Surface area calculator with working

• Cuboid: 5 × 4 × 3 cm. Faces of 20, 15 and 12 cm², each twice: $2 \times 47 = 94$ cm².
• Cylinder: radius 3 cm, height 10 cm. Ends $2\pi \times 9 = 18\pi$; curved surface $2\pi \times 3 \times 10 = 60\pi$; total $78\pi \approx 245.0$ cm².
• Cone: radius 3 cm, slant height 5 cm. Base $9\pi$ plus curved surface $15\pi$ = $24\pi \approx 75.4$ cm².
• Sphere: radius 3 cm. $4\pi \times 9 = 36\pi \approx 113.1$ cm².

Unroll the curved surface of a cylinder and it's a rectangle: its width is the height $h$ and its length is the circumference of the end, $2\pi r$. So the curved area is $2\pi r h$. An open-topped cylinder (like a tin can without a lid) has only one circular end: $\pi r^2 + 2\pi r h$.

The cone's curved surface uses the slant height $l$, measured along the side from the tip to the edge of the base. If you only know the perpendicular height $h$, use Pythagoras: $l = \sqrt{r^2 + h^2}$. For radius 3 and height 4, $l = 5$. Our Pythagoras calculator does this step.

Surface area is in square units (cm²); volume is in cubic units (cm³). Doubling every length multiplies surface area by 4 and volume by 8. This is why small animals lose heat faster than large ones: they have more surface area for their volume. For volume, use our volume calculator.

• Counting only three faces of a cuboid (each appears twice).
• Using the perpendicular height in $\pi r l$ instead of the slant height.
• Forgetting the flat face of a solid hemisphere (the total is $3\pi r^2$, not $2\pi r^2$).

Practise with our volume and surface area questions.

• Sketch the net and label each face's area.
• Read whether the shape is open or closed (a box with no lid has five faces).
• Leave answers in terms of $\pi$ when asked for an exact value.
• The curved surface area formulas for a cone and a sphere are usually given; check your exam's formula sheet.