Interactive 3D Solid Explorer: Surface Area and Volume
What is the difference between surface area and volume? Why does the volume formula for a prism involve its cross-sectional area? Which height should you use when calculating the volume of a pyramid or cone?
Use the Interactive 3D Solid Explorer below to investigate these questions visually.
This learning tool is suitable for Primary and Secondary Mathematics students, including G1, G2 and G3 learners. You can rotate each solid, display hidden measurements, change its dimensions and follow the calculation step by step.
How to use the 3D Solid Explorer
Select a 3D solid from the shape menu.
Enter the required measurements.
Select the unit of measurement: mm, cm, m or km.
Drag the model to rotate it.
Scroll with a mouse or pinch the screen to zoom.
Show or hide the measurements, edges and highlighted cross-section.
Select Transparent view to see hidden edges and internal measurements.
Select Cross-sectional area, Surface area or Volume, depending on the chosen solid.
Follow the formula, substitution, calculation and final answer.
Use the labelled measurements rather than estimating dimensions from the model. A Mathematics diagram should never be assumed to be drawn exactly to scale.
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Interactive 3D Solid Explorer
Rotate solids, identify their measurements and calculate surface area and volume.
STEP 1
Set the measurements
STEP 2
Explore the solid
Drag to rotate · Scroll or pinch to zoom
STEP 3
Calculate
Solids included in the explorer
Prisms
Triangular prism
Trapezium prism
Square prism
Rectangular prism
Rhombus prism
Parallelogram prism
Pyramids
Triangular pyramid
Square-based pyramid
Curved solids
Cone
Cylinder
Hemisphere
Sphere
What is cross-sectional area?
A cross-section is the 2D shape produced when a solid is cut straight through.
A prism has the same cross-section throughout its entire length. This is why the volume of any prism can be calculated using:
Volume = cross-sectional area × prism length
V = AL
Here, A represents the cross-sectional area and L represents the prism length.
For example, consider a prism with:
Cross-sectional area = 20 cm²
Prism length = 8 cm
Its volume is:
V = AL
V = 20 × 8
V = 160 cm³
What is surface area?
Surface area is the total area of all the outside surfaces of a 3D solid.
Surface area is measured in square units, such as:
mm²
cm²
m²
km²
To find the total surface area of a solid with flat faces, calculate the area of every exposed face and add the areas together.
For curved solids, the curved surface and any circular bases must be considered.
What is volume?
Volume is the amount of three-dimensional space occupied by a solid.
Volume is measured in cubic units, such as:
mm³
cm³
m³
km³
The small raised 3 in cm³ tells us that the measurement represents a volume.
For example, the volume of a cuboid measuring 6 cm by 4 cm by 3 cm is:
Volume = length × width × height
Volume = 6 × 4 × 3
Volume = 72 cm³
Formula reference for prisms
For any prism:
Volume: V = AL
Total surface area: SA = 2A + PL
Here:
A is the cross-sectional area.
P is the perimeter of the cross-section.
L is the prism length.
2A represents the two matching ends.
PL represents the total area of the rectangular side faces.
Triangular prism
Cross-sectional area:
A = ½bh
Volume:
V = ½bhL
Total surface area:
SA = 2A + PL
The perimeter P is the sum of the three sides of the triangular cross-section.
Trapezium prism
Cross-sectional area:
A = ½(a + b)h
Volume:
V = ½(a + b)hL
Total surface area:
SA = 2A + PL
The measurements a and b are the lengths of the parallel sides of the trapezium.
Square prism
Cross-sectional area:
A = s²
Volume:
V = s²L
Total surface area:
SA = 2s² + 4sL
If the prism length is also equal to s, the solid is a cube.
Rectangular prism
Cross-sectional area:
A = wh
Volume:
V = whL
Total surface area:
SA = 2wh + 2wL + 2hL
A rectangular prism is also commonly called a cuboid.
Rhombus prism
Cross-sectional area:
A = ½d₁d₂
Volume:
V = ½d₁d₂L
Total surface area:
SA = 2A + PL
The symbols d₁ and d₂ represent the diagonals of the rhombus.
Parallelogram prism
Cross-sectional area:
A = bh
Volume:
V = bhL
Total surface area:
SA = 2A + PL
The height h must be perpendicular to the selected base.
Formula reference for pyramids
For any pyramid:
Volume = ⅓ × base area × perpendicular height
V = ⅓AH
The perpendicular height runs from the base to the apex at an angle of 90° to the base.
Triangular pyramid
Volume:
V = ⅓AH
Here, A is the area of the triangular base.
To find its total surface area, calculate and add:
The area of the triangular base
The areas of the three triangular side faces
A general triangular pyramid may have side faces with different areas, so there is not always one short surface-area formula.
Square-based pyramid
Base area:
A = s²
Volume:
V = ⅓s²H
Total surface area:
SA = s² + 2sℓ
Here:
s is the side length of the square base.
H is the perpendicular height.
ℓ is the slant height of a triangular face.
The slant height and perpendicular height are not interchangeable.
Formula reference for curved solids
Cone
Slant height:
ℓ = √(r² + H²)
Curved surface area:
CSA = πrℓ
Total surface area:
SA = πr² + πrℓ
Volume:
V = ⅓πr²H
The perpendicular height H is used for volume. The slant height ℓ is used for surface area.
Cylinder
Area of one circular base:
A = πr²
Curved surface area:
CSA = 2πrH
Total surface area:
SA = 2πr² + 2πrH
Volume:
V = πr²H
The term 2πr² represents the two circular bases.
Hemisphere
Curved surface area:
CSA = 2πr²
Total surface area, including the circular base:
SA = 3πr²
Volume:
V = ⅔πr³
Check whether a question asks for curved surface area or total surface area. The total surface area includes the circular base.
Sphere
Surface area:
SA = 4πr²
Volume:
V = ⁴⁄₃πr³
A sphere has one continuous curved surface and no flat faces, edges or vertices.
Important measurements to recognise
Cross-sectional area
The cross-sectional area is the area of the 2D shape repeated throughout a prism.
Do not confuse cross-sectional area with surface area. Cross-sectional area is used to calculate the volume of a prism.
Prism length
The prism length is the distance between its two matching cross-sectional ends.
It may be drawn horizontally, vertically or diagonally. Its direction on the page does not change its mathematical meaning.
Perpendicular height
The perpendicular height meets the base at an angle of 90°.
The perpendicular height is used when calculating the volume of a pyramid or cone.
Slant height
The slant height is measured along the surface of a pyramid or cone.
It is normally used when calculating surface area.
Slant edge
A slant edge joins the apex of a pyramid to a vertex of its base.
For a square-based pyramid, the slant edge and slant height are different measurements. The slant height meets the midpoint of a base side, while the slant edge reaches a base vertex.
Radius
The radius is the distance from the centre of a circle or sphere to its edge or surface.
Diameter
The diameter passes through the centre and is twice the radius:
d = 2r
Face, edge and vertex
A face is a flat surface of a solid.
An edge is where two faces meet. Curved solids may also have curved edges where a flat face meets a curved surface.
A vertex is a corner where edges meet.
Why use Transparent view?
A solid model can hide measurements located inside or behind it.
Transparent view allows you to see:
Hidden edges
The perpendicular height of a pyramid
The central height of a cone
Radius and diameter lines
The repeated cross-section of a prism
Switch between solid and transparent views to understand both the outside surfaces and the internal measurements.
Common mistakes to avoid
Confusing surface area with volume.
Writing square units for volume.
Writing cubic units for surface area.
Using slant height instead of perpendicular height in a volume formula.
Using perpendicular height instead of slant height in a surface-area formula.
Forgetting the factor of ⅓ for a pyramid or cone.
Forgetting one or both circular bases when calculating total surface area.
Using 2πr² as the total surface area of a hemisphere.
Treating the prism length as part of its cross-section.
Confusing radius with diameter.
Rounding intermediate values too early.
Assuming that a 3D diagram is drawn exactly to scale.
Keep unrounded values in your calculator and round only the final answer unless the question tells you otherwise.
Try these investigations
Use the explorer to answer the following questions:
What happens to the volume of a prism when its length is doubled?
What happens to the volume of a cylinder when its height is doubled?
What happens to the volume of a sphere when its radius is doubled?
Does doubling the radius of a cylinder merely double its volume?
Compare a cone and cylinder with the same radius and perpendicular height. How are their volumes related?
Compare a square-based pyramid and square prism with the same base and perpendicular height. How are their volumes related?
Which measurements affect the cross-sectional area of a prism?
Which measurement affects the volume but not the cross-sectional area?
What is the difference between the curved surface area and total surface area of a hemisphere?
Does rotating a solid change its surface area or volume?
Make a prediction before changing the measurements. Then use the explorer to test it.
Questions students often ask
Does rotating a solid change its surface area or volume?
No. Rotation changes only the orientation of the solid. Its measurements, surface area and volume remain unchanged.
Why is surface area measured in square units?
Surface area measures the combined areas of the outside surfaces. Each surface is two-dimensional, so square units are used.
Why is volume measured in cubic units?
Volume measures the number of unit cubes that can fit inside a solid. This is why cubic units such as cm³ and m³ are used.
Why do pyramids and cones have a factor of ⅓?
A pyramid or cone has one-third of the volume of a corresponding prism or cylinder with the same base area and perpendicular height.
Is prism length the same as height?
Not always. The prism length is the distance through which the cross-section is repeated. The word height may instead refer to a measurement within the cross-section.
Is slant height used to calculate volume?
No. The volume of a pyramid or cone uses its perpendicular height. Slant height is used when calculating surface area.
Does the total surface area of a hemisphere include its base?
Yes. The total surface area is 3πr² because it includes the curved surface and circular base. Its curved surface area alone is 2πr².
Why might my answer be slightly different?
Calculations involving π, square roots or derived lengths may need to be rounded. Keep the full calculator value during your working and round only the final answer.
For teachers and parents
This explorer can be used to:
Demonstrate the difference between surface area and volume.
Show how a prism’s cross-section relates to its volume.
Compare prisms and pyramids with the same base and height.
Compare cylinders and cones with the same radius and height.
Distinguish perpendicular height, slant height and slant edge.
Reveal internal measurements using Transparent view.
Investigate how changing one dimension affects a result.
Check calculations after students complete their own working.
Encourage students to make and test mathematical predictions.
Students should still identify the correct formula, substitute the measurements and show their working. The explorer is designed to support mathematical understanding—not to make the calculator do all the thinking.
Want to revise flat figures first? Visit the Interactive 2D Shape Explorer to investigate area, perimeter and circumference.
You can also return to the Geometry Explorer to choose another learning tool.
