Surface Area & Volume Calculator Math

Select a shape
Surface Area: -- Volume: --

What is the Surface Area & Volume Calculator?

Quick Answer: An interactive mensuration tool that calculates the surface area and volume of common 3D shapes: cube, cuboid, sphere, hemisphere, cylinder, cone, and frustum of a cone. Just select the shape, enter the required dimensions, and see the results along with a 3D visualization.

Theory of Mensuration

Mensuration deals with the measurement of geometric figures. For each 3D shape, standard formulas apply:
Cube: SA = 6a2, V = a3
Cuboid: SA = 2(lb + bh + hl), V = l × b × h
Sphere: SA = 4πr2, V = (4/3)πr3
Hemisphere: SA = 3πr2 (curved + base), V = (2/3)πr3
Cylinder: SA = 2πr(r + h), V = πr2h
Cone: SA = πr(r + l) (l = √(r2+h2)), V = (1/3)πr2h
Frustum: SA = π(R+r)l + πR2 + πr2, V = (1/3)πh(R2 + r2 + Rr)
All calculations assume consistent units; the result is unit2 for area and unit3 for volume. The canvas draws a rough 3D sketch of the selected shape with its dimensions labelled.

Step-by-Step Examples

Example 1: Cube of side 5 cm

  1. Select "Cube". Enter side a = 5.
  2. SA = 6 × 25 = 150 cm2, V = 125 cm3.

Example 2: Cylinder with r=3 cm, h=10 cm

  1. Choose "Cylinder", enter r=3, h=10.
  2. SA = 2π×3×(3+10) = 2π×3×13 ≈ 245.04 cm2, V = π×9×10 ≈ 282.74 cm3.

Example 3: Frustum with R=5, r=3, h=4

  1. Pick "Frustum", enter R=5, r=3, h=4.
  2. Slant height l = √((5-3)2+42) = √(4+16)=√20≈4.472, SA = π(5+3)×4.472 + π×25 + π×9 ≈ 112.4 + 78.5 + 28.3 = 219.2 cm2, V = (1/3)π×4×(25+9+15) = (1/3)π×4×49 ≈ 205.3 cm3.

Frequently Asked Questions

What units are used?

Enter dimensions as numbers; the results are in square units and cubic units respectively. Any consistent unit (cm, m, inches) can be used.

How is the frustum of a cone different from a truncated cone?

They are the same. A frustum is the portion of a cone between two parallel planes. The calculator handles both radii (R and r) and height h.

Does the hemisphere include the base circle?

Yes, the surface area formula includes the curved surface plus the flat circular base (3πr2).

Can I see the formulas?

The result card shows the computed values. The theory section above lists all formulas. Hover over the canvas to see dimension labels.

Is the 3D drawing accurate?

The canvas draws a simplified 3D representation to help identify the shape and dimensions; it is not to scale.