Geometry, Perimeter & Area
From a single point to a 3-D pyramid: learn lines and angles, the angles made by clock hands, shapes and their properties, how to find perimeter and area — and how to spot every line of symmetry.
1Points, lines, rays and line segments
Geometry starts with a tiny dot. Everything else is built from it.
- Point
- An exact position, shown by a dot and named with a capital letter (point A). It has no length or width.
- Line segment
- A straight path between two end points. It has a fixed length that you can measure. Written AB (or BA — same segment).
- Ray
- Starts at one end point and goes on forever in one direction. Ray AB starts at A and passes through B. Ray AB ≠ ray BA!
- Line
- Goes on forever in both directions, so it has no end points and cannot be measured.
Types of lines
Four points A, B, C, D lie on one straight line in that order. How many line segments can be named?
- From A: AB, AC, AD → 3.
- From B (don't repeat BA): BC, BD → 2.
- From C: CD → 1.
- Total = 3 + 2 + 1 = 6. With 5 points it would be 4 + 3 + 2 + 1 = 10.
Which has exactly one end point and goes on forever in one direction?
2Angles
When two rays start from the same point, they make an angle. The common point is the vertex; the two rays are the arms. Angles are measured in degrees (°) with a protractor. The size of an angle depends on how much one arm has turned away from the other — not on how long the arms are drawn.
| Turn | Degrees | Right angles |
|---|---|---|
| Quarter turn | 90° | 1 |
| Half turn | 180° | 2 |
| Three-quarter turn | 270° | 3 |
| Full (complete) turn | 360° | 4 |
- How many 45° angles make a three-quarter turn? ¾ turn = 270°; 270 ÷ 45 = 6.
- Riya faces North and turns 90° clockwise. She now faces East. Another half turn (180°) and she faces West.
An angle drawn with short arms can be bigger than one drawn with long arms. Only the opening between the arms matters. Also: a right angle is exactly 90° — "nearly 90°" is still acute or obtuse.
A three-quarter turn is equal to how many right angles?
3Angles on a clock
A clock face is a full circle of 360° shared equally by 12 numbers. So the angle between two neighbouring numbers is 360° ÷ 12 = 30°. (Each minute mark is 360° ÷ 60 = 6°.)
| Time | Angle between hands | Kind |
|---|---|---|
| 12:00 | 0° (hands together) | — |
| 1:00, 2:00 | 30°, 60° | acute |
| 3:00, 9:00 | 90° | right |
| 4:00, 5:00 | 120°, 150° | obtuse |
| 6:00 | 180° | straight |
At 3:30 the angle is not 90°. The minute hand is on 6, but the hour hand has moved halfway from 3 to 4. The angle is 2½ gaps × 30° = 75°. The simple "gaps × 30°" rule is exact only on the hour.
What angle do the hands of a clock make at 4 o'clock?
42-D shapes: polygons, triangles, quadrilaterals and circles
A polygon is a flat, closed shape made only of straight line segments. In every polygon, number of sides = number of vertices (corners) = number of angles. A regular polygon has all sides equal and all angles equal. A circle is not a polygon (it has no straight sides).
Triangles
| By sides | Meaning | By angles | Meaning |
|---|---|---|---|
| Equilateral | all 3 sides equal (all angles 60°) | Acute-angled | all 3 angles acute |
| Isosceles | 2 sides equal | Right-angled | one angle is 90° |
| Scalene | no sides equal | Obtuse-angled | one angle is obtuse |
The three angles of any triangle add up to 180°, so a triangle can never have two right angles.
Quadrilaterals
| Shape | Sides | Angles |
|---|---|---|
| Square | all 4 equal; opposite sides parallel | all 90° |
| Rectangle | opposite sides equal and parallel | all 90° |
| Parallelogram | opposite sides equal and parallel | opposite angles equal |
| Rhombus | all 4 equal; opposite sides parallel | opposite angles equal |
| Trapezium | only one pair of parallel sides | — |
| Kite | two pairs of equal neighbouring sides | — |
Every square is also a rectangle (4 right angles) and also a rhombus (4 equal sides). But not every rectangle is a square!
Parts of a circle
- Radius: from the centre to any point on the circle. All radii of a circle are equal.
- Diameter: a line segment through the centre joining two points on the circle. Diameter = 2 × radius; radius = diameter ÷ 2.
- Chord: any segment joining two points on the circle. The diameter is the longest chord.
- A circle has one centre but countless radii and diameters.
The diameter of a circle is 18 cm. What is its radius?
53-D solids and nets
Solids take up space and have length, breadth and height. We describe them by their faces (flat or curved surfaces), edges (where two faces meet) and vertices (corners where edges meet).
| Solid | Faces | Edges | Vertices | Real-life example |
|---|---|---|---|---|
| Cube | 6 (equal squares) | 12 (all equal) | 8 | dice, ice cube |
| Cuboid | 6 (rectangles) | 12 | 8 | brick, book, matchbox |
| Cylinder | 3 (2 flat circles + 1 curved) | 2 (curved) | 0 | pipe, can, drum |
| Cone | 2 (1 flat circle + 1 curved) | 1 (curved) | 1 (apex) | ice-cream cone, birthday cap |
| Sphere | 1 curved surface | 0 | 0 | ball, marble |
| Square pyramid | 5 (1 square + 4 triangles) | 8 | 5 | the pyramids of Egypt |
| Triangular pyramid | 4 triangles | 6 | 4 | — |
| Triangular prism | 5 (2 triangles + 3 rectangles) | 9 | 6 | a tent, a Toblerone box |
A cylinder has 2 flat circular faces, but that does not give it 2 vertices. A vertex is a sharp corner where straight edges meet. Cylinder: 0 vertices. Sphere: 0 vertices, 0 edges. Cone: just 1 vertex (the tip).
For solids with only flat faces: Faces + Vertices − Edges = 2. Cube: 6 + 8 − 12 = 2 ✓. Square pyramid: 5 + 5 − 8 = 2 ✓. If your counts don't give 2, count again! Cube asked as faces + edges + vertices = 6 + 12 + 8 = 26.
Nets
A net is a flat pattern that folds up into a solid. A cube's net has 6 squares. There are 11 different cube nets — but not every group of 6 squares works.
Which solid has one flat face, one curved face and one vertex?
6Perimeter
Perimeter is the total length of the boundary of a closed shape — how far an ant walks to go once around it. It is measured in units of length: cm, m, km.
Rectangle
Perimeter = length + breadth + length + breadth
= 2 × (length + breadth)
Square
All 4 sides equal:
Perimeter = 4 × side
so side = perimeter ÷ 4
Any polygon
Add up the lengths of all the sides. Equilateral triangle: 3 × side.
A rectangular field is 30 m long and 20 m wide. Fencing costs ₹15 per metre. Find the cost of fencing it.
- Fence goes around the boundary → perimeter = 2 × (30 + 20) = 2 × 50 = 100 m.
- Cost = 100 × ₹15 = ₹1,500.
- Trap: multiplying 30 × 20 gives area, not the length of fence!
For a shape made only of horizontal and vertical sides (an L-shape or a staircase), push the notch's sides out to make a rectangle. The perimeter does not change! So you only need the total width and total height: P = 2 × (width + height).
- The perimeter of a square is 56 cm. Side = 56 ÷ 4 = 14 cm.
- A rectangle has perimeter 40 cm and length 12 cm. Length + breadth = 40 ÷ 2 = 20, so breadth = 20 − 12 = 8 cm.
- A wire 36 cm long is bent into an equilateral triangle. Each side = 36 ÷ 3 = 12 cm. The same wire bent into a square: side = 36 ÷ 4 = 9 cm.
What is the perimeter of a rectangle 15 cm long and 8 cm wide?
7Area
Area is the amount of surface a flat shape covers. We measure it by counting unit squares: a square of side 1 cm has an area of 1 square centimetre (1 sq cm or 1 cm²). Big areas use sq m.
Counting squares
Count full squares as 1 and exact half squares as ½ (two halves make one). If a shape covers "more than half" or "less than half" squares, count more-than-half as 1 and ignore less-than-half — this gives an estimate.
Formulas
Rectangle
Area = length × breadth
6 cm × 4 cm = 24 sq cm
Square
Area = side × side
7 cm × 7 cm = 49 sq cm
The perimeter of a square is 56 cm. The unshaded part is 2/7 of the square. What is the shaded area?
- Side = 56 ÷ 4 = 14 cm.
- Area = 14 × 14 = 196 sq cm.
- Shaded part = 1 − 2/7 = 5/7 of the square.
- 5/7 of 196 = 196 ÷ 7 × 5 = 28 × 5 = 140 sq cm.
A rectangle has an area of 72 sq cm and a length of 12 cm. Find its perimeter.
- Breadth = area ÷ length = 72 ÷ 12 = 6 cm.
- Perimeter = 2 × (12 + 6) = 36 cm.
- A room is 5 m by 4 m. How many square tiles of side 1 m cover it? Area = 20 sq m → 20 tiles.
- The L-shape above: whole rectangle 10 × 8 = 80 sq cm, minus the cut-out corner 4 × 3 = 12 sq cm → area = 68 sq cm.
A square has an area of 81 sq cm. What is its perimeter?
8Area vs perimeter — the big traps
Perimeter
The boundary (a length). Units: cm, m. Think: fence, border, lace, frame, walking around.
Area
The surface inside. Units: sq cm, sq m. Think: carpet, paint, tiles, grass, covering.
Two shapes can cover the same area but have different perimeters (and the other way round). Never assume one from the other — always calculate both.
A square of side 8 cm is cut into 4 equal smaller squares. Compare the total perimeter before and after.
- Before: 4 × 8 = 32 cm.
- Each small square has side 4 cm → perimeter 4 × 4 = 16 cm.
- Total of 4 small squares = 4 × 16 = 64 cm — double! Cutting creates new edges.
- But the total area stays the same: 8 × 8 = 64 sq cm = 4 × (4 × 4).
If the side of a square is doubled, the perimeter doubles, but the area becomes 4 times as big. Side 3 → area 9; side 6 → area 36.
9Symmetry and patterns
A shape has a line of symmetry if you can fold it along that line so the two halves match exactly — one half is the mirror image of the other.
| Capital letters | Lines of symmetry |
|---|---|
| A, M, T, U, V, W, Y | 1 (vertical) |
| B, C, D, E, K | 1 (horizontal) |
| H, I, O, X | 2 (vertical and horizontal) |
| F, G, J, L, N, P, Q, R, S, Z | 0 |
| Shape | Lines of symmetry |
|---|---|
| Circle | countless (every diameter) |
| Square | 4 |
| Rectangle, rhombus | 2 |
| Equilateral triangle | 3 |
| Isosceles triangle, kite | 1 |
| Scalene triangle, parallelogram | 0 |
| Regular pentagon / hexagon / octagon | 5 / 6 / 8 (same as number of sides) |
S, N and Z look "balanced" because they look the same when turned upside down, but no fold line makes the halves match. They have 0 lines of symmetry. Similarly, a parallelogram has 0.
Patterns
Shape and matchstick patterns grow by the same amount each step. Find what is added each time, then use it.
- Triangles in a row: 3, 5, 7, … (add 2). For 10 triangles: 2 × 10 + 1 = 21 sticks.
- Squares in a row: 4, 7, 10, … (add 3). For 8 squares: 3 × 8 + 1 = 25 sticks.
- Square numbers: 1, 4, 9, 16, 25 … (1×1, 2×2, 3×3 …). The gaps are 3, 5, 7, 9 — odd numbers!
Which capital letter has exactly two lines of symmetry?
Explore
- 🎮Geoboardtoytheater.com · stretch bands into shapes, count area and perimeter
- 🎮Measuring Anglesmathplayground.com · practise with a virtual protractor
- 🎮Polypadmathigon.org · build polygons, nets and symmetric designs
- 🎮Plane figureskhanacademy.org · lines, angles, shapes and symmetry practice
- 📖SymmetrySimple English Wikipedia · symmetry in nature and art
10Quick recap
- Segment: 2 end points; ray: 1 end point; line: none (goes on both ways).
- Parallel lines never meet; perpendicular lines meet at 90°.
- Acute < 90° ; right = 90° ; obtuse between 90° and 180° ; straight = 180° ; reflex between 180° and 360°.
- Quarter turn 90°, half turn 180°, full turn 360°.
- Clock: each gap between numbers = 30°. 3:00 → 90°, 4:00 → 120°, 6:00 → 180°.
- Polygon: sides = vertices = angles. Pentagon 5, hexagon 6, octagon 8.
- Diameter = 2 × radius; the diameter is the longest chord.
- Cube/cuboid: 6 faces, 12 edges, 8 vertices. Cylinder: 0 vertices. Cone: 1 vertex.
- Perimeter of rectangle = 2 × (l + b); square = 4 × side.
- Area of rectangle = l × b; square = side × side (sq cm, sq m).
- Fence/border → perimeter; carpet/tiles/paint → area.
- Same area can have different perimeters; cutting a shape adds perimeter.
- Square 4 lines of symmetry, rectangle 2, equilateral triangle 3, circle countless.
- H, I, O, X have 2 lines; F, G, J, L, N, P, Q, R, S, Z have none.