3D Volume & Surface Area Calculator

Volumetric calculation is a fundamental pillar of mechanical engineering, architecture, geological planning, and physics. Volume measures the absolute quantity of three-dimensional space enclosed by a closed boundary surface, expressed in cubic units ($u^3$, $cm^3$, or $m^3$). Conversely, surface area measures the total boundary space exposed to the external environment, expressed in square units ($u^2$). Historically, determining these dimensions required complex mathematical integrals; today, standard algebraic formulations allow us to analyze these shapes instantly.

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🧬 Integral Calculus Proofs for Curved Shapes

The formulas for curved solids such as spheres and cones are direct results of **integral calculus**.

Consider a **Sphere** of radius $r$. To calculate its volume, we can model it as a solid of revolution. By revolving the semi-circle equation $y = \sqrt{r^2 - x^2}$ around the x-axis from $x = -r$ to $x = +r$, we integrate thin cylindrical slices of volume $dV = \pi y^2 dx$:
V = ∫_{-r}^{+r} \pi (r² - x²) dx = \pi [ r²x - x³/3 ]_{-r}^{+r} = \frac{4}{3} \pi r³
This elegant proof establishes the absolute constant $\frac{4}{3}$ in spherical calculations.

Similarly, a **Cone** can be modeled as a series of stacked circular discs whose radii scale linearly from 0 at the apex to $r$ at the base at height $h$. Integrating these discs along the height axis demonstrates that a cone occupies exactly **one-third** of the volume of a cylinder with the exact same height and base radius:
V = ∫_{0}^{h} \pi (\frac{r}{h} x)² dx = \frac{1}{3} \pi r² h
This relationship shows how closely related different 3D shapes are in spatial geometry.

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🧊 Orthographic & Isometric 3D SVG Projection

Representing a 3D solid on a flat 2D computer screen is a classic problem in computer graphics. Because SVGs operate strictly in a 2D Cartesian coordinate plane, our engine utilizes specialized **orthographic isometric projections** to render wireframes.

For a three-dimensional point $(x, y, z)$, we translate it to 2D screen coordinates $(X, Y)$ using standard isometric angles (typically $\theta_x = 30^\circ$ and $\theta_y = -30^\circ$ to the horizontal):
X = x · cos(30°) - y · cos(30°)
Y = z - x · sin(30°) - y · sin(30°)
By applying this translation to the vertices of cubes, cylinders, and pyramids, we draw highly accurate, rotating-like wireframes. This creates structural depth, allowing you to visualize the shape's proportions in real time as you adjust length, width, and height.

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🌋 Geothermal and Geological Applications in Iceland

In Iceland, volumetric equations are of extreme practical importance across our energy and geological landscapes:

• Hot Water Cylindrical Silos: The iconic **Perlan** in Reykjavík features massive cylindrical hot water storage tanks. To calculate the thermal capacity and volume of geothermally heated water stored to supply households in the Capital Area, engineers apply the cylinder formula $V = \pi r^2 h$, ensuring the municipal grid has massive reserves.
• Volcanic Cone Excavations: When analyzing lava flow or calculating gravel deposits from conical volcanic hills (like those in the Reykjanes peninsula), geologists use the cone volume formula. This helps calculate the tonnage of basalt or pumice available for construction and road safety barriers.
• Geothermal Pressurized Fuel Cells: Pressurized gases are typically stored in **capsules** (cylinders with hemi-spherical caps). Using the capsule formulas allows engineers to calculate maximum structural loads and interior pressure distributions, preventing failure under tectonic shifts.