Geometry: Surface Areas and Volumes

Surface Areas and Volumes

• Surface Area of a Cube: 6a^2, where a is the side length
• Surface Area of a Cuboid: 2(lb + bh + lh), where l, b, and h are the length, breadth, and height respectively
• Surface Area of a Sphere: 4πr^2, where r is the radius
• Volume of a Cube: a^3, where a is the side length
• Volume of a Cuboid: lbh, where l, b, and h are the length, breadth, and height respectively
• Volume of a Sphere: (4/3)πr^3, where r is the radius
• Volume of a Cylinder: πr^2h, where r is the radius and h is the height
• Volume of a Cone: (1/3)πr^2h, where r is the radius and h is the height

Circles

• Circumference of a Circle: 2πr, where r is the radius
• Area of a Circle: πr^2, where r is the radius
• Sector of a Circle: A sector is a region bounded by an arc and two radii
• Segment of a Circle: A segment is a region bounded by an arc and a chord
• Circle Theorems:
  • The angle at the center of a circle is twice the angle at the circumference
  • The angle in a semicircle is a right angle
  • Angles in the same segment are equal
  • The angle of a cyclic quadrilateral is supplementary to the angle at the center of the circle

Quadratic Equations

• Quadratic Equation: A polynomial equation of degree two, in the form ax^2 + bx + c = 0
• Roots of a Quadratic Equation: The values of x that satisfy the equation
• Factorization Method: Factorizing the quadratic equation into the product of two binomials
• Quadratic Formula: x = (-b ± √(b^2 - 4ac)) / 2a
• Graph of a Quadratic Equation: A parabola that opens upwards or downwards

Trigonometry

• Trigonometric Ratios:
  • sine (sin): opposite side / hypotenuse
  • cosine (cos): adjacent side / hypotenuse
  • tangent (tan): opposite side / adjacent side
• Trigonometric Identities:
  • sin^2A + cos^2A = 1
  • tanA = sinA / cosA
• Trigonometric Values:
  • sin(30°) = 1/2
  • sin(45°) = 1/√2
  • sin(60°) = √3/2
• Graphs of Trigonometric Functions:
  • sine wave: periodic and oscillatory
  • cosine wave: periodic and oscillatory
  • tangent wave: periodic and oscillatory

Surface Areas and Volumes

• Cube: Surface area = 6a^2, volume = a^3, where a is the side length
• Cuboid: Surface area = 2(lb + bh + lh), volume = lbh, where l, b, and h are the length, breadth, and height respectively
• Sphere: Surface area = 4πr^2, volume = (4/3)πr^3, where r is the radius
• Cylinder: Volume = πr^2h, where r is the radius and h is the height
• Cone: Volume = (1/3)πr^2h, where r is the radius and h is the height

Circles

• Circumference: 2πr, where r is the radius
• Area: πr^2, where r is the radius
• Sector: Region bounded by an arc and two radii
• Segment: Region bounded by an arc and a chord
• Circle Theorems:
  • Angle at the center of a circle is twice the angle at the circumference
  • Angle in a semicircle is a right angle
  • Angles in the same segment are equal
  • Angle of a cyclic quadrilateral is supplementary to the angle at the center of the circle

Quadratic Equations

• Definition: Polynomial equation of degree two, in the form ax^2 + bx + c = 0
• Roots: Values of x that satisfy the equation
• Factorization Method: Factorizing the quadratic equation into the product of two binomials
• Quadratic Formula: x = (-b ± √(b^2 - 4ac)) / 2a
• Graph: Parabola that opens upwards or downwards

Trigonometry

• Trigonometric Ratios:
  • sine (sin) = opposite side / hypotenuse
  • cosine (cos) = adjacent side / hypotenuse
  • tangent (tan) = opposite side / adjacent side
• Trigonometric Identities:
  • sin^2A + cos^2A = 1
  • tanA = sinA / cosA
• Trigonometric Values:
  • sin(30°) = 1/2
  • sin(45°) = 1/√2
  • sin(60°) = √3/2
• Graphs of Trigonometric Functions:
  • sine wave: periodic and oscillatory
  • cosine wave: periodic and oscillatory
  • tangent wave: periodic and oscillatory

This quiz covers the formulas for surface areas and volumes of various three-dimensional shapes including cubes, cuboids, and spheres.