3D Figures & Measurement

GE.I.1Prisms & cylinders01/04

Volume of prisms and cylinders.

Step01of 04

Universal volume formula for prisms and cylinders — anything with two parallel congruent bases connected by perpendicular sides:

V = B \cdot h

$B$ = area of the base, $h$ = height (perpendicular distance between the two bases). Just multiply.

GE.I.1 — Volume of prisms and cylinders.

Step 01 of 04

Universal volume formula for prisms and cylinders — anything with two parallel congruent bases connected by perpendicular sides:

V = B \cdot h

$B$ = area of the base, $h$ = height (perpendicular distance between the two bases). Just multiply.

Step 02 of 04

Specific bases. Just plug the right area formula in for $B$ .

Rectangular prism (box)	→ $V = l w h$ (length × width × height)
Triangular prism	→ $V = \frac{1}{2} bh \cdot H$ (triangle area × prism height)
Cylinder	→ $V = π r^{2} h$ (circle area × height)

Step 03 of 04

Worked example. A cylinder with radius 4 and height 10:

V = π (4)^{2} (10) = 160 π \approx 502.65

Triangular prism with base triangle of base 6, height 4, prism length 12:

V = \frac{1}{2} (6) (4) \cdot 12 = 12 \cdot 12 = 144

Step 04 of 04

Volume scales with the cube of any linear dimension.

If you double every dimension of a box, volume goes up 8×. Triple every dimension, volume goes up 27×. This is the same scale factor logic as similar polygons: linear ratio cubed.

Key insight

Prism / cylinder volume = base area × height. Different shapes only change the formula for B. Watch units — a "height" of 5 cm with a base area of 12 sq cm gives volume 60 cubic cm.

GE.I.2 — Volume of cones and pyramids.

Step 01 of 04

Cones and pyramids have a single apex point — like prisms and cylinders squished to a point. Their volume is exactly ONE-THIRD of the corresponding prism / cylinder with the same base and height.

V = \frac{1}{3} B \cdot h

Step 02 of 04

Specific shapes.

Cone (circular base)	→ $V = \frac{1}{3} π r^{2} h$
Square-based pyramid	→ $V = \frac{1}{3} s^{2} h$
Rectangular pyramid	→ $V = \frac{1}{3} l w h$
Triangular pyramid (tetrahedron)	→ $V = \frac{1}{3} (\frac{1}{2} b h_{t}) \cdot h$

In every case, $h$ is the PERPENDICULAR height from base to apex — NOT a slant height.

Step 03 of 04

Worked examples.

Cone, radius 6, height 9:

V = \frac{1}{3} π (6)^{2} (9) = \frac{1}{3} \cdot 36 π \cdot 9 = 108 π

Square pyramid, side 10, height 12:

V = \frac{1}{3} (10)^{2} (12) = \frac{1}{3} \cdot 100 \cdot 12 = 400

Step 04 of 04

Slant height vs. perpendicular height — don't confuse them.

Perpendicular height	→apex straight down to the base — used for VOLUME
Slant height	→apex along the surface to the edge of the base — used for SURFACE AREA

For a right cone with base radius $r$ , perpendicular height $h$ , and slant height $ℓ$ : $ℓ^{2} = r^{2} + h^{2}$ — Pythagorean.

Key insight

Cone or pyramid volume = ⅓ × base area × HEIGHT (perpendicular, NOT slant). The ⅓ factor is the only thing that distinguishes this from prism/cylinder volume.

GE.I.3 — Volume of a sphere.

Step 01 of 04

Sphere volume. Memorize this one:

V = \frac{4}{3} π r^{3}

The " $r^{3}$ " reminds you it's a 3D measurement (cubic units). The $\frac{4}{3} π$ coefficient is a mathematical constant that comes from integrating the sphere's profile (you'll prove this in calculus).

Step 02 of 04

Worked examples.

Sphere with radius 3:

V = \frac{4}{3} π (3)^{3} = \frac{4}{3} π \cdot 27 = 36 π

Sphere with radius 6 (double the radius — volume goes up by 8×):

V = \frac{4}{3} π (6)^{3} = \frac{4}{3} π \cdot 216 = 288 π

Step 03 of 04

Find r given V. Solve backwards.

Volume $V = \frac{32}{3} π$ :

\frac{32}{3} π = \frac{4}{3} π r^{3} \Rightarrow r^{3} = 8 \Rightarrow r = 2

Step 04 of 04

Hemisphere volume = half a sphere.

V_{hemi} = \frac{1}{2} \cdot \frac{4}{3} π r^{3} = \frac{2}{3} π r^{3}

Often appears in problems combining a hemisphere on top of a cylinder (think a silo or a domed building) — compute each volume separately and add.

Key insight

Sphere volume =

\frac{4}{3} π r^{3}

. Hemisphere = half that. Volume scales as

r^{3}

, so doubling the radius gives 8× the volume.

GE.I.4 — Surface area of solids.

Step 01 of 04

Surface area is the total area of all the faces / curved surfaces of a 3D solid. Measured in SQUARE units (not cubic, like volume).

Strategy	→unfold the solid into a "net" of flat shapes, sum their areas
Practical	→memorize the most common formulas

Step 02 of 04

Common surface area formulas.

Rectangular prism (box)	→ $S A = 2 (l w + l h + w h)$
Cube	→ $S A = 6 s^{2}$
Cylinder	→ $S A = 2 π r^{2} + 2 π r h$ (two circles + the rectangle that wraps around)
Cone (closed)	→ $S A = π r^{2} + π r ℓ$ (base circle + lateral surface, with slant height $ℓ$ )
Sphere	→ $S A = 4 π r^{2}$

Step 03 of 04

Worked example — cylinder. Radius 5, height 8.

S A = 2 π (5)^{2} + 2 π (5) (8) = 50 π + 80 π = 130 π

Imagine cutting the cylinder open and unrolling it: two end-cap circles plus a rectangle whose width is the circumference ( $2 π r$ ) and whose height is the cylinder's height. Add the three areas.

Step 04 of 04

Cone surface area uses SLANT height, not perpendicular height. If you're given the perpendicular height instead, recover the slant via Pythagorean:

ℓ = r^{2} + h^{2}

Worked example: cone with radius 6 and perpendicular height 8. Slant $ℓ = 36 + 64 = 10$ . So:

S A = π (6)^{2} + π (6) (10) = 36 π + 60 π = 96 π

Key insight

Surface area = sum of every face / curved surface, in square units. Cylinder = two circles + an unrolled rectangle. Sphere has its own special constant:

4 π r^{2}

. Cone surface area uses SLANT height — that's the most-tested gotcha.

GE.I.5 — Cross-sections of 3D figures.

Step 01 of 04

A cross-section is the 2D shape you see when you slice a 3D figure with a plane. The shape depends on (a) the original solid, (b) the angle and position of the slice.

Step 02 of 04

Slicing a cube.

Slice parallel to a face	→square (same size as the face)
Slice parallel to an edge but tilted	→rectangle
Diagonal slice through 3 vertices	→triangle
Slice through 4 face midpoints	→rhombus or hexagon (depending on angle)

Step 03 of 04

Slicing a cylinder.

Horizontal (parallel to base)	→circle (same size as the base)
Vertical through the axis	→rectangle (height × diameter)
Tilted	→ellipse

Slicing a cone.

Horizontal	→circle
Vertical through the apex	→triangle (isosceles)
Tilted (not too steep)	→ellipse
Parallel to the slant	→parabola
Steep (cuts both halves)	→hyperbola

The "conic sections" — circle, ellipse, parabola, hyperbola — are literally the cross-sections of a cone.

Step 04 of 04

Slicing a sphere. Always a circle, no matter the angle. The circle's size depends only on how close the slice passes to the center.

Through the center	→great circle (radius = sphere radius)
Off-center	→smaller circle
Tangent to surface	→a single point (degenerate "circle" of radius 0)

Key insight

Cross-section shape depends on the solid AND the slice angle. Cube → can be triangle, rectangle, square, hexagon. Cylinder → circle, rectangle, or ellipse. Cone → all four conic sections. Sphere → always a circle.

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GE.I.1 – GE.I.5 · CCSS G.GMD.A.3 / G.MG.A.1 / G.GMD.B.4·Full curriculum →