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Surface Area — Definition, Formula & Examples

Surface Area

The total area of the exterior surface of a solid. Many formulas for the area of a surface are given below.

For the following tables,
h = height of solid s = slant height P = perimeter or circumference of the base
l = length of solid B = area of the base r = radius of sphere
w = width of solid R = radius of the base a = length of an edge

Figure Volume Lateral Surface Area Area of the Base(s) Total Surface Area
Box (also called rectangular parallelepiped, right rectangular prism) lwh 2lh + 2wh 2lw 2lw + 2lh + 2wh
Prism Bh Ph 2B Ph + 2B
Pyramid Formula: one-third times B times h - B -
Right Pyramid Formula: one-third times B times h Formula: one-half times P times s B B + (1/2)Ps, formula for total surface area of a right regular pyramid, where B is base area, P is perimeter, s is slant height.
Cylinder Bh - 2B -
Right Cylinder Bh Ph 2B Ph + 2B
Right Circular Cylinder πR2h 2πRh 2πR2 2πRh + 2πR2
Cone Formula: one-third times B times h - B -
Right Circular Cone Formula: (1/3)πR²h πRs or Formula: pi times R times the square root of (R squared plus h squared) πR2 πRs + πR2 or Formula: πR√(R² + h²) + πR²

Figure Volume Total Surface Area
Sphere Formula: (4/3)πr³ Formula: 4πr²
Regular Tetrahedron The formula: (√2 / 12) a³ Mathematical expression: a squared times square root of 3
Cube (regular hexahedron) a3 6a2
Regular Octahedron Formula for the volume of a regular octahedron: (√2/3)a³, where a is the edge length. The formula: 2a²√3
Regular Dodecahedron Surface area formula for a regular dodecahedron: (15 + 7√5) / 4 × a³, where a is edge length. Surface area formula: 3a² times the square root of (25 + 10√5), where a is edge length.
Regular Icosahedron Volume formula for a regular icosahedron: V = 5(3 + √5)/12 × a³, where a is edge length 5a²√3

For the tables above,
h = height of solid s = slant height P = perimeter or circumference of the base
l = length of solid B = area of the base r = radius of sphere
w = width of solid R = radius of the base a = length of an edge

 

See also

Lateral surface, right prism, right regular pyramid, frustum of a cone or pyramid, torus, surface area of a surface of revolution

Key Formula

SAbox=2lw+2lh+2whSAcylinder=2πRh+2πR2SAsphere=4πr2SAcone=πRs+πR2\begin{gathered}\text{SA}_{\text{box}} = 2lw + 2lh + 2wh\\\text{SA}_{\text{cylinder}} = 2\pi Rh + 2\pi R^2\\\text{SA}_{\text{sphere}} = 4\pi r^2\\\text{SA}_{\text{cone}} = \pi R s + \pi R^2\end{gathered}
Where:
  • ll = Length of the solid (for a box)
  • ww = Width of the solid (for a box)
  • hh = Height of the solid
  • RR = Radius of the base (for a cylinder or cone)
  • rr = Radius of a sphere
  • ss = Slant height (for a cone or pyramid)
  • π\pi = The constant pi, approximately 3.14159

Worked Example

Problem: Find the total surface area of a rectangular box with length 5 cm, width 3 cm, and height 4 cm.
Step 1: Write the formula for the surface area of a box.
SA=2lw+2lh+2wh\text{SA} = 2lw + 2lh + 2wh
Step 2: Substitute the given values: l = 5, w = 3, h = 4.
SA=2(5)(3)+2(5)(4)+2(3)(4)\text{SA} = 2(5)(3) + 2(5)(4) + 2(3)(4)
Step 3: Calculate each pair of opposite faces separately.
SA=30+40+24\text{SA} = 30 + 40 + 24
Step 4: Add the three terms to get the total surface area.
SA=94 cm2\text{SA} = 94 \text{ cm}^2
Answer: The total surface area of the box is 94 cm².

Another Example

This example involves a curved surface (the lateral side of a cylinder) rather than only flat faces, showing how π enters the formula. It also demonstrates leaving an answer in exact form (78π) before approximating.

Problem: Find the total surface area of a right circular cylinder with a base radius of 3 cm and a height of 10 cm. Use π ≈ 3.14.
Step 1: Write the formula for the total surface area of a right circular cylinder. It includes the lateral (side) surface plus two circular bases.
SA=2πRh+2πR2\text{SA} = 2\pi Rh + 2\pi R^2
Step 2: Substitute R = 3 and h = 10.
SA=2π(3)(10)+2π(3)2\text{SA} = 2\pi(3)(10) + 2\pi(3)^2
Step 3: Simplify each part. The lateral surface area is 60π and the area of both bases is 18π.
SA=60π+18π=78π\text{SA} = 60\pi + 18\pi = 78\pi
Step 4: Approximate the result using π ≈ 3.14.
SA≈78×3.14=244.92 cm2\text{SA} \approx 78 \times 3.14 = 244.92 \text{ cm}^2
Answer: The total surface area of the cylinder is 78π ≈ 244.92 cm².

Frequently Asked Questions

What is the difference between surface area and lateral surface area?
Surface area (total surface area) includes every outer face of a solid — the sides and the bases. Lateral surface area counts only the side faces, excluding the top and bottom bases. For a cylinder, for example, the lateral surface area is 2πRh, while the total surface area adds the two circular bases: 2πRh + 2πR².
What is the difference between surface area and volume?
Surface area measures the total area covering the outside of a 3D shape, expressed in square units (like cm²). Volume measures the space enclosed inside the shape, expressed in cubic units (like cm³). Think of surface area as the amount of wrapping paper needed and volume as how much the box can hold.
How do you find the surface area of a sphere?
Use the formula SA = 4πr², where r is the radius of the sphere. For instance, a sphere with radius 5 cm has surface area 4π(25) = 100π ≈ 314.16 cm². Note that a sphere has no separate base or lateral surface — its entire surface is one curved area.

Surface Area vs. Volume

Surface AreaVolume
What it measuresTotal area of the outer surface of a solidAmount of space enclosed inside a solid
UnitsSquare units (cm², m², ft²)Cubic units (cm³, m³, ft³)
Box formula2lw + 2lh + 2whlwh
Sphere formula4πr²(4/3)πr³
Real-world analogyAmount of paint to cover the outsideAmount of water the shape can hold

Why It Matters

Surface area shows up whenever you need to know how much material covers the outside of an object — painting a room, wrapping a gift, or manufacturing a container. In science classes, surface area affects rates of heat transfer and chemical reactions. Standardized tests and geometry courses regularly ask you to compute and compare surface areas of prisms, cylinders, cones, and spheres.

Common Mistakes

Mistake: Confusing surface area with volume by using the wrong units or the wrong formula.
Correction: Surface area always uses square units (like cm²) because you are measuring area. Volume uses cubic units (like cm³). Double-check that your formula outputs an area, not a space measurement.
Mistake: Forgetting to include the base(s) when total surface area is required.
Correction: Lateral surface area covers only the sides. If a problem asks for total surface area, you must add the area of all bases. For a cylinder, that means adding 2πR² to the lateral area 2πRh.

Related Terms