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Shape & measurement

Mensuration 2D: boundary versus surface

Distinguish perimeter from area, understand the triangle's half-rectangle rule, and measure circles and composite shapes.

IntermediateAbout 15 min3 worked examples
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01 / THE STARTING POINT

Before you begin

What you need: Multiplication, squares and measurement units.

  • Separate length measurements from area measurements
  • Choose formulas for rectangles, triangles and circles
  • Split composite shapes into familiar parts
02 / BUILD THE CONCEPT

Understand the idea

Perimeter measures the boundary

Perimeter is the distance around a shape and uses length units such as centimetres. Area measures the surface covered and uses square units such as cm².

Triangle height is perpendicular

Two copies of a triangle can form a parallelogram with the same base and perpendicular height. That is why triangle area is half of base × height. A sloping side is not generally the height.

Curved and composite shapes

For a circle, circumference = 2πr and area = πr², where r is the radius. For a composite shape, add areas of non-overlapping pieces or subtract a cut-out from an outer shape.

03 / A DIFFERENT WAY TO SEE IT

A triangle is half its enclosing rectangle

h = 5 cmb = 8 cm
Both shapes have base 8 cm and perpendicular height 5 cm. Rectangle area is 40 cm²; triangle area is 20 cm².
04 / FROM IDEA TO ANSWER

A method you can reuse

KEEP THIS HANDY

Rectangle: A = lb; triangle: A = ½bh; circle: A = πr²

Rectangle perimeter = 2(l + b); circle circumference = 2πr. Use perpendicular height h.
  1. Sketch and label the measurements, converting them to one unit.
  2. Choose the correct shape formula or split the shape into simpler pieces.
  3. Calculate, retain π if an exact answer is wanted, and attach the correct units.
A useful insight

If every length in a plane figure is scaled by k > 0, perimeter scales by k and area by k².

05 / WATCH THE METHOD WORK

Worked examples

Read the question first. Try a step yourself, then compare your reasoning.

EXAMPLE 01

Rectangle measurements

A rectangle is 8 cm by 5 cm. Find area and perimeter.

  1. Area = 8 × 5.
  2. Perimeter = 2(8 + 5).
  3. Area and perimeter describe different measurements.
Answer40 cm²; 26 cm
EXAMPLE 02

Circle with a diameter

A circle has diameter 14 cm. Find area using π = 22/7.

  1. Radius = diameter/2 = 7 cm.
  2. Area = (22/7) × 7².
  3. The result is a square-unit measurement.
Answer154 cm²
EXAMPLE 03

A rectangular cut-out

A 10 m by 8 m garden has a 4 m by 3 m rectangular pond. Find the remaining area.

  1. Outer area = 10 × 8 = 80 m².
  2. Pond area = 4 × 3 = 12 m².
  3. Subtract the non-overlapping cut-out.
Answer68 m²
06 / YOUR TURN

Check your understanding

Quick questions, with explanations. These are for self-study and do not affect your account score.

0 of 3 questions checked

QUESTION 1Triangle base 10 cm, perpendicular height 6 cm. Area?

See the explanation

30 cm². ½ × 10 × 6 = 30 cm².

QUESTION 2A circle has radius 3 m. Exact circumference?

See the explanation

6π m. Circumference = 2πr = 6π m.

QUESTION 3A square's side doubles. Its area becomes?

See the explanation

4 times. Area scales with the square of length: 2² = 4.

07 / TAKE THE NEXT STEP

Put your understanding to work

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EXPLORE THE NUMBERS

Rectangle Calculator

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Mensuration 2D Practice Quiz 1

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Mensuration 2D Practice Quiz 2

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Mensuration 2D Practice Quiz 3

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Area and perimeter practice

Try a fresh set of ten questions.

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NEXT LESSONMensuration 3D
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