Class 10 Maths Important Formulas – NCERT Chapters 6 to 10 (Board Exam Edition)
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NCERT Chapters 6–10 of Class 10 Mathematics account for approximately 35–40 marks in the CBSE board examination, based on the official CBSE sample paper pattern. These chapters test a student's ability to apply geometric reasoning, algebraic coordinates, and trigonometric ratios — skills that also lay the foundation for Class 11 and 12 Mathematics.
Students who score 90+ in Class 10 Maths consistently report that formula recall was not their problem — knowing when and how to apply each formula was. That distinction is exactly what this page addresses.
Each section below gives you the formula, the condition under which it applies, and a quick exam tip from faculty who have trained thousands of students at Kota and now teach on eSaral with IIT Bombay-verified credentials.
- Triangles
- Coordinate Geometry
- Introduction to Trigonometry
- Some Applications of Trigonometry
- Circles
India's Best Exam Preparation for Class 10th - Download Now
Chapter-Wise Formula Quick Reference Table
Chapter | Key Formulas | Approx. Marks | Difficulty |
|---|---|---|---|
Ch 6: Triangles | BPT, Similarity ratios, Area ratio, Pythagoras | 8–10 | Medium |
Ch 7: Coordinate Geometry | Distance Formula, Section Formula, Midpoint Formula, Area of Triangle | 6–8 | Low–Medium |
Ch 8: Trigonometry | Six Trigonometric Ratios, Three Identities, Standard Values Table | 8–10 | Medium–High |
Ch 9: Applications of Trigonometry | tan θ = H/D, Angle of Elevation, Angle of Depression | 5–6 | Medium |
Ch 10: Circles | Tangent ⟂ Radius, PA = PB, Angle Sum Properties | 5–6 | Low–Medium |
How to Memorise Class 10 Maths Formulas Effectively?
Use the 3-Day Spaced Repetition Method
Write each formula on Day 1. Review without looking on Day 2. Test yourself again on Day 4. This method, backed by memory research, embeds formulas in long-term memory far more effectively than reading them repeatedly the night before an exam.
Formula Cards vs Mind Maps — Which Works Better?
Both tools work, but for different learners:
- Formula cards suit students who prefer isolated recall (strong for multiple-choice and fill-in-the-blank questions).
- Mind maps (like the downloadable one on this page) suit visual learners and help recall the relationship between formulas — essential for long-answer proofs.
The eSaral mind map for Chapters 6–10 was designed by faculty with experience training students at Kota — the same Kota-quality structured thinking, now accessible online.
Practice Backward From Answer
For formula retention in Coordinate Geometry and Trigonometry, work backward: start with the answer, identify which formula produces it, and reverse-engineer the question. This trains your brain to pattern-match formulas to question types — exactly what board exams reward.
For additional practice, explore the NCERT Solutions Class 11 Maths and the NCERT Books Class 11 pages to see how Class 10 foundations connect to higher-level content.
Class 10 Mathematics NCERT (Chapters 6–10)
Important Formulas + Quick Mind Maps
Chapter 6: Triangles
Important Formulas
1. Basic Proportionality Theorem (BPT)
If a line is drawn parallel to one side of a triangle:
$$ \frac{AD}{DB} = \frac{AE}{EC} $$
2. Similar Triangles
For similar triangles:
$$ \frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} $$
3. Areas of Similar Triangles
$$ \frac{\text{Area of } \triangle ABC}{\text{Area of } \triangle DEF} = \left(\frac{AB}{DE}\right)^2 $$
4. Pythagoras Theorem
$$ (\text{Hypotenuse})^2 = (\text{Base})^2 + (\text{Perpendicular})^2 $$
$$ c^2 = a^2 + b^2 $$
Mind Map – Triangles
TRIANGLES
|
--------------------------------
| | |
BPT Similarity Pythagoras
| | |
Parallel Equal Angles Right Triangle
Line Proportional c²=a²+b²
Sides
Chapter 7: Coordinate Geometry
Important Formulas
1. Distance Formula
$$ d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} $$
2. Section Formula
For point dividing line internally in ratio m:n
$$ \left( \frac{mx_2+nx_1}{m+n}, \frac{my_2+ny_1}{m+n} \right) $$
3. Midpoint Formula
$$ \left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} \right) $$
Mind Map – Coordinate Geometry
COORDINATE GEOMETRY
|
---------------------------------
| | |
Distance Section Midpoint
Formula Formula Formula
| | |
Length of Ratio Centre Point
Line Segment Division
Chapter 8: Introduction to Trigonometry
Important Formulas
1. Trigonometric Ratios
$$ \sin \theta = \frac{P}{H} $$ $$ \cos \theta = \frac{B}{H} $$ $$ \tan \theta = \frac{P}{B} $$ $$ \csc \theta = \frac{H}{P} $$ $$ \sec \theta = \frac{H}{B} $$ $$ \cot \theta = \frac{B}{P} $$
2. Important Identities
$$ \sin^2 \theta + \cos^2 \theta = 1 $$ $$ 1 + \tan^2 \theta = \sec^2 \theta $$ $$ 1 + \cot^2 \theta = \csc^2 \theta $$
3. Complementary Angles
$$ \sin(90^\circ-\theta)=\cos\theta $$ $$ \cos(90^\circ-\theta)=\sin\theta $$
Standard Values Table
θ | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
sinθ | 0 | 1/2 | 1/√2 | √3/2 | 1 |
cosθ | 1 | √3/2 | 1/√2 | 1/2 | 0 |
tanθ | 0 | 1/√3 | 1 | √3 | ∞ |
Mind Map – Trigonometry
TRIGONOMETRY
|
---------------------------------
| | |
Ratios Identities Values
| | |
sin cos tan sin²+cos²=1 0° to 90°
sec cosec cot tan²+1=sec²
Chapter 9: Some Applications of Trigonometry
Important Formulas
Height and Distance
$$ \tan \theta = \frac{\text{Height}}{\text{Distance}} $$ $$ \sin \theta = \frac{\text{Perpendicular}}{\text{Hypotenuse}} $$ $$ \cos \theta = \frac{\text{Base}}{\text{Hypotenuse}} $$
Terms Used
- Line of sight
- Angle of elevation
- Angle of depression
Mind Map – Applications of Trigonometry
APPLICATIONS OF TRIGONOMETRY
|
---------------------------------
| | |
Height & Angle of Angle of
Distance Elevation Depression
|
tanθ = H/D
Chapter 10: Circles
Important Formulas & Theorems
1. Tangent Theorem
The tangent at any point of a circle is perpendicular to the radius.
2. Equal Tangents
Tangents drawn from an external point are equal.
$$ PA = PB $$
Mind Map – Circles
CIRCLES
|
--------------------------------
| |
Tangent Radius
| |
Perpendicular Centre to Point
to Radius of Contact
|
Equal Tangents
PA = PB
Quick Revision Sheet
Chapter-wise Key Formulas
Triangles
- $c^2=a^2+b^2$
- Similarity ratios
- Area ratio theorem
Coordinate Geometry
- Distance Formula
- Midpoint Formula
- Section Formula
Trigonometry
- $\sin^2\theta+\cos^2\theta=1$
- $1+\tan^2\theta=\sec^2\theta$
Applications of Trigonometry
- $\tan\theta=\frac{H}{D}$
Circles
- Tangent ⟂ Radius
- Equal tangents from an external point
Exam Tips
- ✅ Learn all trigonometric identities thoroughly.
- ✅ Practice coordinate geometry numerals daily.
- ✅ Draw neat diagrams in triangles and circles.
- ✅ Memorize standard trigonometric values.
- ✅ Revise formulas before solving MCQs and board questions.
These formulas are best applied through full worked problems — link to the CBSE class 10 Maths hub so students can practise each one in context.
Frequently Asked Questions
Is the area ratio formula different from the similarity ratio in Chapter 6?
Yes. The similarity ratio compares corresponding sides directly (e.g., AB/DE). The area ratio is the square of the similarity ratio. So if two similar triangles have sides in ratio 2:3, their areas are in ratio 4:9. Forgetting to square is the single most common error in Chapter 6 board answers.
How many formulas are there in Class 10 Maths NCERT Chapters 6 to 10?
Across the five chapters, there are approximately 25–30 distinct formulas and theorems that are examinable. Chapter 8 (Trigonometry) has the highest count with 6 ratios, 3 identities, 6 complementary angle pairs, and a full standard values table. Chapter 10 (Circles) has the fewest but they carry consistent 3-mark proof questions every year.
What are the most important formulas in Class 10 Maths Chapters 6–10?
The highest-yield formulas are: Pythagoras Theorem (Ch 6), Distance Formula and Section Formula (Ch 7), the three Pythagorean trigonometric identities and standard value table (Ch 8), tan θ = Height/Distance (Ch 9), and the equal-tangent theorem PA = PB (Ch 10). These appear in 60–70% of board exam questions from this unit.
What is the section formula and how is it different from the midpoint formula?
The section formula finds the coordinates of a point that divides a line segment in any ratio m:n. The midpoint formula is a special case where m = n = 1, giving the exact centre of the segment. Use the section formula when the ratio is specified; use midpoint when you need the centre point or when the problem says "bisects."
Do I need to prove the Pythagoras Theorem for the board exam?
Yes. The CBSE board exam at Class 10 level can ask for a formal proof of the Pythagoras Theorem as a 5-mark question. The standard proof uses similarity of triangles (△ABC ~ △ADB ~ △BDC). Knowing the proof also reinforces your understanding of Chapter 6 similarity concepts, making related application questions easier to solve.
What is the difference between angle of elevation and angle of depression in Chapter 9?
The angle of elevation is measured upward from the horizontal line of sight — used when an observer looks up at an object. The angle of depression is measured downward from the horizontal — used when an observer looks down at an object. Crucially, the angle of depression from point A to point B equals the angle of elevation from point B to point A, which simplifies two-level problems.