The Pythagorean Theorem in Three Dimensions Practice

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An illustration shows a cone that rests flat on the circular base. The visible circumference of the base and sides are in solid lines. The circumference of the base at the back that is not visible is indicated by a dashed line. Parts are labeled. A large point is marked in the center of the base. A solid line passes through this point and extends from one edge of the base to the other to indicate the diameter of the base. The line is labeled d equals 10 centimeters. The right side of the cone extending from where the diameter meets the edge of the base up to the point of the cone shows a slant height of 13 centimeters. A dotted line extends from the interior of the vertex of the cone down to the point marked in the center of the base. A right angle is indicated where the dotted height line meets the diameter line in the center of the base.

Find the length of the height of the cone.

(1 point)
cm

Answer 1

To find the height of the cone, we can apply the Pythagorean Theorem in three dimensions, focusing on the right triangle formed by the height (h), the radius (r), and the slant height (l).

1. Identify the dimensions:

   • Diameter of the base, \( d = 10 \) cm.
   • Radius of the base, \( r = \frac{d}{2} = \frac{10}{2} = 5 \) cm.
   • Slant height, \( l = 13 \) cm.

2. Use the Pythagorean Theorem: The relationship in the right triangle can be expressed as: \[ l^2 = h^2 + r^2 \] We need to solve for \( h \). Rearranging the formula gives: \[ h^2 = l^2 - r^2 \]

3. Calculate \( h \): Substituting the values: \[ l^2 = 13^2 = 169 \] \[ r^2 = 5^2 = 25 \] Now substituting back: \[ h^2 = 169 - 25 = 144 \] Taking the square root of both sides: \[ h = \sqrt{144} = 12 \text{ cm} \]

So, the height of the cone is 12 cm.