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Foundations · Geometry

Measurement, perimeter and area

Units and conversions, perimeter and area of rectangles, triangles and circles, and the angles you meet before trigonometry.

First, the rules

Count the squares, then let the formula count them for you

Perimeter is the distance round the edge; area is how many unit squares fit inside. Every area formula is a shortcut for counting squares.

★ Metric units
Example 1
1 km = 1000 m · 1 m = 100 cm · 1 cm = 10 mm · 1 kg = 1000 g · 1 L = 1000 mL · 1 h = 60 min. Bigger unit → divide; smaller unit → multiply.
Area units
Example 1
1 m² = 100 × 100 = 10 000 cm² — the conversion factor is squared.
★ Perimeter
Example 2
Add every side. Rectangle: 2(l + w). Square: 4s.
★ Areas
Example 2
rectangle l × w · square s² · triangle ½ b h · parallelogram b h · trapezium ½ (a + b) h
★ Circle
Example 3
C = 2πr = πd, A = πr² — π ≈ 3.14; leave π in the answer unless asked.
★ Angle facts
Example 3
Right angle 90°, straight line 180°, full turn 360°; angles on a line add to 180°, around a point to 360°, vertically opposite angles are equal.

See it

Shapes on a unit grid

Resize a rectangle, triangle, parallelogram or circle and count squares against the formula. The triangle shows the dashed rectangle it is half of; the parallelogram shows the slide that makes it a rectangle.

Step by step

Worked examples

1

Units

conversions

Convert (a) 3.25 km to metres, (b) 480 mm to metres, (c) 2.5 m² to cm².

Ask yourself: Which way do you go — to a bigger or a smaller unit?

Show the solution
  1. (a) × 1000: 3 250 m.
  2. (b) mm → cm (÷ 10) → m (÷ 100): 480 ÷ 1000 = 0.48 m.
  3. (c) 1 m² = 10 000 cm²: 2.5 × 10 000 = 25 000 cm².

Answer 3 250 m · 0.48 m · 25 000 cm²

2

A compound shape

perimeter · area

An L-shape is a 10 cm × 6 cm rectangle with a 4 cm × 3 cm rectangle cut from one corner. Find its area and its perimeter.

Ask yourself: Area: big minus small. Perimeter: the cut-out does not change it — why?

Show the solution
  1. Area: 10 × 6 − 4 × 3 = 60 − 12 = 48 cm².
  2. Perimeter: the two new edges of the notch are exactly as long as the parts they replaced, so it stays 2(10 + 6) = 32 cm.

Answer 48 cm² · 32 cm

3

Circle and angles

circle · angle facts

(a) A circular table has diameter 1.4 m. Find its circumference and area. (b) Two lines cross; one of the four angles is 38°. Find the other three.

Ask yourself: Radius is half the diameter. Crossing lines make two pairs of equal angles.

Show the solution
  1. (a) r = 0.7: C = π × 1.4 ≈ 4.40 m; A = π × 0.49 ≈ 1.54 m².
  2. (b) Vertically opposite: another 38°. On a straight line: 180 − 38 = 142°, twice. Check: 38 + 142 + 38 + 142 = 360 ✓.

Answer ≈ 4.40 m · ≈ 1.54 m² · 38°, 142°, 142°

Practice: measure it

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Perimeter and area of rectangles, triangles and circles, unit conversions, and angle facts. Your best score is kept in this browser only.

About this lesson

Measurement and basic shapes for school mathematics: metric units and conversions, perimeter and area of rectangles, triangles, parallelograms and circles, counting unit squares, and the basic angle facts (right, straight, around a point, on a line). A shape you can resize, worked examples and practice questions with explanations. Everything on this page runs in your browser; nothing you type is sent or stored. Spotted an error? Tell us.