10 Maths Formula Booklet PDF

### Numbers - Real Numbers - Rational Numbers - Integers - Whole Numbers - Natural Numbers - Irrational Numbers - Real - Rational - Irrational - Rational Numbers - If *p* and *q* are integers & *q≠0* - The Rational Numbers are...

### Numbers - Real Numbers - Rational Numbers - Integers - Whole Numbers - Natural Numbers - Irrational Numbers - Real - Rational - Irrational - Rational Numbers - If *p* and *q* are integers & *q≠0* - The Rational Numbers are all the numbers that can be expressed in the form of *p/q* where *p* and *q* are integers & *q≠0* - Rational numbers includes: - Fractions - Terminating Decimal Numbers - Non-terminating and Repeating Decimal Numbers - Irrational Number - A number is called an Irrational Number, if it cannot be written in the form *p/q* where *p* and *q* are integers and *q≠0*. - Real Numbers - Rational numbers together with irrational numbers are said to be Real numbers. - That is, a Real number is either rational or irrational. - The Fundamental Theorem of Arithmetic: - Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur. - For any two positive integers *a* and *b*, HCF(*a,b*)* LCM(*a,b*) = *a* *b* (Product of two Numbers) - Let *x* be a rational number whose decimal expansion terminates. Then we can express *x* in the form *p/q*, where *p* and *q* are coprime, and the prime factorisation of *q* is of the form 2*n*5*m*, where *n*, *m* are non-negative integers. ### Integer - Whole - Negative - Proper - Improper - Natural - 0 - Even - Odd - Prime - Co-prime - Composite - Perfect ### Fraction ### All Non-terminating and Non-Repeating decimal numbers are Irrational numbers. - E.g. √2,√3,√5, √7,π, 0.12342678954253419... etc. ### Polynomials - Polynomials are algebraic expressions made up of one or more terms of a particular type. #### Standard form - The terms with highest degree first, then at the last constant term. - *f(x) = a x + a₁xº−1+a2x−2 + ...+a_2x² + a₂-1x+a* #### Coefficients - *a0, a1, a2, an* are Real Numbers - *a≠0* - *n* is whole number - Leading Coefficient - *a0* is a leading coefficient. - A real number *a* is a zero of the polynomial *p(x)* if *p(x) = 0*. - If *x = a* is such that *p(x) = 0*, then *x = a* is said to be a root of polynomial equation *p(x) = 0*. #### Linear Polynomial - *P(x) = ax + b* - *x = -b/a* - Constant Term - Coefficient of *x* #### Quadratic Polynomial - *P(x) = ax²+bx+c* #### Sum of the zeros - *-b/a* - Coeff of *x* - Coeff of *x²* #### Product of the zeros - *c/a* - Cons. term - Coeff. of *x²* #### Cubic Polynomial - *P(x) = ax³ + bx² + cx + d* #### Sum of the zeros - *-b/a* - Coeff of *x* - Coeff. of *x²* #### Product of the zeros - *-d/a* - αβ + βγ + ya = -c/a - αβγ = -d/a #### α, β are taken as roots of any quadratic polynomial. #### In case of cubic polynomial, The roots are taken as α, β, γ. ### Type Of Polynomials #### By Number of Non Zero Terms - Monomial - Single term - Ex: *x*, *2*, *x²* - Polynomial - Binomial - Two terms - Ex: *z + 4, x²+2, x+y, x³ + 3x²* - Polynomial - Trinomial - Three terms - Ex: *x² + x + 1, x² + 2xy + y², x + y+z* - Polynomial #### By Number of Distinct Variable - Univariate - Single Variable - Ex: *x+9, x3 + 3x²* - Polynomial - Bivariate - Two Variable - Ex: *x+y+9, x² + 2xy + y²* - Polynomial - Trivariate - Three Variable - Ex: *x+y+z+9, x² + y2 + z2 + 2xy + 2yz + 2xz* - Polynomial #### By Degree - Constant Polynomial - of degree 0 - Ex: *2, 3, 5 ... 2 = 2x°* - Linear Polynomial - of degree 1 - Ex: *x + 2, y + 5, 3u + 4* - Quadratic Polynomial - of degree 2 - Ex: *2x² +5 , x² + 2/7x , 5x² + 2x + π* - Cubic Polynomial - of degree 3 - Ex: *8x3, 2x³ + x + 1, 6-x³* ### Algebraic Identities - Identity I: (x + y)² = x² + 2xy + y² - Identity II: (x - y)² = x² – 2xy + y² - Identity III: x² - y² = (x + y)(x - y) - Identity IV: (x + a) (x + b) = x²+(a + b)x + ab - Identity V: (x + y + z)² = x² +y² + z² + 2xy + 2yz + 2zx - Identity VI: (x + y)³ = x³ +y³ + 3xy(x + y) - Identity VII: (x - y)³ = x3 -y³-3xy(x - y) - Identity VIII: x³ + y³ + z³ – 3xyz = (x + y + z)(x2 + y2 + z² - xy - yz - zx) ### Linear Equation in Two Variables - 2 Variables - Degree: 1 - *a.x + b.y = c* - Pair of Linear Equation in Two Variables - Two linear equations in the same two variables are called a pair of linear equations in two variables. - The most general form of a pair of linear equations is - *a₁ x + b₁y + c₁ = 0* - *a₂ x + by + c₁ = 0* - where *a1, a2, b1, b2, c1, c2* are real numbers, such that *a² + b² ≠ 0, a² + b² ≠ 0*. - (*i*) intersecting, then *a1/a2 ≠ b1/b2* - (*ii*) coincident, then *a1/a2 = b1/b2 = c1/c2*. - (*iii*) parallel, then *a1/a2 = b1/b2 ≠ c1/c2*. - A pair of linear equations in two variables can be represented, and solved, by the: - (*1*) graphical method - (*2*) algebraic method - Pair of Linear Equation in Two Variables - 1.Graphical Method of solution of a Pair of Linear Equations - Compare the ratios - *a1/a2 = b1/b2* --> Intersecting lines --> Exactly one solution (unique) Consistent - *a1/a2 = b1/b2 = c1/c2* --> Coincident lines --> Infinitely many solutions Consistent - *a1/a2 = b1/b2 ≠ c1/c2* --> Parallel lines --> No solutions Inconsistent - 2. Algebraic Method of solving pair of LE - Substitution Method - Elimination method - Cross Multiplication - *b1c2-b2c1* - *x = a1b2 – a2b1* - *y = c1a2 - c2a1* - *a1b2 – a2b1* - Case-1 If *a₁b2-a2b₁ ≠ 0* - *x* and *y* have some finite values, with unique solution for the system of equations. - Case-2 If *a₁b2 - a2b₁ = 0* - (*a*) if *a1/a2 = b1/b2 = c1/c2* (*λ ≠ 0*) --> Infinite no. of sol. - (*b*) *a1/a2 = b1/b2 ≠ c1/c2* --> So system of equations is inconsistent. ### Quadratic Equation - General form of the quadratic equation is ax² + bx + c = 0, where *a, b, c* are real numbers and *a≠0*. #### Roots of the Quadratic Equation - A real number *a* is said to be a root of the quadratic equation *ax² + bx + c = 0,* if *aa² + ba + c = 0*. - The zeroes of the quadratic polynomial *ax² + bx + c* and the roots of the quadratic equation *ax² + bx + c = 0* are the same. #### Quadratic formula - For *ax²+bx+c=0, a≠0;* then - *x = -b±√b²-4ac / 2a* - *x = -b + √D / 2a* OR *x = -b-VD / 2a* - where, *D = b² - 4ac*, known as its discriminant & is denoted by *D* or *D*. #### Nature of Roots - 1. Two distinct real roots, - If *D>0* & is a perfect square the roots will be rational and unequal. - If *D>0* & is not a perfect square The roots will be irrational and occur in a pair of conjugate surds - 2. Two equal roots (i.e., coincident roots), if *b² - 4ac = 0,* - 3. no real roots, if *b² - 4ac < 0*. #### Solution of a Quadratic Equation by Factorisation - If we can factorise *ax² + bx + c, a ≠ 0,* into a product of two linear factors, then the roots of the quadratic equation *ax² + bx + c = 0* can be found by equating each factor to zero. #### Solution of a Quadratic Equation by completing the Square - A quadratic equation can also be solved by the method of completing the square. ### Quadratic Equation - Quadratic Equation (degree 2) - General Form (*ax² + bx + c = 0*) - Solutions - METHOD 1 - Factorisation - Solution of a Quadratic Equation by Factorisation - If we can factorise *ax² + bx + c, a ≠ 0*, into a product of two linear factors, then the roots of the quadratic equation *ax² + bx + c = 0* can be found by equating each factor to zero. - METHOD 2 - Completing Square - Solution of a Quadratic Equation by completing the Square - A quadratic equation can also be solved by the method of completing the square. - METHOD 3 - Quadratic Formula - Nature Of Roots - Discriminant (D) - *D > 0* Real distinct roots - *D = 0* Equal Roots - *D < 0* No real roots ### Sequence - A systematic arrangement of numbers according to a given rule is called a sequence. - The numbers in a sequence are called its terms. - We refer the first term of a sequence as *T1*, second term as *T 2* and so on. - The *nth* term of a sequence is denoted by *T n*, which may also be referred to as the general term of the sequence. ### Progression - Sequences of numbers which follow specific patterns are called progressions - It is a special case of sequence in which it is possible to express its *nth* -term in terms of n, mathematically. - *a1, a2, a3,...., an* or *by T1, T2, T3, Τη* - *T₁ = Term at the nth place of a progression = General term or nth th – term* ### Arithmetic Progression - Arithmetic Progression (A.P.) - If difference between two consecutive terms is constant throughout the sequence, then it is known as Arithmetic Progression and this constant difference is known as its Common Difference (*d*). - Types of Arithmetic Progression - Finite AP - Finite number of terms. - Eg.: 1, 4, 9, 16, 25 - Infinite AP - Infinite number of terms. - Eg.: 1, 4, 9, 16, 25, 36.......... - General Form of an A.P. - If 'a' is the first term and 'd' is the common difference, then the standard appearance of an A.P. is - *a a+d a+2d a+3d .....* - General Term of an A.P. - *T = 1 = nth term or last term* - *Tn = a+(n-1)d* - Summation of *n* terms of an A.P. - *Sn = n/2 [2a + (n-1)d]* - *Sn = n/2 [a + 1]* where, *l* (last term) = *a + (n-1)d* - Frequently used summations - (*1*)Sum of first 'n' natural numbers : - *1+2+3+ ...+ n = n(n + 1)/2* - (*2*) Sum of first 'n' odd natural numbers : - *1+3+5+ ...... + (2n - 1) = n(1 + 2n - 1)/2 = n²* - (*3*) Sum of first 'n' even natural numbers : - *2 + 4 + 6 + ...... + 2n = n(2 + 2n)/2 = n(n + 1)* - Arithmetic Mean - If *a, b, c* are in AP, then - *b-a=c-b* - *2b = a + c* - *A. M. b = a + c / 2* ### Series - If the terms of sequence are connected by sign of addition or subtraction, then it is called as Series. - If *a1, a2, a3, ...., an* is a sequence, then the expression *a₁ + a2 + a3+.....+ an* is a series. ### Congruent Shapes - Two figures are Congruent if they have same shape and same size. - Same Shape - Same Size - Corresponding angles are congruent ### Similar Figures - Two figures are similar if they have the same shape but not size. - Same Shape - Different Size - Corresponding angles are congruent ### Congruency & Similarity Symbol - Same Shape - Congruency - Same Size - Similarity - Same Shape ### Triangle - Thales Theorem or BPT - *DE || BC* - *AD/DB = AE/EC* - *AD/DB = AE/EC* - Criteria For Similarity of Triangles - AAA or AA Similarity - SSS Similarity - SAS Similarity - AAA Similarity ### AAA Similarity - Theorem - If in two triangles, the corresponding angles are equal, then their corresponding sides are in the same ratio and hence the two triangles are similar. - This criterion is referred to as the AAA (Angle-Angle-Angle) criterion of similarity of two triangles. ### SSS Simil'arity - Theorem - If in two triangles, sides of one triangle are proportional to (i.e., in the same ratio of) the sides of the other triangle, then their corresponding angles are equal and hence the two triangles are similar. - This criterion is referred to as the SSS (Side-Side-Side) similarity criterion for two triangles. ### SAS Similarity - Theorem - If one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional then the two triangles are similar. ### Triangle - This criterion is referred to as the SAS (Side-Angle-Side) similarity criterion for two - Theorem 6.7: If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse then triangles on both sides of the perpendicular are similar to the whole triangle and to each other - Pythagoras Theorem - In a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. ### In a ∆ABC - ∠B = 90° - AC² = AB² + BC² ### Converse of Pythagoras Theorem - In a triangle if the square of one side is equal to the sum of the squares of the other two sides, then the triangle is right angled. ### Area related theorem of Similar Triangles - Theorem: The areas of two similar triangles are proportional to the squares of their corresponding sides - *Area of ∆ АВС / Area of A DEF = AB² / DE²* - *AABC ~ A DEF* ### Coordinate System - A coordinate system is a system of assigning addresses for positions in the plane (2D) or in space (3D). - They are called Cartesian because the idea was developed by the mathematician and philosopher Rene Descartes who was also known as Cartesius. - Rectangular Coordinate system - The rectangular coordinate system for the plane consists of a rectangular grid where each point in the plane is addressed by an ordered pair of numbers (x, y) - Coordinate System - The horizontal line is called the x-axis, and the vertical line is called the y-axis. - The coordinate axes divide the plane into four parts called quadrants. ### The Origin - The coordinates of the origin are (0, 0). - The point of intersection of the axes is called the origin. ### Abscissa - The distance of a point from the y-axis is called its x-coordinate, or abscissa. ### Ordinate - The distance of the point from the x axis is called its y-coordinate, or ordinate. - If the abscissa of a point is *x* and the ordinate is *y*, then (*x, y*) are called the coordinates of the point. - The coordinates of a point on the x axis are of the form (*x, 0*) and that of the point on the y-axis are (*0,y*). ### Distance Formula - *AB = √(x2 - x1)² + (Уг – У1)2* - *d = √(x2 - x1)² + (Уг – У1)2* ### Section Formula - *X = mx2 + nx1 / m+n* - *y = my2 + ny1 / m+n* ### Coordinate Geometry - Internal Division - *P = (mx2 + nx₁ / m+n , my2 + ny1 / m+n)* - External Division - *P = (mx2-nx1 / m-n , my2-ny1 / m-n)* ### Mid point Formula - If *R* is the mid-point, then *m₁ = m2* and the coordinates of *R* are *R(X1+X2/2 , Y1+Y2/2)* ### Co-ordinates of the centroid of triangle - * (x1+x2+x3/3 , y1+y2+y3/3)* ### Area of a Triangle - *1/2 [×1(У2 - Уз) + Х2 (Уз - У2) + Х3 (У1 - У2)]* ### Trigonometry - A branch of mathematics in which we study the relationships between the sides and angles of a triangle, is called trigonometry. - In a right triangle *ABC*. - *sinθ = side opposite to angle θ / hypotenuse* - *cosθ = side adjacent to angle θ / hypotenuse* - *tanθ = side opposite to angle θ / side adjacent to angle θ* - *cosecA = 1 /sinA* - *secA = 1 /cosA* - *tanA = tanA / cotA*. - *cotA = cosA / tanA* ### Trigonometric Ratios of specific angles | Angle (θ) | 0° | 30° | 45° | 60° | 90° | |---|---|---|---|---|---| | sin θ | 0 | 1/2 | 1/√2 | √3/2 | 1 | | cos θ | 1 | √3/2 | 1/√2 | 1/2 | 0 | | tan θ | 0 | 1/√3 | 1 | √3 | N.D. | | cot θ | N.D. | √3 | 1 | 1/√3 | 0 | | sec θ | 1 | 2/√3 | √2 | 2 | N.D. | | cosec θ | N.D. | 2 | √2 | 2/√3 | 1 | ### Important trigonometric identities - *sin²A + cos²A = 1* - *sin²A = 1 - cos² A* - *cos²A = 1 − sin²A* - *1 + tan²A = sec²A* - *sec²A - tan²A = 1* - *tan²A = sec²A - 1* - *1 + cot²A = cosec²A* - *cosec²A- cot²A = 1* - *cot2A = cosec²A-1* ### Remarks: - *1. sin A. cosec A = 1* - *2. cos A. sec A = 1* - *3. tan A. cot A = 1* ### Trigonometric ratios of complementary angles. - (*i*) sin(90° – θ)=cost - (*ii*) cos(90° – θ) = sinθ - (*iii*) tan(90° -0): = cotθ - (*iv*) cot(90° - 0) = tanθ - (*v*) sec(90° – 0) = coseco - (*vi*) cosec (90° – 0) = secθ ### SOME APPLICATIONS OF TRIGONOMETRY - (*1*) The line of sight is the line drawn from the eye of an observer to the point in the object viewed by the observer. - (*2*) The angle of elevation θ1:angle formed by the line of sight with the horizontal when it is above the horizontal level - (*3*) The angle of depression θ2:angle Formed by the line of sight with the horizontal when it is below the horizontal level - Note: - (*1*) The angle of elevation as well as angle of depression are measured with reference to a horizontal line. - (*2*) All objects such as towers, trees, mountains etc. shall be considered as linear for mathematical convenience. - (*3*) The height of the observer, is neglected, if it is not given in the problem. - (*4*) Angle of depression of *P* as seen from *O* is equal to the angle of elevation of *O*, as seen from *P*. i.e., *∠AOP= ∠OPX* - (*5*) The angle of elevation increases as the object moves towards the right of the line of sight. - (*6*) The angle of depression decreases as the object moves towards the left of the line of sight. ### Trigonometric Ratios of specific angles | Angle (θ) | 0° | 30° | 45° | 60° | 90° | |---|---|---|---|---|---| | sin θ | 0 | 1/2 | 1/√2 | √3/2 | 1 | | cos θ | 1 | √3/2 | 1/√2 | 1/2 | 0 | | tan θ | 0 | 1/√3 | 1 | √3 | N.D. | | cot θ | N.D. | √3 | 1 | 1/√3 | 0 | | sec θ | 1 | 2/√3 | √2 | 2 | N.D. | | cosec θ | N.D. | 2 | √2 | 2/√3 | 1 | ### Circle - 1. Circle: A circle is a collection of all points in a plane which are at a constant distance (radius) from a fixed point (centre). - 2. Tangent to a Circle: It is a line that intersects the circle at only one point. There is only one tangent at a point of the circle. The tangent to a circle is a special case of the secant, when the two end points of its corresponding chord coincide. - 3. Number of tangents from a point on a circle. - (*i*) No tangent (when a point lying inside the circle) - (*ii*) One tangent (when point lying on the circle) - (*iii*) Two tangent (when point lying outside the circle) - 4. Theorems: - (*i*) The tangent at any point of a circle is perpendicular to the radius through the point of contact. - (*ii*) The lengths of tangents drawn from an external point to circle are equal. - *OQ = OR* (Radii of the same circle) - *OP=OP* (Common) - *PAOQPA ORP* (RHS) - *PQ=PR* (CPCT) - Remarks: - 1. By theorem above, we can also conclude that at any point on a circle there can be one and only one tangent. - 2. The line containing the radius through the point of contact is also sometimes called the normal to the circle at the point. - 1. The theorem can also be proved by using the Pythagoras Theorem as follow: *PQ² = OP² - OQ = OP-OR² = PR¹ (As OQ = OR) Which gives PQ = PR.* - 2. Note also that *∠OPQ = ∠OPR.* Therefore, *OP* is the angle bisector of *QPR*, i.e., the centre lies onthe bisector of the abngle between the tow tangents. ### Extra Points - 1. In two concentric circles, the chord of the larger circle, which touches the smaller circle, is bisected at the point of contact. *AP = BP* - 2. Two tangents TP and TQ are drawn to a circle with centre *O* from an external point *T* than ∠ *PTQ = 2 ∠ OPQ*. - 3. The tangents drawn at the ends of a diameter of a circle are parallel. *AB || CD.* - 4. A quadrilateral *ABCD* is drawn to circumscribe a circle, then *AB + CD=AD + BC* - 5. An angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the centre. *∠AOB + ∠APB = 180°* - 6. A parallelogram circumscribing a circle is a rhombus. - *AP = AS* [Tangents from an external point are equal] - *BP=BQCR=CQ DR=DS* - *AB = BC = CD = DA* - *ABCD* is a rhombus. ### CONSTRUCTION - To divide a line segment in a given ratio. - Method - 1 - Given a line segment *AB*, we want to divide it in the ratio *m: n*, where both *m* and *n* are positive integers. - 1. Draw any ray *AX*, making an acute angle with *AB*. - 2. Locate 5 (= *m + n*) points *X1,X2,X3, X4,* and *X5*, on *AX* such that *AX1 = X1X2=X2X3=X3X4=X4X5.* - 3. Join *BX5*. - 4. Through the point *X3* (*m = 3*), draw a line parallel to *X5B* (by making an angle equal to *D AX5B*) at *X3* intersecting *AB* at the point *C*. - Method - 2 - 1. Draw any ray *AX* making an acute angle with *AB*. - 2. Draw a ray *BY* parallel to *AX* by making ∠*ABY* equal to / *BAX*. - 3. Locate the points *A1, A2, A3,* (*m = 3*) on *AX* and *B1, B2,* (*n = 2*) on *BY* such that *AA1,=A1A2 = A2A3= BB1 = B1 B2* - 4. Join *A3 B2*. Let it intersect *AB* at a point *C*. - To construct a tangent to a Circle - 1. Join *PO* and bisect it. Let *M* be the midpoint of *PO*. - 2. Taking *M* as centre and *MO* as radius, draw a circle. Let it intersect the given circle at the points *Q* and *R*. - 3. Join *PQ* and *PR*. - *PQ* and *PR* are the required tangents. ### Perimeter and Area of a Circle - Circumference *C = 2πr* - Area *A = πr²* - Diameter *d = 2r* - Radius *r = d / 2* ### AREA RELATED TO CIRCLE - Length of Arc - Thus, Length of the arc of segment *OAPB* is *θ / 360° x 2πr* - Area of Segment - Area of segment *APB* = Area of sector *OAPB* - Area of *ΔOAB* - Area of sector of angle *θ = θ / 360° x πr²* - ∴ Area segment *APB* = *θ / 360° x πr² - area of ΔOAB* | Name | Figure | Perimeter | Area | |---|---|---|---| | Circle | | 2*π*r or *π*d | *π*r² | | Semi-circle | | *π*r + 2r | *π*r²/2 | | Ring | | *2π(r+R)* | π(R² – r²) | | Sector of a circle | | *θ/360° x 2πr* | *πr²/360° × θ* | | Area of segment of a circle | | *θ/360° x 2πr* | *πr²/360° × θ* | ### Cuboid - Length = *l* units - Breadth = *b* units - Height = *h* units - T. S. A. = *6a²* sq. units - C. S. A. = *4a²* sq. units - Volume = *a³* cubic units - T. S. A. = *2(lb + bh + lh)* sq. units - C. S. A. = *2h × (l + b)* sq. units - Volume = *lxbxh* cubic units - Longest Diagonal = *a√3* - Sum of lengths of all edges = *12a* - Longest Diagonal = *√l² + b² + h²* - Sum of lengths of all edges = *4(l + b + h)* ### Cube - Length = *a* units - Breadth = *a* units - Height = *a* units - T. S. A. = *2nrh + 2πr² = 2πr(h + r)* sq. units - C. S. A. = *2πrh* sq. units - Volume = *πr²h = πr²h* cubic units ### Hollow Cylinder - Outer Radius = *R* - Inner Radius = *r* - Thickness = *(R-r)* units - Height = *h* - C. S. A = *(External C. S. A.) + (Internal C. S. A.) = (2πRh + 2πrh) sq. units* - T. S. A = *(C. S. A.) + (Area of 2 base rings) = (2mRh + 2nrh) + (2πR²- 2πr²) = 2π(Rh + rh + R²- r²) sq. units* - Volume = *π²h – πr²h = π(R² – r²)h* cubic units ### Right Circular Cylinder - Radius: *r* units - Height : *h* units ### Right Circular Cone - Radius: *r* units - Height: *h* units - Slant Height: *l* units - T. S. A. = *πrl + πr² = πr(l + r)* sq. units - C. S. A. = *πrl* sq. units - Volume = *1/3 πr²h* cubic units - Slant Height = *√r² + h²* units ### Hemisphere - Total Surface Area= *2πr² + πr² = 3πr² sq. units* - Curved Surface Area = *2πr² + πr²* sq. units - Volume = *2/3 πr³* cubic units - (Half of that of a sphere) ### Hollow Hemisphere - Thickness = *(R-r)* units - Curved Surface Area= *(External C. S. A.) + (Internal C. S. A.) = (2nR2 + 2πr²)sq. units* - Total Surface Area = *(C. S. A.) + (Area of upper ring) = (2πR² + 2πr²) + (πR² – πι²) = 3nR2 + πr² sq. units* - Volume = *R3 - πr3 = 2/3π(R3 – r3)* cubic units ### Sphere - Radius (Sphere) =