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Area, surface area and volume

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Notes de leçon

What is Area?

  • Area is the amount of space inside a flat shape. We measure it in square units, like square centimetres (cm2) or square metres (m2).
  • The area of a rectangle is length × width. For a square, it's side × side.
  • You can find the area of a shape by counting how many unit squares fit inside it.
  • Area is additive: if you cut a shape into pieces, the total area is the sum of the areas of the pieces.

Area of Triangles

  • A triangle is half of a rectangle or parallelogram.
  • The base (b) is one side of the triangle, and the height (h) is the perpendicular distance from the base to the opposite corner.
  • Area of a triangle = \frac{1}{2} \times b \times h.
  • Example: If b = 6 cm and h = 4 cm, then area = \frac{1}{2} \times 6 \times 4 = 12 cm2.

Area of Parallelograms

  • A parallelogram is a four-sided shape with opposite sides parallel.
  • Its area is base × height, where the height is the perpendicular distance between the base and its opposite side.
  • You can cut a parallelogram and rearrange it into a rectangle to see why the formula works.
  • Example: base = 5 cm, height = 3 cm, area = 5 × 3 = 15 cm2.

Area of Trapeziums

  • A trapezium (or trapezoid) has one pair of parallel sides. Those are called the bases: b1 and b2.
  • The height (h) is the perpendicular distance between the two bases.
  • Area = \frac{1}{2} \times (b1 + b2) \times h.
  • You can split a trapezium into a rectangle and two triangles, or double it to make a parallelogram, to find the formula.

Compound Shapes

  • A compound shape is made of two or more simple shapes, like rectangles and triangles.
  • To find its area, decompose it into simple shapes, find each area, then add them together.
  • Sometimes you can also compose a shape by adding a piece to make a rectangle, then subtract that piece's area.
  • Always use the same units for all measurements before calculating.

Volume of Cuboids

  • Volume is the amount of space inside a 3-D shape. We measure it in cubic units, like cubic centimetres (cm3) or cubic metres (m3).
  • A cuboid is a box with six rectangular faces. A cube is a special cuboid with all edges equal.
  • Volume of a cuboid = length × width × height (V = l × w × h).
  • For fractional edge lengths, multiply the fractions. Example: \frac{1}{2} \times \frac{1}{3} \times \frac{1}{4} = \frac{1}{24} cm3.

Nets and Surface Area

  • A net is a flat pattern that folds up to make a 3-D shape.
  • The surface area (A) of a solid is the total area of all its faces. For a shape with flat faces, surface area = sum of the areas of the faces.
  • To find surface area using a net, calculate the area of each face on the net and add them up.
  • For a cuboid, there are 6 faces: opposite faces are equal. So surface area = 2 × (lw + lh + wh).
  • For a cube with side length s, surface area = 6 × s2.

Choosing the Right Units

  • Use square units (like cm2) for area, and cubic units (like cm3) for volume.
  • Always match the units: if lengths are in metres, area is in m2 and volume is in m3.
  • When solving problems, check if you need area (like painting a wall) or volume (like filling a box).
  • Real-world examples: wrapping a present uses surface area; filling a fish tank uses volume.

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  1. 1.What does the surface area of a solid object measure?

    Easy
    • AThe total area that the surface of the object occupies
    • BThe space inside the object
    • CThe length of the object's edges
    • DThe weight of the object

  2. 2.Surface area is the sum of the areas of its faces for objects with flat polygonal faces.

    Easy

    True or false?

  3. 3.Which formula correctly gives the area of a triangle with base b and height h?

    Easy
    • AA = (b × h) / 2
    • BA = b × h
    • CA = 2 × b × h
    • DA = b + h

  4. 4.A rectangle has length 8 cm and width 5 cm. What is its area?

    Medium
    • A40 cm²
    • B13 cm²
    • C26 cm²
    • D45 cm²

  5. 5.A rectangular prism has length 5 cm, width 3 cm, and height 4 cm. What is its volume?

    Medium
    • A60 cm³
    • B12 cm³
    • C20 cm³
    • D45 cm³

  6. 6.A triangular prism has a base area of 10 cm² and a height of 7 cm. What is its volume?

    Medium
    • A70 cm³
    • B17 cm³
    • C35 cm³
    • D140 cm³

  7. 7.Which of the following shapes have flat faces? (select all that apply)

    Medium
    • ACube
    • BSphere
    • CCylinder
    • DPyramid
    • ECone

  8. 8.The surface area of a solid is the same regardless of its orientation in space.

    Easy

    True or false?

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