📐Geometry & Stereometry

Geometry Calculator: Areas & 3D Solids (2D / 3D)

Compute surface areas, perimeters, volumes, and lateral areas for all standard geometric shapes and solids with interactive 2D/3D visualizer and level trainer.

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Unit:

Radius (r in cm)

Area (A)

78.54cm²

Exakt:

A = 25\pi\,\text{cm²}

Perimeter (U)

31.416cm

Exakt:

U = 10\pi\,\text{cm}

📝Step-by-Step Derivation & Formulas

1Durchmesser

d = 2 \cdot r

d = 2 \cdot 5

d = 10

2Flächeninhalt

A = \pi \cdot r^2

A = \pi \cdot 5^2 = \pi \cdot 25

A \approx 78,5398

3Umfang

U = 2 \cdot \pi \cdot r

U = 2 \cdot \pi \cdot 5

U \approx 31,4159

📐Interactive 2D / 3D Geometry Visualizer

True-to-scale vector diagrams & 3D projections with live parameter adjustments

Radius (r):5

Historical Origin & The Challenge Solved

Why was this mathematics developed, where does it come from, and what real-world problem did it solve?

📜

Epoche & ZeitalterAncient Greece & Alexandria (c. 300 – 212 BCE)

Pioniere: Euclid of Alexandria, Archimedes of Syracuse

1. King Hieron's Golden Crown & Granary Capacity

King Hieron II of Syracuse suspected his goldsmith of adulterating a consecrated gold crown with silver. Archimedes had to determine the precise volume of the irregular crown without melting or damaging it.

2. Axiomatic Geometry & The Method of Exhaustion

Euclid's "Elements" established axiomatic plane and solid geometry. Archimedes calculated exact formulas for spheres, cones, cylinders, and parabolic segments using infinitesimal exhaustion.

3. 3D Printing, Architectural BIM & Virtual Reality (VR)

Additive 3D printer slicers, architectural CAD models, and virtual reality physics engines run on geometric volume, surface area, and perimeter algorithms.

„Do not disturb my circles!“ — Archimedes of Syracuse (212 BCE)

Real-World Applications & Practical Use

Where geometric formulas are essential across architecture, engineering, and everyday projects

Architecture & Construction

Material Estimations & Floor Areas

Determining concrete volume, paint coverage, and roofing tile counts for pitched roofs and pyramids.

Praxis-Beispiel:

Calculating the lateral surface area of a grain silo to estimate weatherproof coating costs.

Logistics & Packaging

Volume Maximization & Shipping

Designing cans, boxes, and containers for maximal volume while minimizing raw material costs.

Praxis-Beispiel:

Computing the holding capacity of conical hoppers and cylindrical tanks.

DIY & Gardening

Excavation, Pool Volumes & Patio Pavers

Accurate estimates for trapezoidal garden beds, circular swimming pools, and excavation earth.

Praxis-Beispiel:

Finding the water capacity of a 4 m round swimming pool with 1.5 m depth (approx. 18,850 liters).

📚Formulas & Didactic Foundations of Geometry & Stereometry

1. Two-Dimensional Planimetry (2D)

Planimetry covers flat figures. Fundamental parameters include the area A (measure of 2D extent) and the perimeter U (boundary length).

Kreis & Sektor: $A = \pi r^2, \; U = 2\pi r, \; b = 2\pi r \frac{\alpha}{360^\circ}$
Ellipse: $A = \pi a b$ (Ramanujan-Umfangsnäherung)
Trapez & Parallelogramm: $A = \frac{a+c}{2} h, \; A = a \cdot h_a$
Regelmäßiges Vieleck: $A = \frac{n a^2}{4 \tan(\pi/n)}$

2. Three-Dimensional Stereometry (3D)

Stereometry analyzes solid spatial objects. Key metrics are volume V (capacity), lateral area M, and total surface area O.

Quader & Würfel: $V = a b c, \; O = 2(ab + bc + ac)$
Zylinder: $V = \pi r^2 h, \; M = 2\pi r h, \; O = 2\pi r(r + h)$
Kegel & Pyramide: $V = \frac{1}{3} G h, \; s = \sqrt{r^2 + h^2}$
Kugel: $V = \frac{4}{3}\pi r^3, \; O = 4\pi r^2$

❓Frequently Asked Questions (FAQ)

How do you calculate the perimeter of an ellipse accurately?

Unlike circles, an ellipse does not have a simple exact closed formula for perimeter, but requires complete elliptic integrals. Srinivasa Ramanujan approximation provides higher than 99.99% accuracy.

What is the difference between lateral area and total surface area?

Lateral area M represents only the side envelope of a solid (e.g., unrolled rectangle of a cylinder). Total surface area O adds the base area G and top cap D: O = 2G + M.

How do you convert area and volume units correctly?

Length units scale by factor 10 (1 m = 100 cm). Area units scale by factor 100 (1 m² = 10,000 cm²), and volume units scale by factor 1,000 (1 m³ = 1,000 liters = 1,000,000 cm³).

Why is a pyramid volume one third of a prism with the same base and height?

Any prism can be decomposed into three pyramids of equal volume. In calculus, integrating the quadratic cross-sectional area yields the factor 1/3.