Pythagoras Triplets and Triangles

📑 On this page

⚡ The short answer

In a right-angled triangle, a² + b² = c², where c is the hypotenuse: the longest side, opposite the 90°. Learn 3-4-5, 5-12-13, 8-15-17, 7-24-25 and their multiples, and most sums need no square root. Area = ½ × base × perpendicular height. The classic trap: using a slant side as the height.

🎬 The 30-second story

The lawn shortcut

“Lecture at 9. Path goes 80 m along the lawn, then 60 m up. I'm walking it.”

🏫

Aarav

8:58 AM. Aarav respects footpaths.

“I'm cutting straight across.”

🌱

Zoya

Zoya takes the diagonal.

“How are you already sitting down?”

⏱️

Aarav

Aarav arrives after 140 m, breathing hard.

“100 m. It's 3-4-5, times twenty.”

📐

Zoya

She did the maths on the way.

Tara's maths: Legs 60 and 80 make a right triangle. Diagonal = √(60² + 80²) = √10,000 = 100 m. Zoya saved 40 m, about 28.6% of Aarav's walk.

What does the Pythagoras theorem say, and why is it true?

Take a triangle with one 90° corner. The two sides forming it are the legs; the side across from it is the hypotenuse. Build a square on each side. The two small squares together exactly fill the big one. That's the whole theorem.

🟦 Picture the percent

A 6-8-10 triangle, counted in little squares

6² = 36

short leg

8² = 64

long leg

10² = 100

hypotenuse

36 + 64 = 100, every time the angle is 90°.

1
Draw a square of side a + b
Area (a + b)² = a² + 2ab + b².
2
Put four copies of the triangle in its corners
They take 4 × ½ab = 2ab.
3
The hole in the middle
A tilted square of side c. Area c².
4
Balance
a² + 2ab + b² = 2ab + c², so a² + b² = c².

It runs backwards too. If the sides satisfy a² + b² = c², there is a right angle. If not, the gap tells you the shape.

🎚️ The scale

Longest side squared vs the other two

← c² smallerc² bigger →

Acute

c² < a² + b²

e.g. 4, 5, 6: 36 < 41

Right

c² = a² + b²

e.g. 5, 12, 13: 169 = 169

Obtuse

c² > a² + b²

e.g. 4, 5, 7: 49 > 41

Which Pythagoras triplets should you know by heart?

A triplet is three whole numbers that fit a² + b² = c². Setters love them because the root comes out clean. Any multiple is also a triplet, decimals included.

Triplet	Check	Multiples you'll meet
3, 4, 5	9 + 16 = 25	6-8-10, 9-12-15, 60-80-100
5, 12, 13	25 + 144 = 169	10-24-26, 0.5-1.2-1.3
8, 15, 17	64 + 225 = 289	16-30-34
7, 24, 25	49 + 576 = 625	14-48-50
1, 1, √2	half a square	diagonal = side × 1.414

→ scroll to see more

📌 Make your own triplets

Pick an odd number n. The others are (n² − 1)/2 and (n² + 1)/2. So 3 gives 4, 5; 5 gives 12, 13; 7 gives 24, 25; 9 gives 40, 41.

How do you find the area of any triangle?

Box a triangle in a rectangle with the same base and height: it fills exactly half. So area = ½ × base × height, where height is the perpendicular drop to the base, not a slanted side.

📐 The figure

Three triangles, three area formulas

Any triangle

Area = ½ × b × h

Right triangle

Area = ½ × a × b

One leg is already the height.

Equilateral

Area = (√3/4) × a²

Not to scale.

🧭 Decision flow

Which area formula?

❓ Base and perpendicular height given?

Yes →

½ × base × height

No ↓

❓ Right-angled?

Yes →

½ × leg × leg

No ↓

❓ Equilateral?

Yes →

(√3/4) × a² ≈ 0.433a²

No →

Three sides: isosceles → height by Pythagoras; else Heron

⚠️ The slant side is not the height

Sides 10, 10 and base 12: the height is 8, not 10. ½ × 12 × 10 = 60 is the planted option; the real area is 48.

Triangle questions solved: warm-up to tricky

✍️ Worked example Warm-up ⏱ 30 sec

What 'inch' does your phone mean?

Meher's screen is 15 cm tall and 8 cm wide. Phones are sold by the diagonal. What is it, in inches (1 inch = 2.54 cm)?

1
The diagonal is the hypotenuse.
2
8, 15 → 17 (64 + 225 = 289).
3
17 ÷ 2.54 ≈ 6.69.

Answer: 17 cm: a 6.7-inch phone

⚡ The exam-hall shortcut

Two numbers from a triplet? Write the third. No squaring.

✍️ Worked example CLAT level ⏱ 60 sec

The slipping ladder

A 10 m ladder leans on a wall, its foot 6 m out. The top slips 2 m down. How far does the foot slide?

1
6, ?, 10 is 3-4-5 doubled: the top is 8 m up.
2
After the slip it's 6 m up; the ladder is still 10 m.
3
Foot distance: 6, ?, 10 again, so 8 m. It moved 8 − 6 = 2 m.

Answer: 2 m

⚡ The exam-hall shortcut

The ladder is the hypotenuse in both pictures. Find each leg, then subtract. The equal 2 m moves are luck of this triplet, not a rule.

✍️ Worked example CLAT level ⏱ 60 sec

The fest banner

A triangular banner has sides 10 m, 10 m and 12 m. Cloth costs ₹40 per m². Total cost?

1
Isosceles: the height hits the 12 m base at its midpoint, 6 m from each end.
2
Height = √(100 − 36) = 8 m.
3
Area = ½ × 12 × 8 = 48 m². Cost = 48 × 40.

Answer: ₹1,920

⚡ Check

Heron agrees: s = 16, √(16 × 6 × 6 × 4) = 48. Halving the base is faster.

✍️ Worked example Tricky ⏱ 90 sec

Two riders, one right angle

At noon Aarav rides north at 24 km/h and Rohan east at 18 km/h from the same crossing. When are they 45 km apart?

1
After t hours the gap is the hypotenuse of 18t and 24t.
2
18 : 24 = 3 : 4, so the gap grows at 30 km per hour.
3
45 ÷ 30 = 1.5 h. Check: 27-36-45.

Answer: 1:30 PM

⚡ The exam-hall shortcut

Combine right-angled speeds with Pythagoras once. Adding 18 + 24 = 42 is for opposite directions.

What is the fastest way to crack a right-triangle sum?

1
Shrink
Divide by the common factor. 15 and 36 become 5 and 12, so c = 13 × 3 = 39.
2
Spot the triplet
Two numbers from the table? Write the third.
3
Box it
c is longer than the longer leg, shorter than both legs added. Legs 5 and 9: c is between 9 and 14.
4
Check last digits
A square never ends in 2, 3, 7 or 8. c² = 147? Strike every whole-number option.

🎛️ Try it yourself — change any number

Try it: the lawn shortcut

Path up the side (m) Path along the lawn (m)

Hypotenuse

100

Area of the right triangle

2,400

c = √(60² + 80²) = 100

60 and 80 give 100. Try 18 and 24 (the riders' hourly gap), then 1 and 1 to see √2.

How do triangles hide inside a CLAT Quant caselet?

CLAT never says 'find the hypotenuse'. It says a station is 1.2 km east and 0.5 km north of a college gate, and asks how much a straight skywalk saves. Find the right angle: roads meeting at a corner, a wall and the ground, a point on a semicircle.

📊 The data

College gate to metro station

Roads (east, then north) · 1.2 + 0.5

1.7 km

Skywalk · √(1.2² + 0.5²)

1.3 km

0.5-1.2-1.3 is 5-12-13 shrunk. Saving = 0.4 ÷ 1.7 ≈ 23.5%. Dividing by 1.3 (30.8%) is the wrong-base option.

✓No right angle, no Pythagoras. 4, 5, 7 has no 90° corner.
✓Finding a leg? Subtract. 17 m ladder, foot 8 m out: √(289 − 64) = 15.
✓Forgetting ½. Base × height is the box, not the triangle.
✓Impossible sides. Two sides must add to more than the third: 3, 4, 8 is no triangle.

How did CLAT 2020 ask about triangles?

CLAT 2020 tied a goat at the centre O of a semicircular plot. Q146–147 (rope and area) are on the circles page. The paper's text lost the figure: AC is the diameter, O its midpoint, B the top of the arc above O, so OA = OB = OC = r. The key's rope is 2 + 2/π ≈ 2.64 (correct algebra gives 3.27, not offered). Q148–150 only fit the key's radius, so we use it.

📐 The figure

The 2020 goat plot and triangle ABC

The plot

OA = OB = OC = r

Triangle ABC

∠B = 90°, AB = BC = r√2

Not to scale. A triangle on a diameter with its third corner on the arc is right-angled at that corner.

📝 Asked in CLAT 2020 · Q148 Side of a right triangle

Direction: In the figure given below, a goat is tied to a pole (at point O) which is the center of the semi-circular plot. The area of the plot is equal to its perimeter. BO is the length of the rope through which the goat is tied to the pole. Answer to the closest decimal. [Figure, described: semicircle on diameter AC, centre O; B is the top of the arc, directly above O. Official answer to Q146: rope = 2 + 2/π.]

If a triangle is created by joining points A, B, and C. What will be the length of AB?

A3.73

B7.33

C1.86

DCan't be calculated with the given data

Reveal answer + reasoning

Answer: A

In triangle AOB, OA = OB = r at 90°. AB = √(r² + r²) = r√2 ≈ 2.637 × 1.414 ≈ 3.73. Box it: AB is longer than r (2.64) and, as a chord, shorter than the diameter (5.27). Only 3.73 fits. About 40 seconds.

Why the others fail

B — Longer than the diameter. No chord can be.
C — Half of AB; shorter than the rope.
D — Two equal legs at 90° fix the hypotenuse.

📝 Asked in CLAT 2020 · Q149 Area from base and height

Same figure as Q148: semicircle on diameter AC, centre O, B the top of the arc; rope BO = 2 + 2/π.

What will be the area of the triangle ABC?

A6.96

B9.66

C3.66

DCan't be calculated with the given data

Reveal answer + reasoning

Answer: A

Base AC = 2r, height BO = r, so area = ½ × 2r × r = r². Via the right angle at B: ½ × 3.73 × 3.73 ≈ 6.96 (unrounded r² ≈ 6.95). About 30 seconds.

Why the others fail

B — Too big: B at the top gives the tallest possible triangle on AC, area r² ≈ 6.95.
C — About half the true area. No reading of the figure gives it.
D — Base and height are both known.

📝 Asked in CLAT 2020 · Q150 Perimeter (disputed key)

Same figure as Q148: semicircle on diameter AC, centre O, B the top of the arc; rope BO = 2 + 2/π.

What will be the perimeter of the triangle ABC?

A40.4

B30.3

C20.2

D10.1

Reveal answer + reasoning

Answer: D

AB + BC + AC = 3.73 + 3.73 + 5.27 ≈ 12.73, which isn't offered. The key's D, 10.1 is 3.73 + 3.73 + 2.64: it adds the radius instead of the diameter. We show the key's answer, but the true perimeter is 12.73.
In the hall: AC must be under AB + BC = 7.46, so the perimeter is under 14.92. Only 10.1 is. About 45 seconds.

Why the others fail

A — Four times 10.1; far above the 14.92 ceiling.
B — Three times 10.1; above the ceiling.
C — Twice 10.1; still above the ceiling.

Which triangle myths cost you marks?

💥 Myth vs fact

Triangle myths

🙅 Myth

a² + b² = c² works for every triangle.

Tap to bust it →

✅ Fact

Only with a 90° angle. Otherwise compare c² with a² + b².

🙅 Myth

Any side can be c.

Tap to bust it →

✅ Fact

c is the longest side, opposite the right angle. For a leg, subtract: 17² − 8² = 225.

🙅 Myth

Area = ½ × base × the other side.

Tap to bust it →

✅ Fact

Only in a right triangle. Otherwise use the perpendicular height.

🙅 Myth

Triplets are just 3-4-5.

Tap to bust it →

✅ Fact

Any multiple works: 0.5-1.2-1.3 is 5-12-13.

Can you crack these five triangle questions?

🎯 Quick quiz

Five triangles, five answers

0 / 5

Q1. A 17 m ladder has its foot 8 m from a wall. How high does it reach?

Q2. Which lengths make a right-angled triangle?

Q3. An equilateral triangle has perimeter 24 cm. Its area?

Q4. A triangle with sides 4, 5 and 7 cm is:

Q5. A rhombus has diagonals 24 cm and 10 cm. Its perimeter?

🎯 Triangles in a nutshell

Right angle → a² + b² = c², c the longest side.
Know 3-4-5, 5-12-13, 8-15-17, 7-24-25 and multiples.
Shrink, spot the triplet, scale back.
Area = ½ × base × perpendicular height.
Diameter + point on the arc = 90° at that point.

📎 Formula card

c² = a² + b²; leg = √(c² − a²)
Triplets: 3-4-5, 5-12-13, 8-15-17, 7-24-25, 9-40-41
Square diagonal = side × √2 ≈ 1.414 × side
Equilateral: height ≈ 0.866a; area = (√3/4)a²
Area = ½ × b × h
Heron: s = (a + b + c)/2, area = √[s(s − a)(s − b)(s − c)]
c² < a² + b² acute; = right; > obtuse

Frequently asked questions

What are the most common Pythagorean triplets?

Memorise 3-4-5, 5-12-13, 8-15-17 and 7-24-25, with 9-40-41 as a bonus. Any multiple also works, so 6-8-10, 30-40-50 and 0.5-1.2-1.3 are triplets too. If two sides of a right triangle match a triplet, the third side is the remaining number, and you skip the square root entirely.

How do you check whether a triangle is right-angled?

Square the longest side and compare it with the sum of the squares of the other two. Equal means right-angled: 5, 12, 13 gives 169 = 25 + 144. Smaller means acute; bigger means obtuse. For 4, 5, 7, 49 is more than 41, so that triangle is obtuse.

How do you find a triangle's area from its three sides?

Use Heron's formula: s = (a + b + c)/2, then area = √[s(s − a)(s − b)(s − c)]. For sides 5, 5 and 6, s = 8 and area = √(8 × 3 × 3 × 2) = √144 = 12. For an isosceles triangle, halving the base and finding the height by Pythagoras is usually quicker.

What is the height of an equilateral triangle?

Height = (√3/2) × side, about 0.866 times the side. The height splits the triangle into two right triangles with hypotenuse a and base a/2, so Pythagoras gives it. A 10 cm side has height 5√3 ≈ 8.66 cm and area about 43.3 cm².

Is Pythagoras asked in CLAT?

Yes, inside a caselet rather than as a bare formula. CLAT 2020 asked three questions on triangle ABC drawn in a semicircular goat plot: a side, the area and the perimeter. Expect it wherever a figure hides a right angle: a diagonal, a ladder, two roads meeting, or a point on a semicircle.