3D Geometry Calculator
Got a photo of your Geometry homework?
Upload it and let AI solve it instantly.
Result
Base triangle sides a, b, c and prism length L.
Quick Examples
3D Geometry Calculator: Volume & Surface Area
Your complete guide to solving 3D geometry problems, from basic sphere and cylinder calculations to pyramids, prisms, torus, and ellipsoids.
How to Use This 3D Geometry Calculator
Choose from Sphere, Cube, Rectangular Prism, Cylinder, or Cone. Enter the radius, height, and side lengths as required. Press Solve to see volume, surface area (lateral and total), slant height, and space diagonal where applicable.
Example: Select Cylinder, enter r = 4 and h = 10 to get volume = 502.65 cubic units and total surface area = 351.86 sq units with every step shown.
Choose from Square Pyramid, Triangular Prism, Torus, Hemisphere, or Ellipsoid. For the triangular prism, enter all three base triangle sides and the prism length. Heron's formula is used automatically for the base area.
Example: Select Square Pyramid, enter b = 6 and h = 8 to get volume = 96 cubic units, slant height = 8.54 units, and total surface area = 139.08 sq units.
3D Geometry Calculator by Student Level
Topics: Volume and surface area of rectangular prisms, cubes, and basic cylinders. Common problems ask you to find how much water a tank holds or how much wrapping paper covers a box.
How to use: Select Rectangular Prism or Cube from the Common Solids tab. Enter the three dimensions and press Solve. Read the volume formula V = l × w × h and each multiplication step in the output.
Tip: Volume is always in cubic units (cm³) and surface area is in square units (cm²). Students often mix these up on exams.
Topics: Spheres, cones, pyramids, and composite solids. GCSE Higher and IB SL regularly test cone slant height, pyramid lateral area, and hemisphere volume alongside total surface area.
How to use: Use the Cone and Pyramid shapes in Common Solids and More Solids respectively. This solver shows slant height and separates lateral area from total area so you can answer both parts of a multi-mark question easily.
Tip: For GCSE composite solid problems, solve each part separately, then add or subtract volumes as needed.
Topics: Torus, ellipsoid, surface area via integration, volume of revolution. These appear in Calculus II and Multivariable Calculus. The torus volume and surface area involve double integrals solved using Pappus's theorem.
How to use: Use the Torus and Ellipsoid options in More Solids. The exact formulas are shown step by step. Use these results to verify integral calculations in your Calculus coursework.
Tip: For the ellipsoid, this calculator uses the Knud Thomsen approximation for surface area, which is accurate to within 1.061% for typical axis ratios.
Using This Calculator for 3D Geometry Assignments
Homework Problems
Solve each 3D problem by hand first, then enter the values here to verify. Common mistakes include using diameter instead of radius in sphere and cylinder formulas, forgetting to add the base area when computing total surface area, and using the slant height instead of the vertical height in volume formulas.
Case Studies and Projects
Engineering and architecture projects often require calculating the volume of cylindrical tanks, spherical domes, or conical roofs. Enter the real-world measurements from your project and use the step-by-step output to write up your solution clearly in your report.
Quiz and Exam Practice
Click Random to load a random 3D shape with preset dimensions. Try to calculate the volume and surface area in your head or on paper first, then press Solve to check. Spheres and cones are the most commonly tested shapes on GCSE and IB Higher papers.
Problem Sets
For multi-part 3D problem sets, start with the Common Solids tab to establish the basic dimensions, then check composite shapes by adding or subtracting the individual volumes. Use the slant height output from the Cone or Pyramid solver in follow-up surface area calculations.
Sphere Calculator
What is the formula for the volume of a sphere?
Volume = (4/3)πr³. Cube the radius first, then multiply by 4π/3. For r = 5: V = (4/3) × 3.14159 × 125 = 523.60 cubic units. Surface area = 4πr². For r = 5: SA = 4 × 3.14159 × 25 = 314.16 sq units.
How do I find the radius of a sphere if I know the volume?
Rearrange V = (4/3)πr³ to get r = √³(3V / 4π). For V = 523.60: r = √³(3 × 523.60 / 12.566) = √³(125) = 5. Enter the radius in the Sphere selector and use the Random button to see various examples with full working.
Real-world sphere example
A spherical water tank has radius 3 m. How much water can it hold? V = (4/3) × π × 27 = 113.10 m³. Since 1 m³ = 1000 litres, the tank holds 113,100 litres. Enter r = 3 in the Sphere solver to verify this with every step shown.
Cylinder Calculator
How do you calculate the volume of a cylinder?
Volume = πr²h. Square the radius, multiply by π, then multiply by height. For r = 4 and h = 10: V = 3.14159 × 16 × 10 = 502.65 cubic units. The formula is essentially the area of the circular base multiplied by the height.
What is the difference between lateral area and total surface area of a cylinder?
Lateral area = 2πrh. This is just the curved side of the cylinder, like the label on a tin can. Total surface area = 2πr(r + h) = lateral area + 2 × base area. For r = 4 and h = 10: lateral area = 251.33 sq units and total SA = 351.86 sq units.
Explain how to solve cylinder surface area step by step
Step 1: Calculate base area = πr². Step 2: Calculate lateral area = 2πrh. Step 3: Total SA = lateral + 2 × base = 2πrh + 2πr² = 2πr(r+h). For r = 3 and h = 7: base = 28.27, lateral = 131.95, total = 188.50 sq units.
Cone Calculator
How do you find the volume of a cone?
Volume = (1/3)πr²h. A cone holds exactly one-third of the volume of a cylinder with the same base and height. For r = 3 and h = 8: V = (1/3) × 3.14159 × 9 × 8 = 75.40 cubic units. Enter r and h in the Cone selector to verify with steps shown.
What is slant height and why does it matter?
The slant height l is the distance along the slanted surface of the cone from the base edge to the apex. It is not the same as the vertical height h. Formula: l = √(r² + h²). For r = 3 and h = 8: l = √(9 + 64) = √73 ≈ 8.544 units. Slant height is needed to calculate the lateral surface area of the cone.
How do I calculate the total surface area of a cone?
Lateral area = πrl, where l is the slant height. Total SA = πr(r + l) = base area + lateral area. For r = 3 and l = 8.544: total SA = 3.14159 × 3 × (3 + 8.544) = 9.4248 × 11.544 ≈ 108.83 sq units. This solver shows slant height, lateral area, and total area all at once.
Rectangular Prism and Cube Calculator
How do you calculate the volume of a rectangular prism?
Volume = length × width × height. For l = 5, w = 4, h = 3: V = 60 cubic units. Total surface area = 2(lw + lh + wh) = 2(20 + 15 + 12) = 94 sq units. The space diagonal = √(l² + w² + h²) = √(25 + 16 + 9) = √50 ≈ 7.07 units.
How do you calculate the volume of a cube?
Volume = s³ where s is the side length. For s = 4: V = 64 cubic units. Surface area = 6s² = 6 × 16 = 96 sq units. Space diagonal = s√3 = 4 × 1.7321 ≈ 6.93 units. A cube is a special rectangular prism where all three dimensions are equal.
Pyramid Calculator
What is the volume formula for a square pyramid?
Volume = (1/3) × b² × h, where b is the base side length and h is the vertical height. For b = 6 and h = 8: V = (1/3) × 36 × 8 = 96 cubic units. The pyramid holds one-third the volume of a prism with the same base and height.
How do I find the slant height of a pyramid?
Slant height l = √((b/2)² + h²). For b = 6 and h = 8: l = √(9 + 64) = √73 ≈ 8.544 units. The slant height is measured from the midpoint of a base edge to the apex along the triangular face. You need it to calculate lateral area.
How do I calculate the total surface area of a square pyramid?
Total SA = b² + 2bl, where b² is the square base area and 2bl is the lateral area from all four triangular faces. For b = 6 and l = 8.544: total SA = 36 + 2 × 6 × 8.544 = 36 + 102.53 = 138.53 sq units. This solver calculates and separates each part clearly.
Triangular Prism Calculator
How do you find the volume of a triangular prism?
Volume = base triangle area × prism length. The base triangle area is found using Heron's formula from the three side lengths. For a base triangle with sides 3, 4, 5 and prism length 10: base area = 6 sq units (right triangle), V = 6 × 10 = 60 cubic units.
What is the surface area of a triangular prism?
Surface area = 2 × base triangle area + perimeter of base × length. The perimeter is the sum of all three base triangle sides. For sides 3, 4, 5 and length 10: SA = 2 × 6 + (3+4+5) × 10 = 12 + 120 = 132 sq units. Enter all values in the Triangular Prism selector to see full working.
Torus Calculator
What is a torus and what are its formulas?
A torus is a doughnut-shaped 3D solid. It has a major radius R (from the center of the torus to the center of the tube) and a minor radius r (the radius of the tube itself). Volume = 2π²Rr². Surface area = 4π²Rr. For R = 5 and r = 2: V = 2 × 9.8696 × 5 × 4 = 394.78 cubic units.
Hemisphere and Ellipsoid Calculator
How do you calculate the volume of a hemisphere?
A hemisphere is exactly half of a sphere. Volume = (2/3)πr³. For r = 4: V = (2/3) × 3.14159 × 64 = 134.04 cubic units. Curved surface area = 2πr² = 100.53 sq units. Total SA (including flat base) = 3πr² = 150.80 sq units.
What is the volume of an ellipsoid?
An ellipsoid is a stretched sphere with three semi-axes a, b, and c. Volume = (4/3)πabc. For a = 5, b = 4, c = 3: V = (4/3) × 3.14159 × 5 × 4 × 3 = 251.33 cubic units. Surface area has no simple closed form; this solver uses the Knud Thomsen approximation.
Real-World Uses of 3D Geometry
Where is 3D geometry used in real life?
Engineers calculate cylindrical tank volumes for water and fuel storage. Architects design spherical domes and conical roofs using surface area formulas. Packaging designers use rectangular prism calculations to minimize material waste. Astronomers use ellipsoid approximations to model planetary shapes.
How do engineers use cylinder and cone geometry?
Pipes and containers are cylindrical. Silos and funnels are conical. A civil engineer calculating how much concrete to pour into a cylindrical column uses V = πr²h. A chemical engineer sizing a conical tank uses V = (1/3)πr²h. These same formulas this calculator uses appear in every engineering discipline.
How is 3D geometry used in manufacturing?
CNC machining, 3D printing, and injection moulding all require precise volume calculations to estimate material cost and machine time. Surface area calculations determine how much material is needed to coat, paint, or plate a 3D part. Every product you buy was designed using these formulas.
Frequently Asked Questions
- 1. What is the formula for the volume of a sphere?
- V = (4/3)πr³. For r = 5: V = (4/3) × 3.14159 × 125 = 523.60 cubic units. Surface area = 4πr² = 314.16 sq units. Enter r in the Sphere selector and press Solve to see every multiplication step. A useful check: the sphere volume is exactly (2/3) of the circumscribed cylinder's volume.
- 2. How do you calculate the surface area of a cylinder?
- Total SA = 2πr(r + h). This is: two circular bases (2πr²) plus the lateral surface (2πrh). For r = 4 and h = 10: SA = 2 × 3.14159 × 4 × 14 = 351.86 sq units. The Cylinder selector in Common Solids shows lateral area and total area separately so you can answer each part of a multi-mark question.
- 3. How do you find the volume of a cone?
- V = (1/3)πr²h. For r = 3 and h = 8: V = (1/3) × 3.14159 × 9 × 8 = 75.40 cubic units. Remember, the cone volume is always one-third the volume of the matching cylinder. The Cone selector also shows slant height l = √(r²+h²) and the full lateral and total surface area.
- 4. What is the difference between lateral area and total surface area?
- Lateral area covers only the curved or slanted sides of a solid, not the base(s). Total surface area covers all faces including the base(s). For a cylinder, lateral = 2πrh and total = 2πrh + 2πr². For a cone, lateral = πrl and total = πr(r+l). Both values are shown separately in the output of this calculator.
- 5. How do you calculate the volume of a rectangular prism?
- V = l × w × h. For l = 5, w = 4, h = 3: V = 60 cubic units. Total SA = 2(lw+lh+wh) = 2(20+15+12) = 94 sq units. Space diagonal = √(l²+w²+h²) = √50 ≈ 7.07 units. Select Rectangular Prism in Common Solids and enter the three dimensions to get all of these results at once.
- 6. What is the volume formula for a pyramid?
- V = (1/3) × b² × h for a square pyramid. For b = 6 and h = 8: V = (1/3) × 36 × 8 = 96 cubic units. Slant height l = √((b/2)²+h²) = √(9+64) ≈ 8.544 units. Total SA = b² + 2bl = 36 + 102.53 = 138.53 sq units. The Square Pyramid selector shows all of these values.
- 7. How do you find the volume of a triangular prism?
- V = base triangle area × prism length L. The base area is calculated automatically using Heron's formula from the three side lengths. SA = 2 × base area + (a+b+c) × L. For a 3-4-5 right triangle base with L = 10: base area = 6, V = 60 cubic units, SA = 12 + 120 = 132 sq units.
- 8. What is the surface area of a torus?
- SA = 4π²Rr, where R is the major radius and r is the minor radius. Volume = 2π²Rr². For R = 5 and r = 2: SA = 4 × 9.8696 × 5 × 2 = 394.78 sq units and V = 2 × 9.8696 × 5 × 4 = 394.78 cubic units. Note that for this specific R and r, SA and V happen to be equal numerically but have different units.
- 9. How do you calculate the volume of a hemisphere?
- V = (2/3)πr³ (exactly half a sphere). For r = 4: V = (2/3) × 3.14159 × 64 = 134.04 cubic units. Curved SA = 2πr² = 100.53 sq units. Total SA (with flat base) = 3πr² = 150.80 sq units. Select Hemisphere in More Solids to see the calculation broken into each step.
- 10. Is this 3D geometry calculator useful for GCSE and IB exams?
- Yes. This calculator covers every 3D shape formula tested in GCSE Maths (Foundation and Higher), IB SL and HL Mathematics, and AP Calculus. It separates lateral area from total surface area, shows slant height automatically, and uses Heron's formula for triangular bases. Use it to verify your working on past papers and understand where each number in the answer comes from.
Need to solve 2D geometry? Try our 2D Geometry Solver for area, perimeter, coordinate geometry, and angle problems.