Find the surface area of common 3D shapes with formulas, step-by-step calculations, and diagrams.
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Cube
Rectangular Prism
Sphere
Cylinder
Cone
Triangular Prism
Square Pyramid
Ellipsoid
About Surface Area
Surface area is a crucial concept in geometry, engineering, architecture, and everyday life. Whether you're designing a product, wrapping a gift, or studying for a math exam, knowing how to calculate surface area can be extremely useful. But since different shapes have different surface area formulas, the first step is always to select the shape you're working with.
Below, we cover how to calculate the surface area for eight common 3D shapes: Cube, Rectangular Prism, Sphere, Cylinder, Cone, Triangular Prism, Square Pyramid, and Ellipsoid.
1. Cube
A cube has six equal square faces. To calculate the surface area of a cube:
Surface Area = 6a²
Where a is the length of a side.
Example: If a cube has sides of 4 cm,
6 × 4² = 6 × 16 = 96 cm²
2. Rectangular Prism
A rectangular prism (or cuboid) has 6 rectangular faces.
Surface Area = 2(lw + lh + wh)
Where l = length, w = width, h = height.
Example: For a prism with l = 3 cm, w = 4 cm, h = 5 cm:
2(3×4 + 3×5 + 4×5) = 2(12 + 15 + 20) = 94 cm²
3. Sphere
A sphere is a perfectly round 3D object.
Surface Area = 4πr²
Where r is the radius.
Example: For a sphere with r = 7 cm:
4π(7)² = 4π(49) = 196π ≈ 615.75 cm²
4. Cylinder
A cylinder has two circular bases and a curved surface.
Surface Area = 2πr(h + r)
Where r = radius, h = height.
Example: For r = 3 cm, h = 10 cm:
2π × 3(10 + 3) = 6π × 13 = 78π ≈ 245.04 cm²
5. Cone
A cone has a circular base and a pointed top.
Surface Area = πr(r + l)
Where r = radius, l = slant height.
Example: r = 5 cm, l = 10 cm
π × 5(5 + 10) = π × 5 × 15 = 75π ≈ 235.62 cm²
6. Triangular Prism
A triangular prism has two triangular bases and three rectangular sides.
Surface Area = bh + l(a + b + c)
Where b = base of triangle, h = height of triangle, a, b, c are the triangle sides, l is prism length.
This varies based on triangle type (equilateral, right, etc.)
7. Square Pyramid
A square pyramid has a square base and four triangular faces.
Surface Area = a² + 2a·l
Where a = base edge, l = slant height.
Example: a = 6 cm, l = 5 cm
6² + 2 × 6 × 5 = 36 + 60 = 96 cm²
8. Ellipsoid
An ellipsoid is like a stretched sphere with 3 semi-axes: a, b, c.
Surface Area ≈ 4π((apbp + apcp + bpcp)/3)1/p
Where p ≈ 1.6075 and a, b, c are semi-axes.
It's complex, so often a calculator is used.
Final Thoughts
Calculating surface area is all about choosing the correct formula for the shape you're dealing with. Whether it's a simple cube or a complex ellipsoid, understanding the geometric properties and measurements involved makes the task easier.
Disclaimer
This surface area calculator is provided for educational and informational purposes only. While we strive for accuracy, we make no warranties or representations regarding the results. The user assumes all responsibility for the use of these calculations and results. For critical applications, please verify calculations independently.