Area & Perimeter (Edexcel GCSE Maths: Foundation)

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Naomi C

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Naomi C

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Perimeter

What is perimeter?

  • Perimeter is the total distance around the outside of a 2D shape
    • The perimeter of a circle is called the circumference
  • Perimeter is a length in one dimension
    • Units of measure include mm, cm, m etc

How do I find the perimeter of a 2D shape?

  • Add together the lengths of all of the sides of the shape
  • For any regular 2D shape, the perimeter will be the number of sides, multiplied by the length of one side
    • For example, the perimeter of a square of side length x  cm will be 4x  cm

How do I find the perimeter of a compound shape?

  • Shapes may be made up of two or more 2D shapes, these are called compound shapes
    • Compound shapes can usually be split into rectangles, triangles and parts of circles
    • You will need to be confident with the properties of 2D shapes
      • For example, the distance between the centre point of a circle and a point on the circumference is its radius
    • Look out for sides that are equal, for example in a rectangle, parallelogram, or isosceles triangle
      • Dashes may be used to mark the equal sides, or the question may tell you which sides are equal
    • You may need to use certain formulas to calculate lengths
      • For example, you may need to use Pythagoras' theorem to calculate a length on a right-angled triangle
    • You may need to use the given information to find the lengths of some of the sides
      • For example, the L-shape below can be split into two rectangles
      • The sum of the lengths of two shorter sides will be the same as the length of the longer side opposite

A compound shape split into two rectangles

Examiner Tip

  • Understanding properties of different 2D shapes can be essential for missing lengths.

Worked example

Find the perimeter of the compound shape.

A compound shape made up of rectangles and an isosceles triangle

The shape can be split up into a triangle and two rectangles

The dashes on the triangle mean that the lengths are equal
It is an isosceles triangle and the missing length is 6 cm

The horizontal lengths of the two rectangles add up to the length of the longest horizontal line
15 cm and the missing length, are equal to the 18 cm length at the top
The missing horizontal length is therefore 3 cm 

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You can now find the sum of all the sides to find the total perimeter

6 + 6 + 18 + 2 + 3 + 4 + 15

54 cm

Area

What is area?

  • Area is the amount of space within the perimeter of a 2D shape
    • For example, the size of a sports field
  • Area is calculated using lengths in two dimensions
    • Units of measure include mm2, cm2, m2 etc

How do I find the area of a shape on a square grid?

  • Count the total number of whole squares inside the shape
    • You can shade or mark the squares you have counted so far
  • Parts of the shape may not contain whole squares
    • Pair up half squares, or parts of squares, to form whole ones
  • There will be a scale telling you how much area one square represents
    • Multiply the number of squares you have counted by this value to find the total area of the shape

Examiner Tip

  • When counting squares, write down your running total so far in each box.
    • This saves time and ensures you do not count any squares twice.

Worked example

Work out the area of the shaded quadrilateral shown on the grid below. Each square on the grid represents 1 cm2.

cie-igce-core-rn-area-we-diagram

 

Count the number of whole squares, keeping track of which you have counted so far

cie-igce-core-rn-area-we-diagram-2

 

Count the partially shaded squares
Try to pair-up fractions of the squares to make whole squares

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There are 14 whole squares in total, and the shape was drawn on a 1 cm2 grid

Total Area is 14 cm2

Area Formulae

Which area formulae do I need to know?

Area formulae for rectangle, triangle, trapezium and parallelogram

How do I find the area of a rectangle?

  • The area, A, of a rectangle of length, l, and width, w, using the formula
    • A equals l cross times w
      • Multiply together the length and the width

How do I find the area of a triangle?

  • The area, A, of a triangle with base, b, and length, l, can be found using the formula
    • A equals 1 half b h
      • Multiply the length of the base (b) by the perpendicular height (h)
      • Halve the answer
  • The perpendicular height may not be the length of one of the sides of the triangle

How do I find the area of a trapezium?

  • The area, A, of a trapezium with parallel lengths, a and b, and perpendicular height, h, can be found using the formula
    • A equals 1 half open parentheses a plus b close parentheses h
      • Add together the lengths of the parallel sides
      • Multiply the result by the distance between the parallel sides
      • Halve the answer
  • You may be able to work out the area of a trapezium by splitting the shape into a rectangle and triangles if you can't remember the formula

How do I find the area of a parallelogram?

  • You can find the area, A, of a parallelogram of length, l, and perpendicular height, h, by using the formula
    • A equals b h 
      • Multiply the length of the base by the perpendicular height
  • The perpendicular height is not a length of the parallelogram
    • It is the distance between the base and its opposite side
  • You can work the area of a parallelogram out by splitting the shape into a rectangle and triangles if you can't remember the formula

Examiner Tip

  • You may have to do some work to find missing lengths first.
    • For example, you may need to use Pythagoras' Theorem to find a missing length on a triangle.
  • The area of a triangle is given to you in the exam but you will need to remember the other formulae.

Worked example

Calculate the area of the following shapes.

(a)
A trapezium

 

Find the area of the trapezium using A equals 1 half open parentheses a plus b close parentheses h
Remember that a  and b  are the two parallel sides and h  is the perpendicular height

A equals 1 half open parentheses 30 plus 15 close parentheses cross times 20

450 cm2

(b)
A parallelogram 
Find the area of the parallelogram using A equals b cross times h
Remember that b  is the base and h  is the perpendicular height

A equals 15 cross times 12

180 cm2

(c)
A right-angled triangle 
Find the area of the right-angled triangle using A equals 1 half b h
Remember that b  is the base and  is the perpendicular height

A equals 1 half cross times 8 cross times 7

28 cm2

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Naomi C

Author: Naomi C

Expertise: Maths

Naomi graduated from Durham University in 2007 with a Masters degree in Civil Engineering. She has taught Mathematics in the UK, Malaysia and Switzerland covering GCSE, IGCSE, A-Level and IB. She particularly enjoys applying Mathematics to real life and endeavours to bring creativity to the content she creates.