Surface Area Calculator – Find Surface Area of Any 3D Shape

Cube

Six identical square faces. Each face has area s², so total is 6s². A cube with side 5 has surface area 6 × 25 = 150 square units.

Cuboid (Rectangular Prism)

Six rectangular faces – three pairs of opposite faces. Add the areas: front/back (lh), left/right (wh), top/bottom (lw), then double. A 10×6×4 box has SA = 2(60 + 40 + 24) = 248 square units.

Sphere

Perfectly round with no edges. Formula is 4πr² – exactly four times the area of a circle with the same radius. A sphere with radius 5 has surface area about 314 square units.

Cylinder

Two circular bases plus the curved side. The side "unrolls" into a rectangle with height h and width 2πr (the circumference). Total: 2πr² + 2πrh.

Cone

Circular base plus the curved lateral surface. The lateral area uses slant height (s), found via Pythagorean theorem: s = √(r² + h²). Total: πr² + πrs.

Square Pyramid

Square base plus four triangular faces. Each triangle has base b and slant height s. Base area is b², lateral area is 2bs (four triangles, each with area ½bs).

Hemisphere

Half a sphere plus the circular base. Curved surface is 2πr² (half of 4πr²), base is πr². Total: 3πr². Think of a dome or a bowl.

Triangular Prism

Two triangular bases plus three rectangular sides. Find triangle area, double it, then add the perimeter times length. Works for any triangular cross-section.

Example 1: Gift box wrapping

Problem: Find the surface area of a cuboid box 12" × 8" × 4" to determine wrapping paper needed.

Solution: SA = 2(lw + lh + wh) = 2(12×8 + 12×4 + 8×4)

SA = 2(96 + 48 + 32) = 2(176) = 352 square inches

Add 10% for overlap: 352 × 1.1 ≈ 387 sq in of wrapping paper needed.

Example 2: Paint for a water tank

Problem: A spherical water tank has radius 1.5 meters. How much paint is needed if 1 liter covers 10 m²?

Solution: SA = 4πr² = 4 × π × 1.5² = 4π × 2.25 ≈ 28.27 m²

Paint needed: 28.27 / 10 ≈ 2.83 liters. Buy 3 liters for one coat.

Example 3: Label for a can

Problem: A cylindrical can has radius 4 cm and height 12 cm. What's the label area (lateral surface only)?

Solution: Lateral area = 2πrh = 2 × π × 4 × 12

Lateral area = 96π ≈ 301.6 cm². The label should be about 302 square centimeters.

Example 4: Tent fabric

Problem: A square pyramid tent has base 6 ft and slant height 5 ft. How much fabric for the sides (no floor)?

Solution: Lateral area = 2bs = 2 × 6 × 5 = 60 square feet

You need 60 sq ft of fabric for the four triangular sides.

Example 5: Ice cream cone

Problem: An ice cream cone has radius 3 cm and height 10 cm. Find the lateral surface area (the cone part you hold).

Solution: First find slant height: s = √(r² + h²) = √(9 + 100) = √109 ≈ 10.44 cm

Lateral area = πrs = π × 3 × 10.44 ≈ 98.4 cm²

The cone's outer surface is about 98 square centimeters.