Counting Figures

medium importance~0-1 Q in Tier 15 formulas⚡ 6 shortcuts3 subtopics

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Counting triangles

⏱ 4 min read🧩 5 question types🎯 25 practice Q

The idea in one minute

A triangle is any closed shape with exactly three straight sides, even when other lines cross it. Never count by eye. Work size by size: smallest triangles, then bigger ones made of joined pieces, then the whole figure. For a triangle with lines from its top corner, one short formula gives the count.

What counts as a triangle

A triangle is a closed shape with exactly three straight sides. A side can pass through other lines on the way. So a big shape made of joined pieces still counts.

Think of a cut samosa. Each cut makes new pieces. Two small pieces joined together can still form one big triangle. Most wrong answers miss these bigger triangles.

Rule: Count every closed shape whose border has exactly three straight pieces. Extra lines inside it do not matter.

Count size by size

Never count everything at once. Go from smallest to biggest.

Count the smallest triangles. These have no line inside.
Count triangles made of 2 joined pieces, then 3 pieces, and so on.
Count the whole figure if it is itself a triangle.
Add the groups.

A square with both diagonals: 4 smallest triangles meet at the centre. 4 halves use a full side as base. Total $4+4=8$ .

Tip: Write each group's count beside the figure. Adding group totals is safer than one running count.

Name the corners for odd figures

Some figures fit no formula: overlapping triangles, house shapes, mixed lines. Label every corner and every crossing point with a letter.

List triangles one point at a time. For each point, find triangles using only the lines actually drawn there. This is slow but nothing gets counted twice.

The apex formula

One common figure: a triangle with $k$ extra lines from its top corner (the apex) to the base.

Every triangle here uses the apex plus two of the $k+2$ lines through it. Choosing any two of them gives:

T=\dfrac{(k+1)(k+2)}{2}

With 3 apex lines: $4\times5\div2=10$ triangles.

Watch: $k$ counts only the extra lines. The two sides of the big triangle are added on top, giving $k+2$ .

Horizontal cuts multiply

Now add $h$ lines parallel to the base, each crossing every apex line. Each cut is one more base line, so the whole set repeats per base:

T=\dfrac{(k+1)(k+2)}{2}\times(h+1)

With 2 apex lines and 1 cut: $6\times2=12$ triangles.

Watch: A cut counts only if it crosses every apex line. A short cut that stops early breaks the formula.

Famous figures worth memorising

Exam figures repeat every year. Learn these totals; each is verified above.

Figure	Triangles
Square (any rectangle) with both diagonals	8
Same square plus both midlines	16
Triangle with all 3 medians	16
Pentagon with all 5 diagonals (star)	35

A triangle split into rows of small triangles gives 1, 5, 13, 27, 48 for 1 to 5 rows. Upside-down triangles count too: 3 rows give 10 upright and 3 upside-down, total 13.

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1very common3 practice Q

Triangle with lines from the apex (and horizontal cuts)

How to spot it:

One big triangle; several lines run from its top corner to the base; sometimes lines parallel to the base cross them all.

T=\frac{(k+1)(k+2)}{2}\times(h+1)

Method

Count k, the extra lines from the apex.
Count h, the lines parallel to the base that cross every apex line.
One base strip: (k + 1)(k + 2) ÷ 2.
Multiply by (h + 1), one factor per base line.

Why it works:

Every triangle picks two of the k + 2 lines through the apex and one of the h + 1 base lines, so the choices multiply.

Try this

A triangle has 3 lines drawn from its apex to the base, and 1 line parallel to the base crossing all of them. How many triangles are there?

Show solution

$k=3$ : one strip has $\dfrac{4\times5}{2}=10$ triangles.
$h=1$ , so 2 base lines.
$10\times2=20$ .

Answer

Type 2very common3 practice Q

Square or rectangle with diagonals and midlines

How to spot it:

A square or rectangle with one or both diagonals, often with the lines joining the middles of opposite sides.

Method

Count the smallest triangles around the centre.
Count two-piece triangles with a full side as base.
Count the half-squares along each diagonal.
Add the groups.

Why it works:

The figure is symmetric, so each size comes in a set of four and one representative settles the group.

Try this

A square has both diagonals and both midlines drawn. How many triangles are there?

Show solution

Smallest: 8 (two in each quarter).
Full side as base: 4.
Half-squares: 4.
$8+4+4=16$ .

Answer

Type 3common2 practice Q

Triangle divided into rows of small triangles

How to spot it:

A big triangle filled with small triangles, some pointing up and some pointing down.

Method

Count n, the number of rows (parts on each side).
Upright triangles of size s: (n − s + 1)(n − s + 2) ÷ 2, for every s.
Then count the upside-down ones, usually only the small size.
Add both lists.

Why it works:

Upright triangles of one size sit in a triangular number of positions; the upside-down ones hide between them.

Try this

A triangle is divided into 3 rows of small triangles. How many triangles are there?

Show solution

Upright: size 1 = 6, size 2 = 3, size 3 = 1, total 10.
Upside-down: 3 small ones.
$10+3=13$ .

Answer

Type 4common3 practice Q

Triangle with lines from its corners (medians and cevians)

How to spot it:

A triangle with lines from one or more corners to the opposite sides, crossing inside.

Method

Label every corner and inside crossing point.
Count the smallest triangles around each crossing.
Look for joined pieces and half-triangles along each drawn line.
Finish with the whole triangle.

Why it works:

The crossing points split each line into levels, so triangles appear at several sizes; labelling keeps them apart.

Try this

In a triangle, all three medians are drawn. How many triangles are formed?

Show solution

Smallest, around the centre: 6.
A full side as base, centre as top: 3.
Each median halves the big triangle: 6 halves.
The whole triangle: 1.
$6+3+6+1=16$ .

Answer

Type 5common3 practice Q

Irregular composite figures (house, star, pentagon)

How to spot it:

A figure built from several shapes: a house with a roof, a star inside a pentagon, two overlapping triangles.

Method

Break the figure into known parts (square with diagonals = 8, and so on).
Count triangles fully inside each part.
Count triangles that cross from one part into another.
Add and double-check by symmetry.

Why it works:

Known sub-figures give quick totals; only the triangles crossing between parts need fresh work.

Try this

A regular pentagon has all 5 of its diagonals drawn, forming a star inside. How many triangles are there?

Show solution

Triangles using 3 outer corners: choose any 3 of 5 = 10.
Using 2 outer corners: 20.
Using 1 outer corner (star tips): 5.
$10+20+5=35$ .

Answer

Formula sheet

Apex lines

T = \frac{(k+1)(k+2)}{2}

k extra lines drawn from the apex to the base.

Apex lines + horizontal cuts

T = \frac{(k+1)(k+2)}{2}\times(h+1)

h lines parallel to the base, each crossing every apex line.

Shortcuts that save time

⚡ Square with diagonals: 8 or 16

A rectangle with both diagonals always has 8 triangles. Add both midlines and it becomes 16. Memorise both.

Example

How many triangles are there in a square with both diagonals and both midlines drawn?

Show solution

Smallest triangles: 8 (two inside each quarter).
Two-piece triangles, full side as base: 4.
Half-squares along the diagonals: 4.
$8+4+4=16$ .

Answer

⚡ Count the lines, then choose two

In an apex figure, count the lines through the apex (extra lines plus both sides) and choose any two of them.

Example

A triangle has 4 lines drawn from its apex to the base. How many triangles are there?

Show solution

Lines through the apex: $4+2=6$ .
Choose 2 of 6: $\dfrac{6\times5}{2}$ .
$=15$ .

Answer

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Counting only the smallest triangles.

After the smallest, hunt joined pieces of 2 and 3 parts, then the whole figure.

Mistake 02

Counting a four-sided pointed region as a triangle.

Trace the border. Only exactly three straight pieces count.

Mistake 03

Forgetting the big outer triangle.

End every count by asking: is the whole figure itself one more triangle?

Mistake 04

Taking k as the number of base parts.

k is the number of extra apex lines. The base has k + 1 parts, not k.

Mistake 05

Multiplying by h instead of h + 1 for horizontal cuts.

The base itself is also a base line, so the multiplier is h + 1.

Mistake 06

Skipping the upside-down triangles in row figures.

Count upright sizes first, then hunt the upside-down ones between them.

Quick revision

Read this the night before the exam.

Count size by size: smallest, then joined pieces, then the whole figure.
Apex lines: $T=\dfrac{(k+1)(k+2)}{2}$ ; with h horizontal cuts multiply by $h+1$ .
Square with 2 diagonals = 8; add both midlines = 16.
Row figures: 1, 5, 13, 27, 48 triangles for 1 to 5 rows.
Triangle with 3 medians = 16; pentagon with all diagonals = 35.
No formula fits? Label every corner and crossing point, then list triangles.

Practice: 25 questions

Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.

Topic test · 10 questions

Suggested time 7 min · wrong answers go to your mistake notebook automatically.

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