Calculate the surface area of twelve closed solids, including prisms, frustums, spherical caps and ellipsoids, with lateral and base areas and every step shown.
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Total closed surface area94.0000 square units
units
0.000001 – 1,000,000,000 units
units
0.000001 – 1,000,000,000 units
units
For a cone, pyramid or frustum, not the slanted side. For a capsule, only the straight middle part.
Total closed surface area
94.0000 square units
Four side faces70.0000 square units
Top and bottom (l × w)24.0000 square units
Includes bases where stated. Cone, pyramid and frustum heights are perpendicular, not slant heights. Capsule height is its straight cylindrical section, excluding the rounded ends. A frustum with a top radius of 0 is a cone. A spherical cap uses the sphere radius R and the cap height h; h = 2R is a whole sphere. An ellipsoid has no elementary closed form: the Knud Thomsen formula used here is within about 1.061% of the exact area and exact for a sphere.
How this was calculated
Surface area is the total area of all the outside faces, like the amount of wrapping paper needed to cover the shape. The formula for this shape is A = 2(lw + lh + wh). The steps below work it out one piece at a time.
The four side faces are rectangles. Together their area is the height times the distance around the bottom (l + w + l + w).Lateral = 2h(l + w)Put in your numbers: 2 × 5 × (3 + 4)Answer: Lateral = 70 square units
The top and bottom are two more rectangles, each length × width.Bases = 2lwPut in your numbers: 2 × 3 × 4Answer: Bases = 24 square units
Add the parts together to get the whole outside area.Total = lateral + basePut in your numbers: Total = 70 + 24Answer: Total = 94 square units
Use the same unit for every length. The answer is in that unit squared (for example cm² if you measured in cm).
Surface area in simple words
Surface area is the total area of all the outside faces of a solid shape. Think of it as how much wrapping paper you would need to cover the shape with no gaps and no overlaps.
To find it, work out the area of each face and add them up. A box has 6 rectangle faces: top and bottom, front and back, and two sides.
Example: a box 3 cm long, 4 cm wide and 5 cm tall. Top and bottom: 2 × 3 × 4 = 24 cm². Front and back: 2 × 3 × 5 = 30 cm². Two sides: 2 × 4 × 5 = 40 cm². Total: 24 + 30 + 40 = 94 cm².
Curved shapes use π (about 3.14). A ball with a radius of 2 cm has a surface area of 4 × π × 2 × 2 ≈ 50.27 cm². Area is always in square units, like cm² or m².
Formula and method
Add the areas of all exposed faces. A rectangular box uses 2(lw + lh + wh), a sphere uses 4πr² and a closed cylinder uses 2πr(r + h). The result displays the selected solid’s formula.
Worked example
A box measuring 3 × 4 × 5 units has surface area 2(12 + 20 + 15) = 94 square units.
Lateral area, bases, prisms and frustums
Each result splits the total into lateral (side) area and base area. A 3 × 4 × 5 box has side faces 2 × 5 × (3 + 4) = 70 and top and bottom 2 × 3 × 4 = 24, for 94 square units.
A triangular prism adds two triangular ends (Heron's formula) to three rectangles: a 3-4-5 end and length 10 give 2 × 6 + 12 × 10 = 132. A conical frustum with radii r₁ and r₂ has slant s = √((r₁ − r₂)² + h²), lateral area π(r₁ + r₂)s and bases π(r₁² + r₂²). With r₁ = 4, r₂ = 1 and h = 4, s = 5 and the total is 25π + 17π = 42π ≈ 131.95.
Spherical caps and ellipsoids
A spherical cap is the piece of a sphere of radius R cut off by a plane at height h from its top. Its curved area is 2πRh and its flat base is a circle of radius a = √(h(2R − h)). With R = 5 and h = 2, a² = 2 × 8 = 16, so the total is 20π + 16π = 36π ≈ 113.10 square units. h = R gives a hemisphere (3πR²) and h = 2R a whole sphere.
An ellipsoid with semi-axes a, b and c has no elementary formula for its surface area. The calculator uses Knud Thomsen's approximation A ≈ 4π((aᵖbᵖ + aᵖcᵖ + bᵖcᵖ) ÷ 3)^(1/p) with p = 1.6075, whose relative error is at most about 1.061%. It is exact for a sphere: a = b = c = 2 gives 16π ≈ 50.27. Treat ellipsoid results as close estimates, not exact values.
Frequently asked questions
What is the difference between lateral and total surface area?
Lateral area covers only the sides. Total surface area adds the bases or ends. A closed cylinder of radius 2 and height 5 has lateral area 20π and total area 28π.
Which height does a cone or capsule require?
A cone requires perpendicular height, not slant height. A capsule requires the length of its straight cylindrical section, excluding its hemispherical ends.