Geometry and Mensuration
Mensuration: Areas and Volumes formulas
At a glance
Cuboid: lbh, 2(lb + bh + hl), √(l² + b² + h²) Cube: a³, 6a², a√3 Cylinder: πr²h, curved 2πrh, total 2πr(r + h) Cone: ⅓πr²h, curved πrl, l = √(r² + h²) Frustum: ⅓πh(R² + Rr + r²), curved π(R + r)L Sphere: ⁴⁄₃πr³, 4πr² Hemisphere: ⅔πr³, curved 2πr², total 3πr² Prism: base × h Pyramid: ⅓ × base × h, lateral ½ × perimeter × slant Sphere in cube r = a/2 cube in sphere a√3 = 2R Melting: volume before = volume after Scaling: k, k², k³
| Solid | Volume | Curved / lateral surface | Total surface |
|---|---|---|---|
| Cube, side a | a³ | 4a² | 6a² |
| Cuboid l × b × h | lbh | 2h(l + b) | 2(lb + bh + hl) |
| Cylinder | πr²h | 2πrh | 2πr(r + h) |
| Cone, l = √(r² + h²) | ⅓πr²h | πrl | πr(r + l) |
| Sphere | ⁴⁄₃πr³ | 4πr² | 4πr² |
| Hemisphere | ⅔πr³ | 2πr² | 3πr² |
| Frustum, L = √(h² + (R − r)²) | ⅓πh(R² + Rr + r²) | π(R + r)L | π(R + r)L + πR² + πr² |
| Right prism | base area × h | base perimeter × h | lateral + 2 × base |
| Regular pyramid | ⅓ × base area × h | ½ × base perimeter × slant height | lateral + base |
Body diagonals: cuboid √(l² + b² + h²), cube a√3.
Plane figures: area and perimeter
Rectangle: lb, 2(l + b) Square: a², diagonal a√2 Triangle: ½ × base × height Equilateral: (√3/4)a² Heron: √(s(s − a)(s − b)(s − c)), s = (a + b + c)/2 Parallelogram: base × height Rhombus: ½ d₁d₂ Trapezium: ½ (sum of parallel sides) × height Circle: πr², 2πr Sector θ: (θ/360)πr², arc (θ/360)2πr
Shaded regions: add and subtract
Shortcut: Square minus its inscribed circle (or the four corner quarters of radius a/2) = 3/14 of a². The leaf from two quarter circles = 4/7 of a². The four-petal flower is the same 4/7 of a².
Cuboid and cube: unfold the box
Cuboid: V = lbh TSA = 2(lb + bh + hl) four walls = 2h(l + b) Diagonal = √(l² + b² + h²) Cube: V = a³ TSA = 6a² diagonal = a√3
Shortcut: (l + b + h)² = (l² + b² + h²) + 2(lb + bh + hl). So (sum of dimensions)² = diagonal² + total surface area.
Cylinder: peel off the label
Curved surface = 2πrh Total surface = 2πr(r + h) Volume = πr²h (base area × height) Hollow pipe: volume of metal = πh(R² − r²)
Cone and frustum: roll a sector
l = √(r² + h²) Curved surface = πrl Total = πr(r + l) Volume = ⅓πr²h (a third of the cylinder on the same base) Sector of angle θ, radius l → cone with r = l × θ/360 Frustum: V = ⅓πh(R² + Rr + r²), curved = π(R + r)L, L = √(h² + (R − r)²)
Shortcut: If you forget the frustum formula, complete the cone. The cut-off top is a similar cone, so its height and radius are in the same ratio as the big one's.
Sphere and hemisphere
Sphere: surface 4πr² volume ⁴⁄₃πr³ Hemisphere: curved 2πr² total 3πr² volume ⅔πr³ Hollow sphere (shell): ⁴⁄₃π(R³ − r³)
Melting and recasting
Volume before = volume after Number of pieces = big volume ÷ one small volume Sphere → n spheres: n = (R/r)³
Shortcut: Cancel before you multiply. π, ⁴⁄₃ and ⅓ usually appear on both sides of the equation, so strike them out first.
Scaling: k, k², k³
Similar solids, length ratio a : b → area ratio a² : b² → volume ratio a³ : b³
Prisms, pyramids and solids inside solids
Prism: V = base area × h lateral = base perimeter × h total = lateral + 2 × base Pyramid: V = ⅓ × base area × h lateral (regular) = ½ × base perimeter × l total = lateral + base Regular tetrahedron, edge a: V = a³/(6√2), surface = √3 a², height = a√(2/3) Body diagonal: cuboid √(l² + b² + h²), cube a√3
| Fit | What is shared | Result |
|---|---|---|
| largest sphere in a cube of side a | diameter = side | r = a/2; sphere fills π/6 ≈ 52% of the cube |
| cube in a sphere (corners on the sphere) | diameter = body diagonal | 2R = a√3, so a = 2R/√3 |
| sphere touching all 12 edges of a cube | diameter = face diagonal | r = a/√2 |
| largest cylinder or cone in a cube | diameter = side, height = side | r = a/2, h = a |
| sphere in a cylinder (snug) | same radius, h = 2r | volumes 2 : 3; surface = curved surface |
| largest cube in a hemisphere of radius R | top corners on the dome | a² + a²/2 = R², so a = R√(2/3) |
Formulas stick when you use them
The Mensuration: Areas and Volumes lesson has worked problems and quick checks for every formula here. Free to start.
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