Important Formulas of 10th Class Math – New Book 2026‑27 (All Units)

All the important formulas of 10th class mathematics on one page: the new book (2026‑27, Punjab Board) first, then the old book. For solved exercises, open our Class 10 math notes (new book) or the old book notes.

New book 2026‑27 (Punjab Board)1. Complex Numbers 2. Quadratic Equations and Inequalities 3. Matrices and Determinants 4. Functions and Graphs 5. Algebraic Fractions 6. Vectors in Plane 7. Trigonometry 8. Chords and Arcs of a Circle 9. Tangent and Angles of a Circle 10. Practical Geometry of Circles 11. Information Handling 12. Probability

Old bookUnit 1 Unit 2 Unit 3 Unit 5 Unit 6 Unit 7 Unit 8 Unit 10 Unit 12 Unit 13

New Book Formulas (2026‑27)

Unit 1: Complex Numbers

• Imaginary unit \[i=\sqrt{-1}, \quad i^2=-1\]
• Powers of i \[i^3=-i, \quad i^4=1\]
• Even power \[i^{2n}=(-1)^n\]
• Odd power \[i^{2n+1}=(-1)^n\,i\]
• Root of a negative number \[\sqrt{-a}=i\sqrt{a}\]
• Standard form \[z=a+bi, \quad \text{Re}(z)=a, \quad \text{Im}(z)=b\]
• Equality \[a+bi=c+di \;\Rightarrow\; a=c,\; b=d\]
• Addition \[(a+bi)+(c+di)=(a+c)+(b+d)i\]
• Subtraction \[(a+bi)-(c+di)=(a-c)+(b-d)i\]
• Multiplication \[(a+bi)(c+di)=(ac-bd)+(ad+bc)i\]
• Conjugate \[\overline{z}=a-bi, \quad \overline{\overline{z}}=z\]
• Sum and difference with conjugate \[z+\overline{z}=2a, \quad z-\overline{z}=2bi\]
• Product with conjugate \[(a+bi)(a-bi)=a^2+b^2\]
• Modulus \[|z|=\sqrt{a^2+b^2}\]
• Reciprocal \[\frac{1}{a+bi}=\frac{a-bi}{a^2+b^2}\]
• Modulus properties \[z\,\overline{z}=|z|^2, \quad |z|=|\overline{z}|=|-z|\]

• Powers of i: they repeat after every 4 steps: \(i,\; -1,\; -i,\; 1\).

Solved exercises: Unit 1 Complex Numbers solutions (Exercises 1.1 to 1.4) · All units ↑

Unit 2: Quadratic Equations and Inequalities

• Standard form \[ax^2+bx+c=0, \quad a \neq 0\]
• Quadratic formula \[x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}\]
• Discriminant \[D=b^2-4ac\]
• Sum of roots \[\alpha+\beta=-\frac{b}{a}\]
• Product of roots \[\alpha\beta=\frac{c}{a}\]
• Equation from roots \[x^2-(\alpha+\beta)x+\alpha\beta=0\]
• Sum of squares of roots \[\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta\]
• Vertex of the parabola \[x=-\frac{b}{2a}\]
• Inequality, between the roots \[(x-r_1)(x-r_2) \le 0 \;\Rightarrow\; r_1 \le x \le r_2\]
• Inequality, outside the roots \[(x-r_1)(x-r_2) \ge 0 \;\Rightarrow\; x \le r_1 \text{ or } x \ge r_2\]
• Square of a real number \[(x-a)^2 \ge 0 \text{ for every real } x\]

• Nature of roots, \(D \gt 0\): roots are real and unequal.
• Nature of roots, \(D=0\): roots are real and equal.
• Nature of roots, \(D \lt 0\): roots are imaginary.
• \(D \gt 0\) and a perfect square: roots are rational and unequal.
• \(D \gt 0\) and not a perfect square: roots are irrational and unequal.
• Methods of solving: factorization, completing the square, the quadratic formula, or the graph (where the parabola cuts the x-axis).
• Maximum or minimum: the vertex gives a maximum when \(a \lt 0\) and a minimum when \(a \gt 0\).
• Graphical solution: the solution is the point where the two graphs meet. A line and a parabola meet at zero, one or two points.
• Changing the subject of a formula: keep the required variable alone on one side. Whatever is done to one side is done to the other.
• Solving an inequality: bring everything to one side, factorize, and find the critical points by putting each factor equal to zero. The critical points divide the number line into intervals; test a value in each interval. In the formulas above, \(r_1 \lt r_2\).
• Negative multiplier: multiplying or dividing by a negative number reverses the inequality sign.
• End points: included for ≤ and ≥, excluded for < and >.

Solved exercises: Unit 2 Quadratic Equations and Inequalities solutions (Exercises 2.1 to 2.7) · All units ↑

Unit 3: Matrices and Determinants

• The matrix A \[A=\begin{bmatrix} a & b \\ c & d \end{bmatrix}\]
• Determinant \[|A|=ad-bc\]
• Adjoint \[\text{Adj}\,A=\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}\]
• Singular matrix \[|A|=0\]
• Inverse \[A^{-1}=\frac{1}{|A|}\,\text{Adj}\,A, \quad |A| \neq 0\]
• Non-singular matrix \[|A| \neq 0\]
• Checking an inverse \[AA^{-1}=A^{-1}A=I\]
• Inverse of a product \[(AB)^{-1}=B^{-1}A^{-1}\]
• Matrix equation \[AX=B \;\Rightarrow\; X=A^{-1}B\]
• Cramer’s rule \[x=\frac{\Delta_x}{\Delta}, \quad y=\frac{\Delta_y}{\Delta}, \quad \Delta \neq 0\]
• Order of a product \[(m \times n)(n \times p) \rightarrow m \times p\]
• Associative law \[A(BC)=(AB)C\]
• Transpose of a product \[(BC)^t=C^tB^t\]
• Distributive laws \[A(B-C)=AB-AC\] \[(B+C)A=BA+CA\]
• Not commutative \[AB \neq BA \text{ (usually)}\]

• Order of a matrix: rows-by-columns (m-by-n).
• Equal matrices: same order and same corresponding entries.
• Transpose: rows become columns.
• Negative of a matrix: change the sign of every entry.
• Addition and subtraction: possible only for matrices of the same order.
• Multiplication condition: columns of the first matrix = rows of the second.
• Cramer’s rule: replace the first column with the constants for \(\Delta_x\), and the second column for \(\Delta_y\).

Solved exercises: Unit 3 Matrices and Determinants solutions (Exercises 3.1 to 3.6) · All units ↑

Unit 4: Functions and Graphs

• Sum \[(f+g)(x)=f(x)+g(x)\]
• Difference \[(f-g)(x)=f(x)-g(x)\]
• Product \[(fg)(x)=f(x)\,g(x)\]
• Quotient \[\left(\frac{f}{g}\right)(x)=\frac{f(x)}{g(x)}, \quad g(x) \neq 0\]
• Composite functions \[(f \circ g)(x)=f(g(x))\] \[(g \circ f)(x)=g(f(x))\]
• Inverse function \[f(f^{-1}(x))=x, \quad f^{-1}(f(x))=x\]
• Absolute value graph \[y=a|x-h|+k, \quad \text{vertex } (h, k)\]
• Absolute value equation \[|A|=k \;\Rightarrow\; A=k \text{ or } A=-k\]
• Less than \[|A| \lt k \;\Rightarrow\; -k \lt A \lt k\]
• Less than or equal \[|A| \le k \;\Rightarrow\; -k \le A \le k\]
• Greater than \[|A| \gt k \;\Rightarrow\; A \gt k \text{ or } A \lt -k\]
• Greater than or equal \[|A| \ge k \;\Rightarrow\; A \ge k \text{ or } A \le -k\]

• Domain and range of the inverse: domain of \(f^{-1}\) = range of \(f\), and range of \(f^{-1}\) = domain of \(f\).

Solved exercises: Unit 4 Functions and Graphs solutions (Exercises 4.1 to 4.3) · All units ↑

Unit 5: Algebraic Fractions

• Algebraic fraction \[\frac{P(x)}{Q(x)}, \quad Q(x) \neq 0\]
• Difference of squares \[a^2-b^2=(a-b)(a+b)\]
• Sum of cubes \[a^3+b^3=(a+b)(a^2-ab+b^2)\]
• Difference of cubes \[a^3-b^3=(a-b)(a^2+ab+b^2)\]
• Multiplication \[\frac{a}{b} \times \frac{c}{d}=\frac{ac}{bd}\]
• Division \[\frac{a}{b} \div \frac{c}{d}=\frac{a}{b} \times \frac{d}{c}\]
• Speed \[\text{Speed}=\frac{\text{Distance}}{\text{Time}}\]
• Work rate \[\text{Work rate}=\frac{1}{\text{Time}}\]
• Quadratic formula \[x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}\]

• Substitution: if \(x^4\) appears, put \(y=x^2\) to reduce the equation to quadratic form.

Solved exercises: Unit 5 Algebraic Fractions solutions (Exercises 5.1 to 5.5) · All units ↑

Unit 6: Vectors in Plane

• Component form \[\vec{a}=xi+yj\]
• Position vector of P(x, y) \[\overrightarrow{OP}=xi+yj\]
• Vector from A to B \[\overrightarrow{AB}=(x_2-x_1)i+(y_2-y_1)j\]
• Magnitude \[|\vec{a}|=\sqrt{x^2+y^2}\]
• Unit vector \[\hat{a}=\frac{\vec{a}}{|\vec{a}|}\]
• Addition \[\vec{a}+\vec{b}=(x_1+x_2)i+(y_1+y_2)j\]
• Subtraction \[\vec{a}-\vec{b}=(x_1-x_2)i+(y_1-y_2)j\]
• Scalar multiplication \[k\vec{a}=kxi+kyj\]
• Translation \[(x, y) \rightarrow (x+a,\; y+b)\]

• Vector from A to B: here A is the point \((x_1, y_1)\) and B is the point \((x_2, y_2)\).
• Equal vectors: two vectors are equal when their corresponding components are equal.
• Parallel vectors: two non-zero vectors are parallel when one is a scalar multiple of the other, so the ratios of their components are equal.

Solved exercises: Unit 6 Vectors in Plane solutions (Exercises 6.1 to 6.3) · All units ↑

Unit 7: Trigonometry

• Sine ratio \[\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\]
• Cosine ratio \[\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\]
• Tangent ratio \[\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\]
• Pythagoras theorem \[a^2+b^2=c^2\]
• Finding an angle \[\theta=\tan^{-1}\left(\frac{\text{opposite}}{\text{adjacent}}\right)\]
• Angles of a triangle \[\alpha+\beta+\gamma=180^\circ\]
• Sine rule \[\frac{a}{\sin\alpha}=\frac{b}{\sin\beta}=\frac{c}{\sin\gamma}\]
• Cosine rule \[a^2=b^2+c^2-2bc\cos\alpha\] \[b^2=a^2+c^2-2ac\cos\beta\] \[c^2=a^2+b^2-2ab\cos\gamma\]
• Area, two sides and the included angle \[\Delta=\tfrac{1}{2}\,ab\sin\gamma\]
• Area, one side and two angles \[\Delta=\frac{a^2\sin\beta\,\sin\gamma}{2\sin\alpha}\]
• Heron’s formula \[\Delta=\sqrt{s(s-a)(s-b)(s-c)}\] \[s=\frac{a+b+c}{2}\]
• Allied angles with 180° \[\sin(180^\circ-\theta)=\sin\theta\] \[\tan(180^\circ-\theta)=-\tan\theta\] \[\csc(180^\circ-\theta)=\csc\theta\]
• Allied angles with 90° \[\cos(90^\circ+\theta)=-\sin\theta\] \[\tan(90^\circ+\theta)=-\cot\theta\] \[\sec(90^\circ-\theta)=\csc\theta\]
• Components of a displacement d \[\text{east}=d\,\sin(\text{bearing})\] \[\text{north}=d\,\cos(\text{bearing})\]

• ASTC rule: All, Sin, Tan, Cos are positive in quadrants I, II, III and IV.
• Angles 180° ± θ: the function stays the same; the sign depends on the quadrant.
• Angles 90° ± θ: the function changes to its co-function; the sign depends on the quadrant.
• Bearing: measured clockwise from North and written with three figures.
• Reverse bearing: differs from the forward bearing by 180°.
• Angle of elevation: measured above the horizontal line.
• Angle of depression: measured below the horizontal line.

Solved exercises: Unit 7 Trigonometry solutions (Exercises 7.1 to 7.4) · All units ↑

Unit 8: Chords and Arcs of a Circle

• Pythagoras theorem \[a^2+b^2=c^2\]

• Chord bisecting theorem: a perpendicular drawn from the centre of a circle to a chord bisects the chord.
• Converse: the perpendicular bisector of a chord passes through the centre of the circle.
• Equal chords: equal chords of the same circle are equidistant from the centre.
• Equal arcs: equal chords intercept equal arcs, and equal arcs subtend equal angles at the centre.

Solved exercises: Unit 8 Chords and Arcs of a Circle solutions (Exercises 8.1 to 8.2) · All units ↑

Unit 9: Tangent and Angles of a Circle

• Cyclic quadrilateral \[\angle A+\angle C=180^\circ\] \[\angle B+\angle D=180^\circ\]
• Circles touching externally \[d=r_1+r_2\]
• Circles touching internally \[d=r_1-r_2, \quad r_1 \gt r_2\]
• Degrees to radians \[\text{radians}=\text{degrees} \times \frac{\pi}{180}\]
• Arc length \[\ell=r\theta\]
• Area of a sector \[A=\tfrac{1}{2}\,r^2\theta\]
• Perimeter of a sector \[P=2r+\ell\]
• Area of a circle \[A=\pi r^2\]
• Circumference of a circle \[C=2\pi r\]

• Radians: in the arc length and sector formulas, θ is in radians.
• Angle at the centre: twice the angle at the circumference standing on the same arc.
• Radius and tangent: the radius drawn to the point of contact is perpendicular to the tangent (90°).
• Tangents from an external point: tangents drawn from the same external point are equal in length.
• Angle in a semicircle: always a right angle (90°).
• Angles in the same segment: equal.
• Equal chords: equal chords subtend equal angles at the circumference.
• Cyclic quadrilateral: opposite angles are supplementary, and the exterior angle equals the interior opposite angle.
• Tangent-chord theorem: the angle between a tangent and a chord equals the angle in the alternate segment.

Solved exercises: Unit 9 Tangent and Angles of a Circle solutions (Exercises 9.1 to 9.2) · All units ↑

Unit 10: Practical Geometry of Circles

• Perpendicular bisector of a chord: always passes through the centre of the circle.
• Circle through three points: one and only one circle passes through three non-collinear points.
• Tangent: touches the circle at exactly one point.
• Radius and tangent: the radius drawn to the point of contact is perpendicular to the tangent.
• Tangents from an external point: two tangents can be drawn, and they are equal in length.
• Constructing tangents from an external point P: draw a circle on OP as diameter, centred at the midpoint M of OP. It cuts the given circle at the points of contact.
• Angle between two tangents: supplementary to the angle between the two radii drawn to the points of contact.

Solved exercises: Unit 10 Practical Geometry of Circles solutions (Exercises 10.1 to 10.2) · All units ↑

Unit 11: Information Handling

• Median position on the ogive \[\frac{N}{2}\]
• Lower quartile position \[Q_1: \; \frac{N}{4}\]
• Upper quartile position \[Q_3: \; \frac{3N}{4}\]
• Decile position \[D_k: \; \frac{kN}{10}\]
• Percentile position \[P_k: \; \frac{kN}{100}\]
• Value from a grouped table \[l+\frac{\text{position}-c}{f} \times h\]
• Interquartile range \[\text{IQR}=Q_3-Q_1\]
• Mean \[\bar{x}=\frac{\sum x}{n}\]
• Range \[\text{highest value}-\text{lowest value}\]
• Mean for frequency data \[\bar{x}=\frac{\sum fx}{\sum f}\]
• Variance \[\sigma^2=\frac{\sum (x-\bar{x})^2}{n}\]
• Variance, short form \[\sigma^2=\frac{\sum x^2}{n}-\bar{x}^2\]
• Variance for frequency data \[\sigma^2=\frac{\sum fx^2}{\sum f}-\bar{x}^2\]
• Standard deviation \[\sigma=\sqrt{\sigma^2}\]

• Symbols: N is the total frequency. In the grouped-table formula, l is the lower boundary of the class, c is the cumulative frequency before the class, f is the frequency of the class and h is the class width.
• Cumulative frequency: add the frequencies successively.
• Less-than ogive: plot cumulative frequencies against the upper class boundaries.
• Class mark: the midpoint of the class.
• Ungrouped data: the median is the middle value in ascending order. The lower and upper quartiles are the middle values of the lower and upper halves.
• Comparing two data sets: the mean compares performance; the lower standard deviation shows the more consistent set.
• Scatter diagram: positive correlation when both values rise together; negative correlation when one rises as the other falls.

Solved exercises: Unit 11 Information Handling solutions (Exercises 11.1 to 11.2) · All units ↑

Unit 12: Probability

• Probability of an event \[P(E)=\frac{n(E)}{n(S)}\]
• Range of probability \[0 \le P(E) \le 1\]
• Impossible event \[P(E)=0\]
• Sure event \[P(E)=1\]
• Complement \[P(A^{\prime})=1-P(A)\]
• Addition law \[P(A \cup B)=P(A)+P(B)-P(A \cap B)\]
• Mutually exclusive events \[P(A \cap B)=0\] \[P(A \cup B)=P(A)+P(B)\]
• Independent events \[P(A \cap B)=P(A) \times P(B)\]
• Dependent events \[P(A \cap B)=P(A) \times P(B \mid A)\]

• Sample space: S is the set of all possible outcomes of a random experiment.
• Sum of probabilities: the probabilities of all possible outcomes add up to 1.
• Two dice: 6 × 6 = 36 outcomes.
• Standard deck: 52 cards.

Solved exercises: Unit 12 Probability solutions (Exercises 12.1 to 12.3) · All units ↑

Formula images of the new book (tap an image to open it)

Old Book Formulas

The formulas below are for the old 10th class math book (Science Group).

Unit # 1 Quadratic Equations

• \[(a + b)^2 = a^2 + b^2 + 2ab\]
• \[(a – b)^2 = a^2 + b^2 – 2ab\]
• \[x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}\]

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Unit # 2 Theory of Quadratic Equations

• \[\text{Discriminant} = b^2 – 4ac\]
• \[x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}\]
• \[1 + \omega + \omega^2 = 0\]
• \[\omega^3 = 1\]
• \[\omega = \frac{1}{\omega^2} \text { or } \omega^2 = \frac{1}{\omega}\]
• \[(x + y + z)(x^2 + y^2 + z^2 – xy – yz – xz)\] \[= x^3 + y^3 + z^3 – 3xyz\]
• \[S = -\frac{b}{a} = – \frac{\text{Coefficient of }(x)}{\text{Coefficient of }(x^2)}\]
• \[P = \frac{c}{a} = \frac{\text{Constant term}}{\text{Coefficient of }(x^2)}\]
• \[x^2 – Sx + P = 0 \text{ or }\] \[x^2 – (\text{Sum of roots}) x + \text{Product of roots} = 0\]
• \[\text{Function is Symmetric if } f(\alpha, \beta) = f(\beta, \alpha)\]
• \[\alpha^2 + \beta^2 = (\alpha + \beta)^2 – 2\alpha \beta\]
• \[\alpha^3 + \beta^3 = (\alpha + \beta)^3 – 3\alpha \beta (\alpha + \beta)\]
• \[\alpha – \beta = \sqrt{(\alpha + \beta)^2 – 4\alpha \beta}\]

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Unit # 3 Variations

• \[\text{Fourth proportional: } d = \frac{bc}{a}\]
• \[\text{Third proportional: } c = \frac{b^2}{a}\]
• \[\text{Mean proportional: }b^2 = ac\]
• \[\text{Continued proportional: }b^2 = ac\]
• \[\text{Invertendo Theorem: If } \frac{a}{b} = \frac{c}{d} \text{ then }\frac{b}{a} = \frac{d}{c}\]
• Alternendo Theorem \[\text{if }\frac{a}{b} = \frac{c}{d} then (a:c = b:d)\]
• Componendo Theorem \[\text{if }\frac{a}{b} = \frac{c}{d}\text{ then}\] \[\text{(i) } a + b : b = c + d : d\] \[\text{(ii) } a : a + b = c : c + d\]
• Dividendo Theorem \[\text{ If }\frac{a}{b} = \frac{c}{d} \text{ then }\] \[\text{(i) }a – b : b = c – d : d\] \[\text{(ii) }a : a – b = c : c – d\]
• Componendo-Dividendo Theorem \[\text{if }\frac{a}{b} = \frac{c}{d}\text{ then }\] \[\text{(i) }a + b : a – b = c + d : c – d\] \[\text{(ii) }a – b : a + b = c – d : c + d\]
• K Method \[\text{Let }\frac{a}{b} = \frac{c}{d} = k\] \[\text{Then }a = bk, c = dk\]

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Unit # 5 Sets and Functions

• \[\text{Number of all possible subsets} =2^n\]
• \[\text{Number of all possible proper subsets} =2^n – 1\]
• \[\text{Number of improper subsets} = 1\]
• \[\text{Formula for number of elements in the power set} =2^n\]
• \[A – B = \{x \mid x \in A \text{ and } x \notin B\}\]
• \[\text{if }A \cap B = \phi \text{ then A and B are disjoint sets.}\]
• \[A’ = A^c = U – A\]
• Commutative property of Union \[A \cup B = B \cup A\]
• Commutative property of intersection \[A \cap B = B \cap A\]
• Associative property of union \[A \cup (B \cup C) = (A \cup B) \cup C\]
• Associative property of intersection \[A \cap (B \cap C) = (A \cap B) \cap C\]
• Distributive property of union over intersection \[A \cup (B \cap C) = (A \cup B) \cap (A \cup C)\]
• Distributive property of intersection over union \[A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\]
• De-Morgan’s laws \[(A∪B)′=A′∩B′\text{ and }(A∩B)′=A′∪B′\]
• \[A−B=A∩B′\]
• \[(A−B)′=A′∪B\]
• \[\text{Elements in Cartesian product } X×Y=m×n\]
• \[\text{Number of binary relations} = 2^{m \times n}\]
• \[\text{If }A⊆B\text{ then }A∪B=B\]
• \[\text{If } A⊆B\text { and } B⊆A\text{ then }A=B\]
• \[A∩A^c=ϕ\]
• \[A∪A^c=U\]

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Unit # 6 Basic Statistics

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Unit # 7 Introduction to Trigonometry

• \(1^\circ = \frac{\pi}{180}\) radians
• \(x^\circ = \frac{x\pi}{180}\) radians
• 1 radian = \(\frac{180^\circ}{\pi}\)
• \(x\) radians = \(\frac{x(180^\circ)}{\pi}\)
• \(l = r\theta\)
• \(A = \frac{1}{2}r^2\theta\)
• \(\theta \pm 360k = \theta\), \(k \in \mathbb{Z}\)
• \[\sin(-\theta) = -\sin(\theta)\]
• \[\cos(-\theta) = \cos(\theta)\]
• \[\tan(-\theta) = -\tan(\theta)\]
• \[\csc(-\theta) = -\csc(\theta)\]
• \[\sec(-\theta) = \sec(\theta)\]
• \[\cot(-\theta) = -\cot(\theta)\]
• \[\sin \theta = \frac{y}{r}\]
• \[\cos \theta = \frac{x}{r}\]
• \[\tan \theta = \frac{y}{x}\]
• \[\sin^2 \theta + \cos^2 \theta = 1\]
• \[1 + \tan^2 \theta = \sec^2 \theta\]
• \[1 + \cot^2 \theta = \csc^2 \theta\]

θ	0°	30°	45°	60°	90°	180°	270°
sin θ	0	\(\frac{1}{2}\)	\(\frac{1}{\sqrt{2}}\)	\(\frac{\sqrt{3}}{2}\)	1	0	−1
cos θ	1	\(\frac{\sqrt{3}}{2}\)	\(\frac{1}{\sqrt{2}}\)	\(\frac{1}{2}\)	0	−1	0
tan θ	0	\(\frac{1}{\sqrt{3}}\)	1	\(\sqrt{3}\)	—	0	—

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Unit # 8: Projection of a Side of a Triangle

• \[(BC)^2 = (AB)^2 + (AC)^2 – 2(mAB) \cdot (mAD)\]\[\text{when angle opposite to (BC) is acute.} \]
• \[(BC)^2 = (AC)^2 + (AB)^2 + 2(mAB) \cdot (mAD)\]\[\text{when angle opposite to (BC) is obtuse.}\]
• \[\text{If } (a^2 + b^2 = c^2), (\Delta)\text{ is a right angled} (\Delta).\]
• \[\text{If } (a^2 + b^2 < c^2), (\Delta)\text{ is an obtuse angled }(\Delta).\]
• \[\text{If } (a^2 + b^2 > c^2), (\Delta)\text{ is an acute angled }(\Delta).\]
  \[\text{Where, (c) is the longest side.}\]

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Unit # 10: Tangent to a Circle

• If two circles touch each other externally, then the distance between their centers is equal to the sum of radii.
• Two circles with centers \[(C_1) \text{ and }(C_2)\], radii of measure \[(r_1) \text{ and }(r_2)\] \[\text{ such that }(mC_1C_2 = r_1 + r_2).\]
• If two circles touch each other internally, then the distance between their centers is equal to the difference of radii.
• Two circles with centers \[(C_1)\text{ and }(C_2), \text{ radii }(r_1) \text{ and }(r_2)\]
  \[\text{such that }(mC_1C_2 = |r_1 – r_2|).\]
• \[\text{Area of Circle} = (\pi r^2).\]
• \[\text{Area of semi circle} = (\frac{1}{2} \pi r^2).\]
• \[\text{Perimeter or Circumference of Circle} = (2\pi r = \pi d).\]
• \[\text{Semi Perimeter or Half Circumference of Circle} = (\pi r).\]

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Unit # 12: Angle in a Segment of a Circle

• \[(m \angle AOC = 2m \angle ABC)\]
  \[{ where,}(m \angle AOC)\text{ is central angle}\]
  \[\text{ and }(m \angle ABC) \text{ is circum angle.}\]
• The angle in a semi-circle is a right angle,
• In a segment greater than a semi-circle is less than a right angle,
• In a segment less than a semi-circle is greater than a right angle.
• \[(m \angle A + m \angle C = 180^\circ) ,and, (m \angle B + m \angle D = 180^\circ),\]
  \[ \text{where }(ABCD)\text{ is a cyclic quadrilateral.}\]
• \[(m \angle ACB = m \angle ADB), where, (m \angle ACB)\text{ and }(m \angle ADB)\] are angles of the same segment.

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Unit # 13: Practical Geometry of Circles

• Perimeter of a regular polygon = \[(n \times l),\]where (n) is the number of sides and (l) is the length of a side.
• The measure of the external angle of a regular hexagon is \[(\frac{\pi}{3}).\]
• The measure of the external angle of a regular octagon is \[(\frac{\pi}{4}).\]
• Formula for finding the angle subtended by the side of an n-sided polygon at the center of the circle = \[(\frac{360^\circ}{n}).\]

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How to use these maths formulas for class 10

Learn the formulas of one unit at a time, then use them in the exercise questions of that unit. Go through the formula sheet again before each test. For more practice, solve our 10th class math past papers, and follow our Facebook page for more maths content.