A right cone has base radius 8 cm and slant height 17 cm. Find its volume in terms of π.
- Find the vertical height: h = √(172 − 82) = √(289 − 64) = √225 = 15 cm.
- Volume = ⅓πr2h = ⅓ × π × 82 × 15.
- ⅓ × 64 × 15 = 320.
Answer: Volume = 320π cm3.
Three-dimensional geometry
Maths: Analysis and Approaches · Geometry and trigonometry. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
Test this topic freeAll Maths: Analysis and Approaches topics
The idea
Points in space have three coordinates (x, y, z). The distance and midpoint formulas are the two-dimensional ones with a z-term added.
For solids you need the volume and surface area of a right pyramid, a right cone, a sphere and a hemisphere. In a cone, the radius r, the vertical height h and the slant height l form a right-angled triangle, so l2 = r2 + h2. For a combined solid, add the volumes, and for area count only the faces you can touch.
The angle between a line and a plane is the angle between the line and its shadow (projection) on the plane. Find a right-angled triangle that contains this angle and use trigonometry.
Rules to remember
- distance = √((x2 − x1)2 + (y2 − y1)2 + (z2 − z1)2)
- midpoint = ((x1 + x2)/2, (y1 + y2)/2, (z1 + z2)/2)
- volume of a pyramid = ⅓ × base area × vertical height
- cone: volume = ⅓πr2h; curved surface area = πrl, where l = √(r2 + h2)
- sphere: volume = (4/3)πr3; surface area = 4πr2
- hemisphere: volume = (2/3)πr3; curved area = 2πr2; total area with the flat face = 3πr2
- angle between a line and a plane = angle between the line and its projection on the plane
Common mistake
Students put the slant height into the volume formula, or the vertical height into πrl. Volume needs the vertical height h; curved surface area needs the slant height l.
Worked example
Ask about this
Type what you are stuck on. The helper finds the explanation, worked examples and the notes to read, free and instant.
Read the full notes
Keep going
← Transformations of graphsTrigonometry of triangles and its applications →Take a Maths: Analysis and Approaches test