The Composite Figure Surface Area Calculator calculates the outside surface area of a solid made from two or more simple shapes, such as a cylinder topped with a cone or hemisphere.
Composite Figure Surface Area Calculator
Surface area of common composite solids — capsule, silo, rocket, ice-cream cone and house shape — without counting the hidden joined faces.
It adds the exposed surface of each part and leaves out the faces where the parts are joined, because those faces are hidden inside the solid. Choose a capsule, silo, rocket (cylinder + cone), ice-cream cone or house shape; the calculator shows the volume too.
Formula
| Composite solid | Surface area |
|---|---|
| Capsule (cylinder + 2 hemispheres) | A = 2πrh + 4πr² |
| Silo (cylinder + hemisphere top, with floor) | A = 2πrh + 2πr² + πr² |
| Rocket (cylinder + cone, with base) | A = 2πrh + πr² + πr√(r² + k²) |
| Ice-cream cone (cone + hemisphere) | A = πr√(r² + h²) + 2πr² |
| House (box + gable roof, with floor) | A = lw + 2h(l + w) + 2l√((w/2)² + r²) + w·r |
Where:
- r = radius (house: roof height)
- h = cylinder height or cone height
- k = cone height on top of the cylinder
- l, w = house length and width
How the Calculation Works
Step 1: Split the composite solid into simple parts (cylinder, cone, hemisphere, box, prism).
Step 2: Find the surface area of each part, but only the faces you can see from outside.
Step 3: Leave out the joined faces. For example, where a cone sits on a cylinder, neither the cone’s base nor the cylinder’s top is counted. Add the remaining areas.
Variables Explained
| Variable | Meaning | Unit |
|---|---|---|
| r | Radius shared by the joined parts | cm, m, in, ft |
| h | Height or length of the cylinder | cm, m, in, ft |
| k | Height of the cone on a rocket | cm, m, in, ft |
| A | Exposed surface area | cm², m², in², ft² |
Example Calculation
Capsule
Given: radius = 3 cm, cylinder length = 10 cm
Calculation: A = 2 × π × 3 × 10 + 4 × π × 3² = 188.50 + 113.10
Result: A ≈ 301.59 cm²
Cylinder with a cone on top
Given: radius = 3 cm, cylinder height = 10 cm, cone height = 4 cm
Cone slant height = √(3² + 4²) = 5 cm
Calculation: A = 188.50 (side) + 28.27 (base) + π × 3 × 5 (cone) = 188.50 + 28.27 + 47.12
Result: A ≈ 263.89 cm²
House shape
Given: 10 × 6 cm base, walls 3 cm high, roof 2 cm high
Calculation: floor 60 + walls 96 + roof 2 × 10 × √(3² + 2²) ≈ 72.11 + gable ends 12
Result: A ≈ 240.11 cm²
Common Area Conversion Table
| Unit | Equivalent |
|---|---|
| 1 m² | 10,000 cm² |
| 1 ft² | 144 in² |
| 1 m² | 10.7639 ft² |
Shape Properties
Why joined faces are not counted
Surface area measures only what is on the outside. When two solids are glued together, the touching faces disappear inside the shape, so adding the full surface area of each part would overcount.
Dimensions Explained
Shared radius
These composite shapes assume the parts share the same radius, so there is no exposed ring where they meet. If the radii differ, calculate each part separately and add the exposed ring (πR² − πr²).
Other combinations
For shapes not listed here, use the individual calculators (cylinder, cone, sphere, prism) and add only the exposed faces.
Frequently Asked Questions
How do you find the surface area of a composite figure?
Break it into simple solids, find the exposed surface of each, and add them. Do not count faces that are joined together.
Do you subtract the overlapping area in composite figures?
Yes. The hidden faces where parts touch are left out of the total, because they are not on the outside.
How do you find the surface area of a capsule?
A = 2πrh + 4πr²: the curved side of the cylinder plus a complete sphere made by the two hemispheres.
What is the surface area of a cylinder with a cone on top?
A = 2πrh + πr² + πrs: cylinder side, bottom base and cone side, with s the cone slant height.
How do you find the volume of a composite solid?
Add the volumes of the parts. Unlike surface area, nothing is removed for joined faces.