This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here are the common mensuration formulas for various 2D and 3D shapes: 2D Shapes 1. Square Side length: s Area: A = s^2 Perimeter: P = 4s 2. Rectangle Length: l, Width: w Area: A = l × w Perimeter: P = 2(l + w) 3. Triangle Base: b, Height: h Area: A = (1)/(2) × b × h For a triangle with sides a, b, c (Heron's formula): Semi-perimeter: s = (a+b+c)/(2) Area: A = sqrt(s(s-a)(s-b)(s-c)) For an equilateral triangle with side a: Area: A = sqrt(3)4 a^2 Perimeter: P = 3a 4. Circle Radius: r, Diameter: d = 2r Area: A = r^2 Circumference: C = 2 r = d 5. Trapezium (or Trapezoid) Parallel sides: a, b, Height: h Area: A = (1)/(2)(a+b)h 6. Parallelogram Base: b, Height: h Area: A = b × h Perimeter: P = 2(a+b) (where a and b are adjacent sides) 7. Rhombus Diagonals: d_1, d_2 Area: A = (1)/(2) d_1 d_2 Perimeter: P = 4s (where s is the side length) 8. Sector of a Circle Radius: r, Angle (in degrees): Area: A = ()/(360^) × r^2 Arc Length: L = ()/(360^) × 2 r 9. Segment of a Circle Radius: r, Angle (in degrees): Area of segment = Area of sector - Area of triangle Area: A = ()/(360^) × r^2 - (1)/(2) r^2 () 3D Shapes 1. Cube Side length: s Volume: V = s^3 Total Surface Area (TSA): TSA = 6s^2 Lateral Surface Area (LSA): LSA = 4s^2 2. Cuboid (Rectangular Prism) Length: l, Width: w, Height: h Volume: V = l × w × h Total Surface Area (TSA): TSA = 2(lw + lh + wh) Lateral Surface Area (LSA): LSA = 2h(l+w) 3. Cylinder Radius: r, Height: h Volume: V = r^2 h Curved Surface Area (CSA): CSA = 2 r h Total Surface Area (TSA): TSA = 2 r h + 2 r^2 = 2 r(h+r) 4. Cone Radius: r, Height: h, Slant height: l = sqrt(r^2 + h^2) Volume: V = (1)/(3) r^2 h Curved Surface Area (CSA): CSA = r l Total Surface Area (TSA): TSA = r l + r^2 = r(l+r) 5. Sphere Radius: r Volume: V = (4)/(3) r^3 Surface Area: SA = 4 r^2 6. Hemisphere Radius: r Volume: V = (2)/(3) r^3 Curved Surface Area (CSA): CSA = 2 r^2 Total Surface Area (TSA): TSA = 2 r^2 + r^2 = 3 r^2 7. Pyramid (General) Base Area: A_b, Height: h Volume: V = (1)/(3) A_b h For a square pyramid with base side s and slant height l_s: Base Area: A_b = s^2 Lateral Surface Area (LSA): LSA = 2s l_s Total Surface Area (TSA): TSA = s^2 + 2s l_s 8. Prism (General) Base Area: A_b, Height: h, Perimeter of Base: P_b Volume: V = A_b h Lateral Surface Area (LSA): LSA = P_b h Total Surface Area (TSA): TSA = P_b h + 2A_b Got more? Send 'em!