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17.5 Composite 2D Shapes and Cylinders

GCSE Maths Foundation · Edexcel · Chapter 17: Perimeter, Area and Volume 2

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: areas and perimeters of compound shapes, then the volume and surface area of a cylinder. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

  • Find the area and perimeter of shapes made from rectangles and triangles
  • Find areas of shapes that include semicircles or circles, including shaded regions
  • Work out the volume of a cylinder and change cm³ to litres
  • Work out the total surface area of a cylinder

Key words

  • Composite shape: A shape made by joining (or cutting) simpler shapes such as rectangles, triangles and circles.
  • Cylinder: A prism with a circular cross-section, like a tin of beans.
  • Cross-section: The shape you see when you slice straight through a prism. It is the same all the way along.
  • Volume: The space inside a 3D shape, in cm³ or m³. 1 litre = 1000 cm³.
  • Surface area: The total area of all the faces (surfaces) of a 3D shape.

Shapes made from rectangles

Split the shape into rectangles with a dashed line, find any missing lengths, then add the areas. Or find a big rectangle and subtract the missing corner.

Worked example: an L-shape

Missing lengths: 10 − 6 = 4 cm (up the step) and 12 − 6 = 6 cm (across the step).
Area: bottom part 12 × 6 = 72 cm², top part 6 × 4 = 24 cm². Total 72 + 24 = 96 cm².
Check by subtracting: 12 × 10 − 6 × 4 = 120 − 24 = 96 cm².
Perimeter: 12 + 6 + 6 + 4 + 6 + 10 = 44 cm.

Answer: Area 96 cm², perimeter 44 cm

Shapes with triangles and circles

Worked example: a gable wall (rectangle + triangle)

A wall is a rectangle 8 m wide and 5 m tall, with a triangle of height 3 m on top.
Rectangle 8 × 5 = 40 m². Triangle ½ × 8 × 3 = 12 m².

Answer: 52 m²

Worked example: an arched window

Rectangle: 1.2 × 1.5 = 1.8 m².
Semicircle: diameter 1.2 m, so r = 0.6 m. ½ × π × 0.6² = 0.565 48… m².
Total = 1.8 + 0.565 48… = 2.365 48… m²

Answer: 2.37 m²

Worked example: a shaded region

A circle fits exactly inside a square of side 10 cm. Find the area outside the circle but inside the square.
Square 10 × 10 = 100 cm². Circle r = 5: π × 5² = 78.539… cm².
Shaded = 100 − 78.539… = 21.460… cm²

Answer: 21.5 cm²

Volume of a cylinder

A cylinder is a prism. Volume of any prism = area of cross-section × length. The cross-section of a cylinder is a circle.

Formula: Volume of a cylinder V = π × r² × h

Worked example: a tin

A tin has radius 5 cm and height 12 cm.
V = π × 5² × 12 = 300π = 942.477… cm³
In litres: 942.477… ÷ 1000 = 0.942… litres

Answer: 942.5 cm³ (about 0.94 litres)

Surface area of a cylinder

Open out a cylinder and you get two circles (top and bottom) and one rectangle. The rectangle's length is the circumference 2πr and its width is the height h.

Formula: Total surface area = 2 × π × r² + 2 × π × r × h (two circles + curved surface)

Worked example: the same tin

Two circles: 2 × π × 5² = 50π. Curved surface: 2 × π × 5 × 12 = 120π.
Total = 170π = 534.070… cm²

Answer: 534.1 cm²

If the cylinder is open (for example a tube with no lid), leave out the missing circle(s).

Common mistakes

  • Adding a dashed (inside) line into the perimeter. The perimeter is only the outside edge.
  • Forgetting to work out missing lengths before splitting a shape.
  • Using the diameter in πr²h. Halve it first.
  • Leaving out one of the circles in the surface area of a closed cylinder.
  • Mixing up units: cm³ for volume, cm² for area, and 1000 cm³ = 1 litre.

Revision questions

Try these, then check your answers below each one.

Question 1. An L-shape is made by cutting a 4 cm by 3 cm rectangle from one corner of a 9 cm by 7 cm rectangle. Work out the area of the L-shape.

Answer: 51 cm²

Working: 9 × 7 - 4 × 3 = 63 - 12 = 51 cm².

Question 2. A cylinder has radius 2 cm and height 15 cm. Work out its volume. Give your answer to 1 decimal place.

Answer: 188.5 cm³

Working: π × 2² × 15 = 60π = 188.495… = 188.5 cm³.

Question 3. A cylinder has radius 3 cm and height 5 cm. What is its volume in terms of π? (Choose from: 45π cm³, 15π cm³, 30π cm³, 90π cm³.)

Answer: 45π cm³

Working: π × 3² × 5 = 45π cm³.

Related lessons

Continue in the full course

This lesson is section 17.5 (EGCMF1705) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.

 

 

 

 

 

 

 

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