Chapter 13 - Surface Areas and Volumes - Surfaces, TSA/LSA, Cylinder/Cone/Sphere, Combined Figures

Sep 18, 2026

Uploaded by Noah

Chapter 13 explains how to compute surface area and volume for common 3D solids using clear definitions and derived formulas. It covers cuboids (l, b, h), including total and lateral surface areas and the space diagonal, then cube relations. It derives cylinder surface area (curved and total) and volume, followed by right circular hollow cylinders. It presents cone curved and total surface area, sphere and hemisphere surface area and volume, then combines shapes to solve composite solids. It also includes frustum of a cone surface area and volume formulas.

LanguageEnglish

CategoryExam

FormatPDF

Pages31

Reading Time1 h 18 min

AccessFree to download

Chapter 13: Surface Areas and Volumes

Surface area and volume definitions

Cuboid and cube formulas

Cylinder formulas

Right circular hollow cylinder formulas

Cone, sphere, and hemisphere formulas

Surface area and volume of combined figures

Frustum of a cone formulas

Brain Builders Coaching Centre Lucknow Prepared By:Sachin Sir,9634156611 MATHEMATICS Chapter 13: Surface Areas and Volumes Surface Areas and Volumes 1. Surface area of a solid is the sum of the areas of all its faces. 2. The space occupied by a solid object is the volume of that object. 3. If l, b, h denote respectively the length, breadth and height of a cuboid, then: Lateral surface area or Area of four walls = 2(ℓ + b) h Total surface area = 2(ℓb + bh + hℓ) Volume = ℓ × b × h Diagonal of a cuboid = Surface Area and Volume of Cuboid A cuboid is the region covered by its six rectangular faces. The surface area of a cuboid is equal to the sum of the areas of its six rectangular faces. Surface area of the cuboid Consider a cuboid whose dimensions are l × b × h, respectively. Cuboid with length l, breadth b and height h The total surface area of the cuboid (TSA) = Sum of the areas of all its six faces TSA (cuboid) = 2(l × b) + 2(b × h) + 2(l × h) = 2(lb + bh + lh) Lateral surface area (LSA) is the area of all the sides apart from the top and bottom faces. The lateral surface area of the cuboid = Area of face AEHD + Area of face BFGC + Area of face ABFE + Area of face DHGC LSA (cuboid) = 2(b × h) + 2(l × h) = 2h(l + b) Length of diagonal of a cuboid =√(l 2 + b2 + h2 ) 4. If the length of each edge of a cube is 'a' units, then: Lateral surface area = 4 × (edge)2 Total surface area = 6 × (edge)2 Volume = (edge)3 Diagonal of a cube =√3 × edge Surface Area and Volume of Cylinder A cylinder is a solid shape that has two circular bases, connected with each other, through a lateral surface. Thus, there are three faces, two circular and one lateral, of a cylinder. Based on these dimensions, we can find the surface area and volume of a cylinder. Surface Area of Cylinder Take a cylinder of base radius r and height h units. The curved surface of this cylinder, if opened along the diameter (d = 2r) of the circular base can be transformed into a rectangle of length 2πr and height h units. Thus, Transformation of a Cylinder into a rectangle. CSA of a cylinder of base radius r and height h = 2π × r × h TSA of a cylinder of base radius r and height h = 2π × r × h + area of two circular bases =2π × r × h + 2πr 2 =2πr (h + r) Volume of a Cylinder Volume of a cylinder = Base area × height = (πr2 ) × h = πr2 h Cylinder with height h and base radius r 5. If r and h respectively denote the radius of the base and the height of a right circular cylinder, then: Area of each end or Base area = 􀟨􀝎 2 Area of curved surface or lateral surface area = perimeter of the base x height = 2􀟨􀝎ℎ Total surface area (including both ends) = 2􀟨􀝎ℎ + 􀟨􀝎 2 + 2􀟨􀝎 (ℎ + 􀝎) Volume = Area of the base × height = 􀟨􀝎2 ℎ 6. If R and r respectively denote the external and internal radii of a right circular hollow cylinder and h denotes its height, then: Area of each end = 􀟨􀜴 2 − 􀟨􀝎 2 Area of curved surface = 2􀟨 (􀜴 + 􀝎)ℎ Total surface area = (Area of curved surface) + 2(Area of each end)= 2􀟨 (􀜴 + 􀝎)ℎ + (􀟨􀜴2 − 􀟨􀝎2 ) 7. If r, h and l respectively denote the radius, height and slant height of a right circular cone, then: Slant height (ℓ) =√ℎ2 +􀝎2 Area of curved surface = 􀟨rℓ = 􀟨r√ℎ2 +􀝎2 Total surface area = Area of curved surface + Area of base = 􀟨rℓ + 􀟨􀝎 2 = 􀟨r(ℓ + r) 8. If r is the radius of a sphere, then: Surface area = 􀟨􀝎 2 Volume = 43 􀟨􀝎 3 Volume = 13 􀟨􀝎 2 ℎ 9. If r is the radius of a hemisphere, then: Area of curved surface = 2􀟨􀝎2 Total surface Area = Area of curved surface + Area of base = 2􀟨􀝎2 +􀟨􀝎2 = 3􀟨􀝎2 2 Volume = 􀟨􀝎 3 3 Surface Area of Combined Figures Areas of complex figures can be broken down and analysed as simpler known shapes. By finding the areas of these known shapes, we can find out the required area of the unknown figure. Example: 2 cubes each of volume 64 cm 3 are joined end to end. Find the surface area of the resulting cuboid. Length of each cube = 64(1/3

FAQs

What is the difference between surface area and volume for a solid?

Surface area is the sum of the areas of all faces of a solid. Volume is the space occupied by the solid object.

How do you calculate the total surface area and lateral surface area of a cuboid?

For a cuboid with length l, breadth b, height h, TSA = 2(lb + bh + hl) and LSA = 2h(l + b), which excludes the top and bottom faces.

What formulas are used for the surface area and volume of a cylinder?

For radius r and height h, CSA = 2πrh and TSA = 2πrh + 2πr² = 2πr(h + r). Volume is V = πr²h.

Sep 18, 2026

Uploaded by Noah