Calculate volume and surface area for cubes, cuboids, cylinders, spheres, cones, prisms, and hemispheres
Volume and surface area calculations are essential topics in GCSE and A-level maths, covering three-dimensional geometry. Volume measures the amount of 3D space inside a solid shape, expressed in cubic units (cm^3, m^3), while surface area measures the total area of all faces and curved surfaces, expressed in square units (cm^2, m^2). Our calculator handles seven common 3D shapes: cubes, cuboids, cylinders, spheres, cones, triangular prisms, and hemispheres. Each calculation displays both the volume and surface area, along with the formula used and step-by-step working that follows the approach expected in GCSE and A-level exam answers. Understanding volume and surface area has numerous practical applications, from calculating the capacity of containers and tanks to determining the amount of material needed to wrap or coat an object. These real-world contexts appear frequently in GCSE exam questions. At GCSE Higher tier, students must know the volume formulas for all standard 3D shapes. The sphere and cone formulas are provided on the formulae sheet, but students need to recognise when to apply each formula and how to substitute values correctly. Cuboid and prism volumes are not on the formulae sheet, as students are expected to memorise these. The National Curriculum requires students to calculate volumes of prisms (including cylinders), pyramids, cones, and spheres. At A-level, students encounter volumes of revolution, where a 2D shape is rotated around an axis to form a 3D solid, and integration is used to find the volume. Our calculator covers the standard shape calculations that form the foundation for these more advanced topics.
To calculate volume and surface area: 1. Select the 3D shape from the dropdown menu: cube, cuboid, cylinder, sphere, cone, triangular prism, or hemisphere. 2. Enter the required measurements for your chosen shape. Each shape uses different inputs. For a cube, enter the side length. For a cylinder, enter the radius and height. For a sphere, enter just the radius. 3. For a triangular prism, enter the base and height of the triangular cross-section, along with the length (depth) of the prism. 4. Review the results showing volume and surface area, both rounded to 2 decimal places. The bar chart provides a visual comparison of the two values. 5. Study the step-by-step working to see how values are substituted into the formulas. This matches the method expected in GCSE and A-level exam answers. 6. For cone calculations, the calculator also shows the slant height, which is found using Pythagoras' theorem (l = sqrt(r^2 + h^2)). This is needed for the surface area calculation. 7. Remember to include correct units in your exam answers. Volume uses cubic units (e.g. cm^3) and surface area uses square units (e.g. cm^2).
Volume and surface area formulas for 3D shapes: Cube: Volume = s^3. Surface Area = 6s^2. Example: side 5. Volume = 125 cm^3. Surface Area = 150 cm^2. Cuboid: Volume = l * w * h. Surface Area = 2(lw + lh + wh). Example: 4 * 3 * 5. Volume = 60 cm^3. Surface Area = 94 cm^2. Cylinder: Volume = pi * r^2 * h. Surface Area = 2 * pi * r^2 + 2 * pi * r * h. Example: radius 3, height 10. Volume = pi * 9 * 10 = 282.74 cm^3. Surface Area = 245.04 cm^2. Sphere: Volume = (4/3) * pi * r^3. Surface Area = 4 * pi * r^2. Example: radius 6. Volume = (4/3) * pi * 216 = 904.78 cm^3. Surface Area = 452.39 cm^2. Cone: Volume = (1/3) * pi * r^2 * h. Slant height l = sqrt(r^2 + h^2). Surface Area = pi * r * l + pi * r^2. Example: radius 4, height 9. Volume = (1/3) * pi * 16 * 9 = 150.80 cm^3. Triangular Prism: Volume = 0.5 * base * triangle_height * length. This is the cross-sectional area multiplied by the length of the prism. Hemisphere: Volume = (2/3) * pi * r^3. Surface Area = 3 * pi * r^2 (curved surface plus the flat circular base). Example: radius 5. Volume = (2/3) * pi * 125 = 261.80 cm^3. The sphere and cone volume formulas are provided on the GCSE formulae sheet. Cuboid and prism formulas should be memorised. All these formulas are tested across AQA, Edexcel, OCR, and WJEC GCSE and A-level maths specifications.