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Cambridge IGCSE0580

Compound shapes and parts of shapes

Mathematics 0580 Chapter Notes

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Compound shapes and parts of shapes
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In this chapter

  1. 1Perimeter and area of compound shapes
  2. 2Parts of a circle
  3. 3Shaded areas by subtraction
  4. 4Volume of compound solids
  5. 5Surface area of compound solids
  6. 6Parts of solids
  7. 7The frustum

1. Perimeter and area of compound shapes

A compound shape is made from simpler shapes joined together. Start by splitting the shape into rectangles, triangles or circles, or by taking one shape away from another.

For the perimeter, add only the edges on the outside. An edge where two parts join is not part of the perimeter. For the area, add the areas of the parts, or subtract the part that is missing.

Write any missing lengths on the sketch before you start.

area of rectangle = length × width

area of triangle = 1/2 × base × height

Examiner insight

“Candidates should be encouraged to write on the diagram to mark in missing lengths and to draw lines to split the compound shape into separate rectangles.” “For the square section, candidates had to remember not to include the forth side and similarly with the semicircle, not to include the diameter; often, candidates did not take account of one or other of these points.”

Worked example 14 marks

Do not use a calculator.

A rectangular field measures 10 m by 6 m. A rectangular corner, 4 m by 3 m, is cut off it. Find the perimeter and the area of the field that is left.

  1. 1

    cutting off a corner takes away a 4 m edge and a 3 m edge and puts back a 3 m edge and a 4 m edge, so the perimeter does not change [1]

  2. 2

    perimeter = 2(10 + 6) = 32 m [1]

  3. 3

    area = 10 × 6 - 4 × 3 [1]

  4. 4

    = 60 - 12 = 48 m² [1]

2. Parts of a circle

A semicircle is half a circle and a quarter circle is a quarter. A sector is the fraction θ/360 of a circle, where θ is its angle at the centre.

A part of a circle has an arc (the curved edge) and straight edges. Its perimeter is the arc plus the straight edges: the diameter for a semicircle, two radii for a sector.

When the question says in terms of π, leave π in the answer. Do not turn it into a decimal.

C = 2π r A = π r²

arc = θ/360 × 2π r sector area = θ/360 × π r²

Examiner insight

“Some candidates calculated the length of the major arc but did not add on the two radii for the perimeter.” “Many worked out a value for the area and even those who did find the correct answer in terms of π often changed it to a numerical value for their final answer.”

Worked example 14 marks

Do not use a calculator.

A semicircle has diameter 12 cm. Find its perimeter and its area. Give both answers in terms of π.

  1. 1

    radius = 12 ÷ 2 = 6 cm [1]

  2. 2

    arc = 1/2 × 2π × 6 = 6π [1]

  3. 3

    perimeter = 6π + 12 cm [1]

  4. 4

    area = 1/2 × π × 6² = 18π cm² [1]

3. Shaded areas by subtraction

A shaded region is often one shape with another cut out of it. Find both areas, then subtract the smaller from the larger.

A diameter must be halved to give the radius before you use π r². Check which formula you need: area, not circumference.

shaded area = area of whole shape - area of removed shape

Examiner insight

“Some did not check the formula page meaning circumference was worked out rather than area.” “Many candidates found both areas but did not subtract one from the other as required.”

Worked example 13 marks

Do not use a calculator.

A small circle lies inside a large circle. The small circle has diameter 8 cm. The large circle has diameter 20 cm. The shaded area is the part of the large circle outside the small circle.

Work out the shaded area. Give your answer in terms of π.

  1. 1

    radii are 10 cm and 4 cm [1]

  2. 2

    shaded area = π × 10² - π × 4² [1]

  3. 3

    = 100π - 16π = 84π cm² [1]

4. Volume of compound solids

A compound solid is made from simpler solids joined together, such as a cylinder with a hemisphere on one end. Find the volume of each part, then add them. If a part has been cut out, subtract it.

For a prism or cylinder, volume is the area of the cross-section times the length. Keep π in your working and round only at the end.

V_cylinder = π r² h

V_sphere = 4/3π r³

Examiner insight

“Many correct answers were seen but some slipped up right at the start by using an incorrect formula for the volume of a cylinder.”

Worked example 13 marks

Do not use a calculator.

A solid is a cylinder of radius 3 cm and height 10 cm with a hemisphere of radius 3 cm joined to one end. Find the volume of the solid in terms of π.

  1. 1

    cylinder = π × 3² × 10 = 90π [1]

  2. 2

    hemisphere = 1/2 × 4/3π × 3³ = 18π [1]

  3. 3

    total = 90π + 18π = 108π cm³ [1]

5. Surface area of compound solids

Surface area is the total area of every outside face. Do not confuse it with volume.

Where two parts are joined, that face is hidden, so leave it out. A closed cylinder has a curved surface and two circular ends.

List the faces so none is missed.

curved surface of cylinder = 2π r h

total surface of closed cylinder = 2π r h + 2π r²

surface of sphere = 4π r²

Examiner insight

“A significant number of candidates found the volume, rather than the surface area.” “Others did not consider the faces that were hidden on the diagram and so only gave half of the total surface area.”

Worked example 14 marks

Do not use a calculator.

A solid is a cylinder of radius 3 cm and height 10 cm with a hemisphere of radius 3 cm joined to one end. The other end of the cylinder is flat. Find the total surface area of the solid in terms of π.

  1. 1

    curved surface of cylinder = 2π × 3 × 10 = 60π [1]

  2. 2

    flat end = π × 3² = 9π [1]

  3. 3

    curved surface of hemisphere = 1/2 × 4π × 3² = 18π [1]

  4. 4

    total = 60π + 9π + 18π = 87π cm² [1]

6. Parts of solids

A hemisphere is half a sphere. Its volume is half the volume of a sphere. Do not use the whole-sphere formula.

A solid hemisphere also has a flat circular face. Count it for the total surface area, but not for the curved surface alone.

The volume of any other part of a solid, such as half a cylinder, is the same fraction of the whole volume.

V_hemisphere = 2/3π r³

A_curved = 2π r² A_total = 3π r²

Examiner insight

“The two most common errors were squaring the radius rather than cubing it and working out the volume of a sphere, and not a hemisphere, by not dividing by 2.”

Worked example 14 marks

Do not use a calculator.

A solid hemisphere has radius 6 cm. Find its volume and its total surface area. Give both answers in terms of π.

  1. 1

    volume = 1/2 × 4/3π × 6³ [1]

  2. 2

    = 144π cm³ [1]

  3. 3

    curved surface = 1/2 × 4π × 6² = 72π [1]

  4. 4

    add the flat face π × 6² = 36π: total = 108π cm² [1]

7. The frustum

A frustum is a cone with the top cut off by a cut parallel to the base. Its volume is the large cone minus the small cone that was removed.

The two cones are similar, so their radii and heights are in the same ratio. Use that ratio to find the height of the small cone. The height of the frustum is not the height of the small cone.

For the surface area, subtract the curved surfaces of the two cones in the same way, then add the two circular ends. Here l is the slant height of a cone.

V_cone = 1/3π r² h

V_frustum = V_large cone - V_small cone

curved surface of cone = π r l

Examiner insight

“In terms of finding the volume of the frustum it appeared that many were familiar with the approach of finding the volume of the larger cone and then subtracting the volume of the smaller, removed, cone.” “Finding the volume of the smaller cone proved more of a challenge, with a significant number not realising that the radius of the smaller cone was not 4.2 cm.”

Worked example 15 marks

A frustum has a circular base of radius 12 cm, a circular top of radius 8 cm and a height of 9 cm.

Calculate the volume of the frustum.

  1. 1

    let the small cone have height h: h/(h + 9) = 8/12, so h = 18 cm [1]

  2. 2

    large cone height = 18 + 9 = 27 cm [1]

  3. 3

    large cone = 1/3π × 12² × 27 = 1296π [1]

  4. 4

    small cone = 1/3π × 8² × 18 = 384π [1]

  5. 5

    frustum = 912π = 2865 cm³ (2870 to 3 significant figures) [1]

Recap

  • Compound shape: split it into simple shapes, or subtract one from another. Perimeter counts outside edges only.
  • Sector: the fraction θ/360 of the circle. Its perimeter is the arc plus two radii.
  • In terms of π: leave π in the answer. Halve a diameter to get the radius.
  • Compound solid: volume is the sum of the parts. Surface area leaves out the joined faces.
  • Cylinder: V = π r² h. Total surface area = 2π r h + 2π r².
  • Hemisphere: volume 2/3π r³, curved surface 2π r², total surface 3π r².
  • Extended: frustum volume is the large cone minus the small cone. Find the small cone's height or radius from similar triangles. Its surface area is the difference of the curved surfaces plus the two circular ends.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Do not use a calculator. A semicircle has diameter 14 cm. Find its perimeter and its area. Give both answers in terms of π.4 marks
  2. Do not use a calculator. A rectangle is 12 cm by 7 cm. A rectangular corner, 5 cm by 2 cm, is cut off it. Find the perimeter of the shape that is left.2 marks
  3. Do not use a calculator. A sector of a circle has radius 5 cm and angle 72°. Find its perimeter in terms of π.3 marks
  4. A circle of radius 4 cm is cut from a square piece of card of side 10 cm. Calculate the percentage of the card that is left. Give your answer to 3 significant figures.4 marks
  5. Do not use a calculator. A closed cylinder has radius 3 cm and height 7 cm. Find its total surface area in terms of π.3 marks
  6. A solid hemisphere has radius 6 cm. Calculate its volume. Give your answer to 3 significant figures.2 marks
  7. Do not use a calculator. A block of wood is a cube of side 6 cm. A hemispherical hollow of radius 3 cm is cut into the top face. Find the volume of wood that is left, in terms of π.3 marks
  8. A water trough is half of a cylinder of radius 30 cm, cut along its length. The trough is 200 cm long. Calculate the capacity of the trough in litres. Give your answer to 3 significant figures.4 marks
  9. A bucket is a frustum. Its circular top has radius 20 cm, its circular base has radius 15 cm and it is 30 cm high. Calculate the capacity of the bucket in litres. Give your answer to 3 significant figures.5 marks
  10. A solid wooden cone has base radius 9 cm and height 12 cm. A cut parallel to the base, 8 cm above the base, removes the top of the cone. The frustum that is left is varnished on every face. Calculate the area that is varnished. Give your answer to 3 significant figures.6 marks

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