Geometry · April 13, 2026
Surface Area, Lateral Area, and Volume Formulas
Surface area and volume formulas apply to every three-dimensional geometrical shape. The surface area of an object is the total region covered by the outer surface of that object, measured in square units. Volume is the amount of space an object occupies, measured in cubic units. This guide covers surface area definitions, Total Surface Area (TSA), Curved Surface Area (CSA), lateral area, and volume formulas for 7 common 3D shapes — cube, cuboid, cylinder, cone, sphere, hemisphere, and pyramid — with interactive diagrams, solved examples, and practice questions.
Surface Area Definition
What is Surface Area?
The area occupied by the outer surface of a three-dimensional object is the surface area of that object. Surface area is measured in square units such as square centimeters (cm²), square inches (in²), square meters (m²), or square feet (ft²).
The space occupied by a two-dimensional flat surface is called area. In geometry, there are different shapes and sizes such as sphere, cube, cuboid, cone, and cylinder — each shape has its own surface area formulas. Two-dimensional (2D) figures such as square, circle, rectangle, and triangle have only area, not surface area, because 2D shapes lack a third dimension.
Surface area has 2 types:
- (i) Total Surface Area (TSA)
- (ii) Curved Surface Area (CSA) / Lateral Surface Area (LSA)
2D shapes have area only. 3D shapes have both surface area and volume.
2D — Square
Area = s²
No volume
3D — Cube
TSA = 6a²
Volume = a³
Total Surface Area
Total Surface Area (TSA) includes the area of every surface of a 3D object — the curved or lateral surface plus the base(s). TSA is the total region covered by all faces of the object. For shapes with a curved surface and flat base(s), TSA equals the curved surface area added to the area of the base(s).
TSA = Curved Surface Area + Area of Base(s)
Curved Surface Area / Lateral Surface Area
Curved Surface Area (CSA) refers to the area of only the curved part of the shape, excluding the base(s). CSA is also referred to as Lateral Surface Area (LSA) for shapes such as cube, cuboid, prism, and pyramid. For shapes like cylinder, cone, and hemisphere, the term Curved Surface Area (CSA) is standard.
Click to toggle between TSA and CSA on a cylinder:
What is Volume?
Volume is the amount of space, measured in cubic units, that a three-dimensional object occupies. Volume is measured in cubic centimeters (cm³), cubic inches (in³), cubic meters (m³), cubic feet (ft³), liters (L), or gallons (gal).
Two-dimensional shapes do not have volume. For example, the Volume of the Circle cannot be found because a circle is a 2D figure. The volume of a sphere, which is a 3D shape, is (4/3)πr³ cubic units.
To calculate the volume of a cylinder, multiply the area of the circular base (πr²) by the height (h), resulting in πr²h cubic units.
Volume = the space inside a 3D shape. Adjust the slider to fill the cylinder:
V = πr²h
Surface Area and Volume Formulas
The table below lists the perimeter, Total Surface Area (TSA), Curved Surface Area (CSA) / Lateral Surface Area (LSA), and volume formulas for 13 common 2D and 3D geometric figures.
2D Shape Area Formulas (Table 6.5.1)
| Shape | Perimeter | Area Formula |
|---|---|---|
| Square | 4b | b² |
| Rectangle | 2(w + h) | w × h |
| Parallelogram | 2(a + b) | b × h |
| Trapezoid | a + b + c + d | ½(a + b) × h |
| Circle | 2πr | πr² |
| Ellipse | 2π√((a² + b²)/2) | π × a × b |
| Triangle | a + b + c | ½ × b × h |
3D Shape Surface Area and Volume Formulas (Table 6.5.2 & 6.5.3)
| Shape | Total Surface Area (TSA) | CSA / LSA | Volume |
|---|---|---|---|
| Cube | 6a² | 4a² | a³ |
| Cuboid | 2(lb + bh + hl) | 2h(l + b) | l × b × h |
| Cylinder | 2πr(r + h) | 2πrh | πr²h |
| Cone | πr(r + l) | πrl | (1/3)πr²h |
| Sphere | 4πr² | 4πr² | (4/3)πr³ |
| Hemisphere | 3πr² | 2πr² | (2/3)πr³ |
| Pyramid | B + ½ × P × s | ½ × P × s | (1/3)Bh |
Where: a = side, l = length, b = breadth, h = height, r = radius, l (cone) = slant height, B = base area, P = base perimeter, s = slant height.
Click a shape to view its formulas:
Cube
Cuboid
Cylinder
Cone
Sphere
Hemisphere
Pyramid
Select a shape above to see its surface area and volume formulas.
Surface Area of Cube and Volume of a Cube
A cube has 6 equal square faces with side length a.
- Lateral Surface Area (LSA) of cube = 4a² square units
- Total Surface Area (TSA) of cube = 6a² square units
- Volume of a cube = a³ cubic units
Surface Area of Cuboid and Volume of Cuboid
A cuboid has 6 rectangular faces with length (l), breadth (b), and height (h).
- Lateral Surface Area (LSA) of cuboid = 2h(l + b) square units
- Total Surface Area (TSA) of cuboid = 2(lb + bh + hl) square units
- Volume of cuboid = l × b × h cubic units
Surface Area of a Cylinder and Volume of a Cylinder
A cylinder has 2 circular bases with radius (r) and height (h). The volume of a cylinder depends on the base radius and height.
- Curved Surface Area (CSA) of cylinder = 2πrh square units
- Total Surface Area (TSA) of cylinder = 2πr(r + h) square units
- Volume of a cylinder = πr²h cubic units
Cylinder: CSA = 2πrh, TSA = 2πr(r+h), Volume = πr²h
Surface Area of a Cone and Volume of a Cone
A cone has 1 circular base with radius (r), height (h), and slant height (l), where l = √(r² + h²).
- Curved Surface Area (CSA) of cone = πrl square units
- Total Surface Area (TSA) of cone = πr(r + l) square units
- Volume of a cone = (1/3)πr²h cubic units
Cone: CSA = πrl, TSA = πr(r+l), Volume = (1/3)πr²h
Surface Area of a Sphere and Volume of Sphere
A sphere is a perfectly round 3D shape where every point on the surface is the same distance (radius r) from the center. A sphere has no flat face, no edge, and no vertex.
- Surface Area of sphere = 4πr² square units
- Volume of sphere = (4/3)πr³ cubic units
Sphere: Surface Area = 4πr², Volume = (4/3)πr³
Surface Area and Volume of Hemisphere
A hemisphere is half of a sphere, cut by a plane passing through the center. A hemisphere has 1 flat circular face and 1 curved face.
- Curved Surface Area (CSA) of hemisphere = 2πr² square units
- Total Surface Area (TSA) of hemisphere = 3πr² square units
- Volume of hemisphere = (2/3)πr³ cubic units
Calculate Cylinder Volume Instantly
Use the free online volume of a cylinder calculator to get instant results for cylinder volume, CSA, and TSA.
🧮 Open Cylinder Volume CalculatorRelated Calculators:
Solved Examples on Surface Areas and Volumes
Example 1: Surface Area of a Cuboid
Find the surface area of a cuboid with length = 4.4 cm (1.73 in), width = 2.3 cm (0.91 in), and height = 5 cm (1.97 in).
Solution:
Given: l = 4.4 cm, w = 2.3 cm, h = 5 cm
Surface area of cuboid = 2(wl + hl + hw)
= 2 × (2.3 × 4.4 + 5 × 4.4 + 5 × 2.3)
= 2 × (10.12 + 22 + 11.5)
= 2 × 43.62
= 87.24 cm² (13.52 in²)
Example 2: Volume of a Cylinder
Find the volume of a cylinder with base radius = 2.1 cm (0.83 in) and height = 30 cm (11.81 in).
Solution:
Given: r = 2.1 cm, h = 30 cm
Volume of cylinder = πr²h
= π × (2.1)² × 30
= π × 4.41 × 30
= π × 132.3
= 415.69 cm³ (25.37 in³)
Use the volume of a cylinder calculator to verify this result instantly.
Example 3: Surface Area of a Cone
Find the surface area of a cone with slant height = 8 cm (3.15 in) and radius = 3 cm (1.18 in).
Solution:
Given: r = 3 cm, s (slant height) = 8 cm
TSA = B + πrs = πr² + πrs
= π(3²) + π(3)(8)
= 9π + 24π
= 33π
= 103.67 cm² (16.07 in²)
Example 4: Volume of a Sphere
Find the volume of a sphere with diameter = 6 meters (19.69 ft).
Solution:
Given: d = 6 m, so r = 3 m
Volume = (4/3)πr³
= (4/3) × π × (3)³
= (4/3) × π × 27
= 36π
= 113.10 m³ (3,993.49 ft³)
Example 5: Surface Area of a Rectangular Pyramid
Find the surface area of a rectangular pyramid with slant height = 10 yards (9.14 m), base width = 8 yards (7.32 m), and base length = 12 yards (10.97 m).
Solution:
Given: s = 10 yd, b = 8 yd, h = 12 yd
TSA = B + ½sP = (b × h) + ½s(2b + 2h)
= (8 × 12) + ½(10)(16 + 24)
= 96 + 5(40)
= 96 + 200
= 296 yards² (247.57 m²)
Practice Questions on Surface Areas and Volumes
- Find the volume of a cube with side length = 5 cm (1.97 in).
- Find the Curved Surface Area (CSA) of a hemisphere with radius = 7 cm (2.76 in).
- Find the surface area of a sphere with radius = 4 cm (1.57 in). Use π = 3.14.
- Find the volume of a cylinder with radius = 3 cm (1.18 in) and height = 10 cm (3.94 in).
- Find the Total Surface Area (TSA) of a cone with radius = 5 cm (1.97 in) and slant height = 13 cm (5.12 in).
- Find the volume of a cuboid with l = 8 cm (3.15 in), b = 6 cm (2.36 in), h = 4 cm (1.57 in).
Show Answers
- Volume of cube = 5³ = 125 cm³ (7.63 in³)
- CSA of hemisphere = 2πr² = 2 × 3.14 × 49 = 307.72 cm² (47.70 in²)
- Surface area of sphere = 4πr² = 4 × 3.14 × 16 = 200.96 cm² (31.15 in²)
- Volume of cylinder = πr²h = 3.14 × 9 × 10 = 282.60 cm³ (17.24 in³)
- TSA of cone = πr(r + l) = 3.14 × 5 × (5 + 13) = 282.60 cm² (43.80 in²)
- Volume of cuboid = 8 × 6 × 4 = 192 cm³ (11.72 in³)
Frequently Asked Questions on Surface Area and Volume
What are the formulas for surface area and volume of a cuboid?
Surface area of a cuboid = 2(lb + bh + hl) square units. Volume of a cuboid = l × b × h cubic units, where l = length, b = breadth, and h = height.
What is the total surface area of a cylinder?
The Total Surface Area (TSA) of a cylinder = 2πr(r + h) square units, where r is the radius of the circular base and h is the height of the cylinder.
How to calculate the volume of a cone-shaped object?
The volume of a cone = (1/3)πr²h cubic units, where r is the radius of the circular base and h is the perpendicular height of the cone.
What is the difference between Total Surface Area and Curved Surface Area?
Total Surface Area (TSA) includes the area of all surfaces of a 3D shape — the curved surface plus the base(s). Curved Surface Area (CSA), also called Lateral Surface Area (LSA), includes only the area of the curved or lateral part, excluding the base(s).
What is the total surface area of a hemisphere?
The Total Surface Area (TSA) of a hemisphere = 3πr² square units. TSA of a hemisphere equals the sum of the curved surface area (2πr²) and the circular base area (πr²).
What is the volume of a sphere?
The volume of a sphere = (4/3)πr³ cubic units, where r is the radius of the sphere.
What are the formulas for surface area and volume of a cube?
Total Surface Area (TSA) of a cube = 6a² square units. Lateral Surface Area (LSA) of a cube = 4a² square units. Volume of a cube = a³ cubic units, where a is the side length.
How to find volume from surface area?
Volume cannot be directly calculated from surface area alone without knowing at least one dimension. For a cube, surface area = 6a², so a = √(SA/6), and volume = a³. For other shapes, at least one measurement (radius, height, or side) is needed along with the surface area to find the volume.
What are the surface area and volume formulas for a cone?
Curved Surface Area (CSA) of a cone = πrl square units, where l is the slant height. Total Surface Area (TSA) = πr(r + l) square units. Volume = (1/3)πr²h cubic units.
Do 2D shapes have volume?
No, two-dimensional (2D) shapes such as squares, circles, rectangles, and triangles do not have volume. Volume exists only in three-dimensional (3D) shapes. 2D shapes have area measured in square units, while 3D shapes have both surface area and volume.
Summary
Surface area and volume formulas are the foundation of mensuration in geometry. Total Surface Area (TSA) covers every face of a 3D object — curved surfaces plus base(s). Curved Surface Area (CSA), or Lateral Surface Area (LSA), covers only the non-base surfaces. Volume measures the 3D space inside the object. The 7 fundamental 3D shapes — cube, cuboid, cylinder, cone, sphere, hemisphere, and pyramid — each have specific TSA, CSA, and volume formulas. Use the volume of a cylinder calculator to compute cylinder volume, CSA, and TSA instantly.