If a cone has a circular base of diameter 10 cm. and a slant height of 10 cm. then the volume is what?

To determine the space a cone occupies, we use the formula for its volume: one-third multiplied by the area of its base, and then by its vertical height. For a cone with a circular base, the base area is calculated using pi times the radius squared. Given a diameter of 10 cm, the radius is half of that, or 5 cm. Therefore, the base area is pi times 5 squared, which equals 25 pi square centimeters. This crucial first step provides the 'B' component for our volume calculation.

However, the problem also provides a slant height of 10 cm, which is not the same as the vertical height needed for the volume formula. To find the actual vertical height, we can visualize a right-angled triangle inside the cone, formed by the radius, the vertical height, and the slant height. The Pythagorean theorem, which states that the square of the hypotenuse (slant height) equals the sum of the squares of the other two sides (radius and vertical height), comes into play. With a radius of 5 cm and a slant height of 10 cm, the vertical height squared is 10 squared minus 5 squared, which is 100 minus 25, resulting in 75. Thus, the vertical height is the square root of 75 centimeters.

Now, with both the base area (25 pi) and the vertical height (square root of 75), we can complete the volume calculation. Multiplying one-third by 25 pi and then by the square root of 75 yields approximately 226.725 cubic centimeters. This method demonstrates how understanding geometric relationships, particularly the Pythagorean theorem, is essential in solving three-dimensional problems. Cones appear in many real-world applications, from traffic cones and ice cream cones to architectural designs and even the shape of certain mountains.