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
Remaining Attempts : 3

Answer 1

To find the height of the cone, we can use the Pythagorean Theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

In this case, we have a cone with:

• The diameter of the base \(d = 10\) cm, which means the radius \(r = \frac{d}{2} = 5\) cm.
• The slant height \(l = 13\) cm (this is the hypotenuse of the right triangle).

We want to find the height \(h\) of the cone. The relationship between the height, radius, and slant height can be expressed as: \[ l^2 = r^2 + h^2. \]

Substituting the known values: \[ 13^2 = 5^2 + h^2, \] \[ 169 = 25 + h^2. \]

Now, subtract 25 from both sides: \[ 169 - 25 = h^2, \] \[ 144 = h^2. \]

Finally, take the square root of both sides to find \(h\): \[ h = \sqrt{144} = 12 \text{ cm}. \]

Thus, the height of the cone is 12 cm.