#### 28800Ein rechtwinkliger Kreiskegel hat einen Basisradius von 5 cm und eine Höhe von 12 cm. Wenn der Kegel mit Wasser gefüllt und dann in ein zylindrisches Gefäß mit einem Basisradius von 10 cm gegossen wird, wie hoch steigt das Wasser im Zylinder?
["Like | Share | Save\nVolumen of a Right Circular Cone Filled with Water: How High Does It Rise in a Cylinder?", "Have you ever wondered how much water a conical vessel holds when filled to the brim, and what happens when that water is poured into a cylindrical container? This straightforward yet fascinating geometry problem combines volume calculation with practical real-life application. In this article, we’ll explore step-by-step how to calculate the water height in a cylinder after pouring from a right circular cone.", "---", "### The Problem at a Glance", "We are given a cone with:\n- Base radius: 5 cm\n- Height: 12 cm\nThis cone is completely filled with water, and then poured into a cylindrical container with a base radius of 10 cm. We need to find out how high the water rises in the cylinder.", "---", "### Step 1: Calculate the Volume of Water in the Cone", "The volume ( V ) of a right circular cone is given by the formula:\n[\nV = \frac{1}{3} \pi r^2 h\n]\nSubstituting the known dimensions:\n- ( r = 5 ) cm\n- ( h = 12 ) cm", "[\nV = \frac{1}{3} \pi (5)^2 (12) = \frac{1}{3} \pi (25) (12) = \frac{1}{3} \pi (300) = 100\pi \ ext{ cm}^3\n]\nSo, the volume of water is ( 100\pi ) cm³.", "---", "### Step 2: Relate That Volume to the Zylinder", "Now, this water is poured into a cylinder with:\n- Base radius: 10 cm\n- Unknown height ( h_{\ ext{cyl}} ), where we want to find the water level.", "The volume of a cylinder is:\n[\nV = \pi R^2 H\n]\nHere, ( R = 10 ) cm, and the known volume ( V = 100\pi ) cm³. Solving for ( h_{\ ext{cyl}} ):", "[\n100\pi = \pi (10)^2 h_{\ ext{cyl}} = \pi (100) h_{\ ext{cyl}}\n]\nDivide both sides by ( \pi ):\n[\n100 = 100 \cdot h_{\ ext{cyl}} \quad \Rightarrow \quad h_{\ ext{cyl}} = 1 \ ext{ cm}\n]", "---", "### Final Answer:\nWhen the water from the cone (with base radius 5 cm, height 12 cm) is poured into the cylinder (base radius 10 cm), it rises to a height of 1 cm.", "---", "### Why This Matters", "Understanding how volumes translate between different shapes helps in real-world applications—from engineering and architecture to cooking and everyday measurements. Despite sharing the same base radius ratio, the cone’s tapered form concentrates far less volume than the cylinder, which explains why the rise is only 1 cm compared to the cone’s original 12 cm height.", "---", "### Summary", "| Step | Formula / Calculation | Result |\n|---------------------|-----------------------------------------|-----------------|\n| Volume of cone | ( \frac{1}{3} \pi r^2 h ) | ( 100\pi ) cm³ |\n| Volume in cylinder | ( \pi R^2 h_{\ ext{cyl}} ) | ( 100\pi ) cm³ |\n| Height in cylinder | ( \frac{100\pi}{\pi \cdot 10^2} ) | ( h_{\ ext{cyl}} = 1 ) cm |", "---", "Keywords: right circular cone volume, cylinder water height, volume conversion, geometry problem, cone-to-cylinder transfer, math tutorial, geometry solved", "Optimize your understanding of volume conversion—complex problems start simply!", "---", "Whether you’re a student, teacher, or math enthusiast, mastering volume relationships like this builds strong problem-solving skills. Next time you think about containers and liquids, remember that geometry turns everyday shapes into powerful calculations!"]
