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Concept · Geometry & Algebra
Solid geometry
solids, surfaces, volumes
Practice this concept→
A Geometry & Algebra set, untimed
Named traps
12 ways this goes wrong.
Each is a named misconception the questions are built to catch. Open one for the move that tempts, and the fix.
- Cut-off ratio wrong referenceCompares a cut-off corner volume to the whole solid instead of to the remaining part.→
- Equilateral-triangle area sqrt3 missingDrops the sqrt3 factor in an equilateral-triangle base area.→
- Frustum vs prism formulaUses a prism formula where the frustum (averaged-base) formula is needed.→
- Impossible cross-section assumedAssumes a cross-section polygon the solid's faces cannot actually produce.→
- Inner solid not subtractedTreats a rotated annulus/ring as a full disk, forgetting to subtract the inner solid.→
- Inscribed vs circumscribed sphereConfuses inscribed-sphere inradius with circumscribed-sphere radius.→
- Lateral vs total surface areaComputes total surface area when only lateral (curved) is required, or vice versa.→
- Oblique-projection area ratioForgets the sqrt2/4 oblique-projection area ratio, or applies it to the side instead of the area.→
- Sector-angle to base-radius wrongRelates the unrolled-sector central angle to the r/l ratio (arc length vs radius) incorrectly.→
- Slant height vs height confusedUses the slant height as the cone/frustum vertical height (or as the radius) or vice versa.→
- Sphere uses wrong diagonalUses a face diagonal instead of the space diagonal for a circumscribed-sphere diameter, or forgets to halve it.→
- Tetrahedron volume factor wrongForgets the 1/6 (scalar triple product) or 1/3 (base x height) factor for a tetrahedron/pyramid.→
Key formulas
9- 3D point P(x,y,z)3D distance AB = √(x2−x1)2 + (y2−y1)2 + (z2−z1)2
- Cuboid volume = lwhsurface = 2(lw+lh+wh)cube volume = s3, surface = 6s2
- CylinderV = πr2 h, S = 2πr2 + 2πrh
- ConeV = (1/3)πr2 h, lateral S = πrl, total S = πrl + πr2, slant l = √r2+h2
- SphereV = (4/3)πR3, S = 4πR2
- Cone lateral unrolls to a sectorarc length 2πr = l·(central angle)base radius r = l·(angle)/(2π)
- Tetrahedron from coordinatesV = (1/6)|scalar triple product|pyramid V = (1/3)·base·height
- Oblique (cavalier) projection area ratioS' = (√2/4) S
- Equilateral triangle of side sarea = (√3/4)s2
Shaky underneath?
If the concept underneath still feels loose, patch it here before coming back to CSCA questions.
FoundationsPrecalculus bridgeAlgebra 2 foundations