Finding Cone Height from Slant Height
The height, radius, and slant height of a right cone form a right triangle where the slant height is the hypotenuse. Use the Pythagorean theorem to find cone height: r squared + h squared = l squared, so h = square root of (l squared - r squared). For a cone with r = 3 cm and slant height l = 5 cm: h = square root of (25 - 9) = square root of 16 = 4 cm. For r = 8 and l = 17: h = square root of (289 - 64) = 15 in. This prerequisite from enVision Mathematics, Grade 8, Chapter 8 is required before calculating cone volume.
Key Concepts
The height ($h$), radius ($r$), and slant height ($l$) of a right cone form a right triangle. To find the height when given the radius and slant height, use the Pythagorean theorem: $$r^2 + h^2 = l^2$$.
Common Questions
How do I find the height of a cone from its slant height?
Use the Pythagorean theorem: r squared + h squared = l squared. Rearrange to h = square root of (l squared minus r squared).
A cone has radius 3 cm and slant height 5 cm. Find the height.
h = square root of (5 squared - 3 squared) = square root of (25 - 9) = square root of 16 = 4 cm.
A cone has radius 8 in and slant height 17 in. Find the height.
h = square root of (17 squared - 8 squared) = square root of (289 - 64) = square root of 225 = 15 in.
Why does h, r, and l form a right triangle?
The radius goes horizontally from the center of the base to the edge. The height goes vertically from the center to the apex. The slant height connects the apex to the base edge, forming the hypotenuse.
Why do I need cone height to find volume?
The cone volume formula is V = (1/3) pi r squared h. If only the slant height is given, you must first find the perpendicular height h using the Pythagorean theorem.
When do 8th graders learn to find cone height from slant height?
Chapter 8 of enVision Mathematics, Grade 8 covers this in the Surface Area and Volume unit.