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4th Std Scholarship · Intelligence Test

Counting Figures

5 concepts · 40 practice questions with answers · Free to read

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Counting segments on a line

From everyday life

Four children stand in a queue at a bus stop. To count how many handshakes are possible you count every pair once. Counting segments on a line works the same way.

Simple definition

Pick any two points on a line and the part between them is one line segment. So counting segments means counting pairs of points.

ABCDE

Points and segments

PointsSegments
21
33
46
510
615
1
Rule 1

Every segment needs two points. One pair of points = one segment.

  • e.g. the segment between A and C is AC.
2
Rule 2

Add the numbers from one less than the points down to 1.

  • e.g. 5 points : 4 + 3 + 2 + 1 = 10.
3
Rule 3

If the parts are given, points = parts + 1.

  • e.g. a line in 4 parts : 5 points, 10 segments.
4
Rule 4

Count in order : of 1 part, of 2 parts, of 3 parts ... and none is missed.

  • e.g. 4 parts : 4 + 3 + 2 + 1 = 10.
Example 1 · easy

A line has 4 points. How many segments?

  1. 3 + 2 + 1
  2. = 6. Answer : 6.
Example 2 · medium

A line is cut into 6 parts. How many segments?

  1. Points = 6 + 1 = 7.
  2. 6 + 5 + 4 + 3 + 2 + 1 = 21. Answer : 21.
Example 3 · exam-style

A line has some points and 15 segments in all. How many points?

  1. Add up : 1 + 2 + 3 + 4 + 5 = 15.
  2. The last number is 5, so points = 5 + 1 = 6. Answer : 6.
Remember

Counting segments means counting pairs of points; do not stop at counting the parts.

Common mistake

Saying only 4 segments for a line of 4 parts. The longer segments (of 2, 3, 4 parts) must be counted too.

Shortcut

Memory clue : "Start one below the points and add down to 1."

Shortcut

5 points : 4 + 3 + 2 + 1 = 10. 6 points : 5 + 4 + 3 + 2 + 1 = 15. Each time the next number is added on top.

Spot the difference

GivenPointsSegments
4 points46
4 parts510
5 points510

Practice questions

Easy
  1. 1A line has 3 points. How many segments?
  2. 2A line has 6 points. How many segments?
  3. 3A line has 8 points. How many segments?
Medium
  1. 4A line is cut into 5 parts. How many segments?
  2. 5A line has 9 points. How many segments?
  3. 6A line is cut into 9 parts. How many segments?
Challenging
  1. 7A line has 12 points. How many segments?
  2. 8A line has 21 segments in all. How many points?
Show answers (try them yourself first)
  1. 13.
  2. 215 (5 + 4 + 3 + 2 + 1).
  3. 328 (7 + 6 + 5 + 4 + 3 + 2 + 1).
  4. 415 (6 points).
  5. 536 (add 8 down to 1).
  6. 645 (10 points).
  7. 766 (add 11 down to 1).
  8. 87 points (6 + 5 + 4 + 3 + 2 + 1 = 21).

Counting angles made by rays

From everyday life

Think of rays of light leaving one lamp. Between any two rays there is an angle.

Simple definition

Two rays leaving the same point make one angle. So counting angles means counting pairs of rays, just like counting segments.

Rays and angles

RaysAngles
21
33
46
510
615
1
Rule 1

Two rays = one angle. Count the pairs of rays.

  • e.g. the angle between rays OA and OC is AOC.
2
Rule 2

Add from one less than the rays down to 1.

  • e.g. 5 rays : 4 + 3 + 2 + 1 = 10 angles.
3
Rule 3

The smallest angles sit next to each other. With 5 rays there are 4 smallest angles.

  • e.g. 6 rays : 5 smallest angles.
4
Rule 4

Count in order : one small angle, two small ones joined, three joined ...

  • e.g. 5 rays : 4 + 3 + 2 + 1 = 10.
Example 1 · easy

4 rays leave one point. How many angles?

  1. 3 + 2 + 1
  2. = 6. Answer : 6.
Example 2 · medium

6 rays leave one point. How many angles?

  1. 5 + 4 + 3 + 2 + 1
  2. = 15. Answer : 15.
Example 3 · exam-style

The figure has 5 rays. How many angles are made by joining exactly two small angles?

  1. There are 4 small angles.
  2. Pairs of neighbouring angles : 4 − 1 = 3. Answer : 3.
Remember

Counting angles is like counting segments; just count rays instead of points.

Common mistake

Counting only the small neighbouring angles. The bigger angles (two or more small ones joined) get missed.

Shortcut

Memory clue : "Rays are the points; angles are the segments."

Shortcut

4 rays = 6 angles, 5 rays = 10 angles, 6 rays = 15 angles. These are the same numbers as the table for points on a line.

Spot the difference

GivenNumber
5 rays, all angles10
5 rays, smallest angles4
5 rays, two small joined3

Practice questions

Easy
  1. 12 rays leave one point. How many angles?
  2. 24 rays leave one point. How many angles?
  3. 36 rays leave one point. How many angles?
Medium
  1. 47 rays leave one point. How many angles?
  2. 58 rays leave one point. How many angles?
  3. 6In a figure of 6 rays, how many angles are made by joining exactly two small angles?
Challenging
  1. 710 rays leave one point. How many angles?
  2. 8There are 28 angles in all. How many rays?
Show answers (try them yourself first)
  1. 11.
  2. 26.
  3. 315.
  4. 421 (add 6 down to 1).
  5. 528.
  6. 64 (5 small angles, pairs 5 − 1).
  7. 745.
  8. 88 rays (7 + 6 + 5 + 4 + 3 + 2 + 1 = 28).

Counting triangles

From everyday life

A slice of pizza or a tent has a triangular shape. Draw lines inside a big triangle and many small and big triangles appear.

Simple definition

Lines drawn from the top corner down to the base cut the triangle into parts. Every triangle is made by two points on the base and the top corner.

Parts of the base and triangles

Parts of the baseTriangles
23
36
410
515
621
1
Rule 1

To count triangles made by lines from the top corner, count the segments on the base.

  • e.g. base in 4 parts : 10 segments, so 10 triangles.
2
Rule 2

Add from the number of parts down to 1.

  • e.g. 4 parts : 4 + 3 + 2 + 1 = 10.
3
Rule 3

If a cross line cuts the triangle, the number doubles (the upper part and the full part).

  • e.g. base in 3 parts with a cross line : 6 + 6 = 12.
4
Rule 4

Count in groups : of 1 part, of 2 parts ... ; do not forget the big full triangle.

  • e.g. 3 parts : 3 + 2 + 1 = 6, and that includes the biggest one.
Example 1 · easy

A triangle has its base in 3 parts. How many triangles?

  1. 3 + 2 + 1
  2. = 6. Answer : 6.
Example 2 · medium

The base is in 3 parts and there is one cross line. How many triangles?

  1. Without the cross line : 3 + 2 + 1 = 6.
  2. The upper part gives 6 more.
  3. 6 + 6 = 12. Answer : 12.
Example 3 · exam-style

The base is in 5 parts with one cross line. How many triangles?

  1. 5 + 4 + 3 + 2 + 1 = 15.
  2. The cross line : 15 more.
  3. 15 + 15 = 30. Answer : 30.
Remember

All the triangles are counted along the base; count the points on the base.

Common mistake

Counting only the small triangles. Count the joined ones and the biggest triangle too.

Shortcut

Memory clue : "One top corner, pairs on the base."

Shortcut

With 6 parts on the base : 6 + 5 + 4 + 3 + 2 + 1 = 21. With a cross line, double it, 42.

Spot the difference

FigureTriangles
base 3 parts6
base 3 parts + cross line12
base 4 parts10

Practice questions

Easy
  1. 1A triangle with the base in 2 parts. How many triangles?
  2. 2A triangle with the base in 3 parts. How many triangles?
  3. 3A triangle with the base in 5 parts. How many triangles?
Medium
  1. 4A triangle with the base in 6 parts. How many triangles?
  2. 5A triangle with the base in 7 parts. How many triangles?
  3. 6The base in 4 parts and one cross line. How many triangles?
Challenging
  1. 7The base in 6 parts and one cross line. How many triangles?
  2. 8There are 45 triangles. In how many parts is the base?
Show answers (try them yourself first)
  1. 13.
  2. 26.
  3. 315.
  4. 421.
  5. 528.
  6. 620 (10 + 10).
  7. 742 (21 + 21).
  8. 89 parts (9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 45).

Counting rectangles and squares

From everyday life

Look at a chessboard or a window grating. You see the small boxes, and also bigger boxes made by joining two or three of them.

Simple definition

In a grid of horizontal and vertical lines, every rectangle is made by two horizontal and two vertical lines. A rectangle with all sides equal is a square.

Grid and rectangles

GridRectanglesSquares
1 × 363
2 × 295
2 × 3188
3 × 33614
3 × 46020
1
Rule 1

To count rectangles multiply the pairs of rows by the pairs of columns.

  • e.g. 2 rows and 3 columns : 3 × 6 = 18.
2
Rule 2

Pairs of rows means adding from the number of rows down to 1.

  • e.g. 2 rows : 2 + 1 = 3. 3 columns : 3 + 2 + 1 = 6.
3
Rule 3

To count squares go by size : of 1×1, of 2×2, of 3×3 ...

  • e.g. 2 × 3 grid : 6 of 1×1 and 2 of 2×2, total 8.
4
Rule 4

Squares are a kind of rectangle, so the squares are always fewer than the rectangles.

  • e.g. 2 × 3 grid : 18 rectangles, 8 squares.
Example 1 · easy

A single row has 3 boxes (1 × 3). How many rectangles?

  1. Pairs of rows : 1. Pairs of columns : 3 + 2 + 1 = 6.
  2. 1 × 6 = 6. Answer : 6.
Example 2 · medium

In a 2 × 2 grid how many rectangles and how many squares?

  1. Rectangles : 3 × 3 = 9.
  2. Squares : 4 of 1×1 and 1 of 2×2 = 5.
  3. Answer : 9 rectangles, 5 squares.
Example 3 · exam-style

How many squares in a 3 × 3 grid?

  1. 9 of 1×1.
  2. 4 of 2×2.
  3. 1 of 3×3.
  4. 9 + 4 + 1 = 14. Answer : 14.
Remember

Rectangles = pairs of rows × pairs of columns; count squares size by size.

Common mistake

Counting only the small boxes. The joined bigger boxes and rectangles must be counted too.

Shortcut

Memory clue : "Pairs of rows times pairs of columns."

Shortcut

In a 3 × 3 grid : 6 × 6 = 36 rectangles and 9 + 4 + 1 = 14 squares.

Spot the difference

GridRectanglesSquares
1 × 4104
2 × 295
3 × 33614

Practice questions

Easy
  1. 1How many rectangles in a 1 × 3 grid?
  2. 2How many rectangles in a 2 × 3 grid?
  3. 3How many squares in a 2 × 2 grid?
Medium
  1. 4How many rectangles in a 3 × 3 grid?
  2. 5How many squares in a 3 × 3 grid?
  3. 6How many rectangles in a 1 × 5 grid?
Challenging
  1. 7How many rectangles in a 3 × 4 grid?
  2. 8How many squares in a 3 × 4 grid?
Show answers (try them yourself first)
  1. 16.
  2. 218 (3 × 6).
  3. 35 (4 + 1).
  4. 436 (6 × 6).
  5. 514 (9 + 4 + 1).
  6. 615 (1 × 15).
  7. 760 (6 × 10).
  8. 820 (12 + 6 + 2).

Mixed figures and exam tricks

From everyday life

Draw the diagonals on a kite paper and triangles appear everywhere. In the exam you need a method so that no shape is missed.

Simple definition

In a mixed figure count in size groups : first the small shapes, then two joined, then three joined. Add the groups at the end.

Figure and triangles

FigureTriangles
square + two diagonals8
rectangle + one diagonal2
rectangle + two diagonals8
square + two diagonals + middle vertical line12
1
Rule 1

Count the small shapes first, then two joined, then the big ones.

  • e.g. square + two diagonals : 4 small, 4 big.
2
Rule 2

Mark or name every shape so that no shape is counted twice.

  • e.g. name a triangle by the letters of its three corners.
3
Rule 3

If the figure is symmetric count one part and multiply.

  • e.g. square with 4 equal parts : 2 in one part, 8 in all.
4
Rule 4

Look at the figure again at the end : did you count the biggest shape?

  • e.g. rectangle + two diagonals : 4 small + 4 halves = 8.
Example 1 · easy

A square with both diagonals. How many triangles?

  1. 4 small triangles.
  2. 4 made of two small ones (half the square).
  3. 4 + 4 = 8. Answer : 8.
Example 2 · medium

Draw the middle vertical line in the same square. How many triangles?

  1. The smallest triangles : 4.
  2. The medium triangles : 4.
  3. The biggest ones (half the square) : 4.
  4. 4 + 4 + 4 = 12. Answer : 12.
Example 3 · exam-style

A rectangle with both diagonals. How many triangles?

  1. 4 small triangles (the diagonals cut the rectangle into 4).
  2. 4 halves of the rectangle (one diagonal each).
  3. 4 + 4 = 8. Answer : 8.
Remember

In a mixed figure make size groups and add them at the end.

Common mistake

Counting one triangle twice or forgetting the biggest shape.

Shortcut

Memory clue : "Small, joined, big; then add."

Shortcut

If the figure is symmetric find the answer for one part and multiply; it saves time.

Spot the difference

FigureTriangles
square + one diagonal2
square + two diagonals8
square + two diagonals + vertical line12

Practice questions

Easy
  1. 1A square with both diagonals. How many triangles?
  2. 2A rectangle with one diagonal. How many triangles?
  3. 3A triangle with the base in 4 parts. How many triangles?
Medium
  1. 4A rectangle with both diagonals. How many triangles?
  2. 5How many rectangles in a 2 × 3 grid?
  3. 6How many squares in a 2 × 3 grid?
Challenging
  1. 7A square with both diagonals and the middle vertical line. How many triangles?
  2. 8A triangle with the base in 5 parts and one cross line. How many triangles?
Show answers (try them yourself first)
  1. 18.
  2. 22.
  3. 310.
  4. 48.
  5. 518.
  6. 68 (6 + 2).
  7. 712.
  8. 830 (15 + 15).

Quick revision of the whole lesson

TopicMemory clue
SegmentsPairs of points; add from one less down to 1
AnglesPairs of rays; same as segments
TrianglesPairs on the base; double if there is a cross line
RectanglesPairs of rows × pairs of columns
SquaresSize by size : 1×1, 2×2, 3×3 ...
Mixed figuresSmall, joined, big; then add

🎯 Most important points for the exam

  • 5 points = 10 segments, 6 points = 15, 7 points = 21.
  • Angles made by rays, and triangles counted along the base, are counted just like segments.
  • A cross line through the triangle doubles the number.
  • Rectangles = pairs of rows × pairs of columns.
  • Count squares size by size : in a 3 × 3 grid 9 + 4 + 1 = 14.
  • Do not forget the bigger, joined shapes.
🧠 Study mantra

"Count the pairs, make groups, do not forget the big shape."