Find the height of the cone if the slant height h is 34cm and base diameter is 32cm.

To find the height of the cone, we can use the relationship between the slant height, the radius of the base, and the height of the cone. This relationship is defined by the Pythagorean theorem, which is applicable here because the height, radius, and slant height form a right triangle. Let's break this down step by step.

Understanding the Components

In a cone:

• The **slant height (h)** is the distance from the top of the cone to any point on the edge of the base, which is given as 34 cm.
• The **base diameter** is the distance across the base of the cone, which is 32 cm. Therefore, the **radius (r)**, which is half of the diameter, is 16 cm.
• The **height (H)** is the vertical distance from the base to the apex of the cone, which we need to find.

Applying the Pythagorean Theorem

According to the Pythagorean theorem, in a right triangle:

h² = r² + H²

Where:

• h = slant height = 34 cm
• r = radius = 16 cm
• H = height (what we are trying to find)

Calculating the Height

Now, we can substitute the known values into the equation:

(34)² = (16)² + H²

This simplifies to:

1156 = 256 + H²

Next, we isolate H²:

H² = 1156 - 256

H² = 900

Now, taking the square root of both sides gives us:

H = √900

H = 30 cm

Final Result

The height of the cone is therefore 30 cm. This method illustrates how geometric relationships can help us find unknown dimensions in three-dimensional shapes like cones.