [4]

View worked solution →

Question 3

Multiple choice Extended 3 marks

A solid metal paperweight is made in the shape of a right circular cone. The cone has a base radius of 6 cm6\text{ cm} and a perpendicular height of 14 cm14\text{ cm}.

Taking π=3.142\pi = 3.142, which of the following is the volume of the paperweight, correct to 33 significant figures?

View worked solution →

Question 4

Structured Core 5 marks

A carpenter is making a flat wooden name-plate in the shape of a capital letter L. The name-plate is a compound shape made from two rectangles joined together to form one solid L-shape, as described below.

The name-plate stands upright. Its outline can be described by starting at the bottom-left corner and moving around the edge:

  • from the bottom-left corner, go up the tall left-hand side, a distance of 10 cm10\text{ cm};
  • then go right along the top edge of the upright part of the L, a distance of 6 cm6\text{ cm};
  • then go straight down the inside step, a distance of 4 cm4\text{ cm};
  • then go right along the top of the foot of the L, a distance of 8 cm8\text{ cm};
  • then go straight down the far right-hand side, a distance of 6 cm6\text{ cm};
  • finally go left along the whole bottom edge, back to the start, a distance of 14 cm14\text{ cm}.

All the corners are right angles.

(a) Work out the perimeter of the name-plate. [2]

(b) Work out the area of the name-plate. Give your answer in cm2\text{cm}^2. [3]

NOT TO SCALE

The L-shaped name-plate

View worked solution →

Question 5

Structured Core 5 marks

A circular ornamental pond in a park has a diameter of 14 m14\text{ m}.

(a) Work out the area of the surface of the pond. Give your answer in terms of π\pi. [3]

(b) A path is to be built around the edge of the pond. Work out the circumference of the pond. Give your answer correct to 3 significant figures. [2]

NOT TO SCALE

The circular pond (diameter 14 m)

View worked solution →

Question 6

Structured Extended 5 marks

A company makes solid closed cylinders out of aluminium to use as rollers.

Each cylinder has radius 4.54.5 cm and height 1111 cm.

(a) Work out the volume of one cylinder. Give your answer correct to 3 significant figures. [3]

(b) Work out the total surface area of one cylinder. Give your answer correct to 3 significant figures. [2]

View worked solution →

Question 7

Structured Extended 7 marks

A designer is cutting a flat metal plate in the shape of a sector of a circle for a fan blade.

The sector has centre OO, radius 12 cm12\text{ cm}, and a sector angle of 210∘210^\circ at OO.

Take π=3.142\pi = 3.142.

(a) Calculate the length of the arc ABAB. Give your answer in centimetres, correct to 3 significant figures. [2]

(b) Calculate the area of the sector. Give your answer in cm2\text{cm}^2, correct to 3 significant figures. [2]

(c) The designer will fit a thin strip of edging around the complete outside of the sector (the two straight radii and the curved arc). Calculate the total length of edging needed, correct to 3 significant figures. [3]

NOT TO SCALE

Sector AOB (radius 12 cm, angle 210°)

View worked solution →

Question 8

Structured Extended 6 marks

A solid trophy is made from a cylinder with a hemisphere fixed on top, so that the flat face of the hemisphere exactly covers the circular top of the cylinder.

The cylinder has a diameter of 12 cm12\text{ cm} and a height of 21 cm21\text{ cm}. The hemisphere has the same diameter as the cylinder.

[The volume of a sphere is 43πr3\frac{4}{3}\pi r^3.]

(a) Show that the radius used for both the cylinder and the hemisphere is 6 cm6\text{ cm}, and work out the volume of the hemisphere. Give your answer in terms of π\pi. [2]

(b) Work out the total volume of the trophy. Give your answer correct to 33 significant figures. [4]

View worked solution →