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MathChapter 20: Surface Areas and Volumes

Notes

Prisms Notes

Model prism surface area and volume from base geometry, then apply the formulas to cubes, removed regions, density, open-top boxes, and water levels.

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Every prism problem starts by choosing a base. Once the base area B, base perimeter P, and perpendicular prism height h are identified, the core formulas describe both the material covering the solid and the space inside it.

Surface area versus volume

Surface area and volume answer different questions
QuantityMeaningUnitsPrism input
Surface areaMaterial covering exposed facesSquare unitsP, h, and B
VolumeSpace contained insideCubic unitsB and h

Prism anatomy

Prism familyTriangular, rectangular, and hexagonal prisms each show two congruent parallel bases, a perpendicular prism height, and a rectangular lateral face.
Prism familyTriangular, rectangular, and hexagonal prisms each show two congruent parallel bases, a perpendicular prism height, and a rectangular lateral face.Triangular prismbaselateral facehRectangular prismbaselateral facehHexagonal prismbaseh
Prism familyTriangular, rectangular, and hexagonal prisms each show two congruent parallel bases, a perpendicular prism height, and a rectangular lateral face.
Prism
A polyhedron with two congruent, parallel faces called bases. In the right prisms used here, the remaining lateral faces are rectangles.
Prism terminology
PartWhat it meansHow it is used
BaseEither congruent parallel faceIts area is B and perimeter is P.
Lateral faceA nonbase rectangular faceAll lateral faces combine to area Ph.
Height hPerpendicular distance between base planesIt multiplies B for volume and P for lateral area.

The three prism formulas

Prism lateral-area netA triangular-prism net shows two congruent triangular bases and a lateral strip whose total width is base perimeter P and whose height is prism height h.
Prism lateral-area netA triangular-prism net shows two congruent triangular bases and a lateral strip whose total width is base perimeter P and whose height is prism height h.height hbase perimeter Plateral strip area = P × h
Prism lateral-area netA triangular-prism net shows two congruent triangular bases and a lateral strip whose total width is base perimeter P and whose height is prism height h.
Prism formulas
QuantityFormulaGeometric reasonUnits
Lateral area\(LA=Ph\)The lateral strip has total width P and height h.square units
Total area\(TA=Ph+2B\)Lateral area plus two congruent basessquare units
Volume\(V=Bh\)Base area extended through perpendicular height hcubic units
Prism lateral area
\[LA=Ph\]

P is the perimeter of one base; h is the perpendicular prism height.

Prism total surface area
\[TA=Ph+2B\]

Include both bases only when both are exposed.

Prism volume
\[V=Bh\]

B is an area, not a perimeter.

Reliable prism workflow

  1. Choose the base

    Name its shape and isolate the dimensions that lie in that base plane.

  2. Calculate B or P

    Use base area for volume; use base perimeter for lateral area.

  3. Identify h

    Find the perpendicular distance between congruent bases.

  4. Apply and label

    Use \(V=Bh\), \(LA=Ph\), or \(TA=Ph+2B\), then attach cubic or square units.

Triangular prisms: two different heights

Triangular-prism dimensionsA triangular prism has a triangular base with base 8 units and triangle altitude 5 units, while the perpendicular distance between the congruent triangular bases is 12 units.
Triangular-prism dimensionsA triangular prism has a triangular base with base 8 units and triangle altitude 5 units, while the perpendicular distance between the congruent triangular bases is 12 units.85prism height 12lateral faceFigure not drawn to scale.
Triangular-prism dimensionsA triangular prism has a triangular base with base 8 units and triangle altitude 5 units, while the perpendicular distance between the congruent triangular bases is 12 units.

Figure not drawn to scale.

Worked example

Build volume from the triangular base

A triangular base has base 8 cm and triangle altitude 5 cm. The prism height is 12 cm. Find the volume.

  1. Area of triangular base

    \(B=\tfrac12(8)(5)=20\text{ cm}^2\).

  2. Extend through the prism

    \(V=Bh=(20)(12)=240\text{ cm}^3\).

The volume is \(240\text{ cm}^3\).

Rectangular prisms and cubes

Rectangular-prism dimensionsA rectangular prism has length 11 units, width 6 units, and perpendicular height 4 units.
Rectangular-prism dimensionsA rectangular prism has length 11 units, width 6 units, and perpendicular height 4 units.4116Figure not drawn to scale.
Rectangular-prism dimensionsA rectangular prism has length 11 units, width 6 units, and perpendicular height 4 units.

Figure not drawn to scale.

Rectangular-prism forms
\[V=\ell wh\qquad TA=2\ell w+2\ell h+2wh\]

These are the general prism formulas after choosing a rectangular base.

Cube formulasA cube has equal positive edge length s on all three displayed dimensions; its volume is s cubed and its surface area is six s squared.
Cube formulasA cube has equal positive edge length s on all three displayed dimensions; its volume is s cubed and its surface area is six s squared.sssV = s³ · SA = 6s²
Cube formulasA cube has equal positive edge length s on all three displayed dimensions; its volume is s cubed and its surface area is six s squared.
Cube formulas
\[V=s^3\qquad SA=6s^2\]

Every edge is s and every exposed face is a square of area s².

Worked example

Reverse a cube formula

A cube has volume \(343\text{ in}^3\). Find its surface area.

  1. Recover the edge

    \(s^3=343\), so the positive physical edge is \(s=7\text{ in}\).

  2. Use six square faces

    \(SA=6s^2=6(7^2)=294\text{ in}^2\).

The surface area is \(294\text{ in}^2\).

Regular-polygon bases

Regular-hexagonal prismA regular hexagonal prism has base side 4 units, its base decomposed into six equilateral triangles, and prism height 9 units.
Regular-hexagonal prismA regular hexagonal prism has base side 4 units, its base decomposed into six equilateral triangles, and prism height 9 units.h = 9side 46 equilateral trianglesFigure not drawn to scale.
Regular-hexagonal prismA regular hexagonal prism has base side 4 units, its base decomposed into six equilateral triangles, and prism height 9 units.

Figure not drawn to scale.

Composite and removed volume

Rectangular prism with two openingsAn outer rectangular prism measuring 30 by 16 by 8 units has two through-openings, each measuring 6 by 5 across the top and extending through the full height; remaining volume is outer volume minus both opening volumes.
Rectangular prism with two openingsAn outer rectangular prism measuring 30 by 16 by 8 units has two through-openings, each measuring 6 by 5 across the top and extending through the full height; remaining volume is outer volume minus both opening volumes.outer length 30height 8opening 6 × 5remaining = outer volume − removed volumeFigure not drawn to scale.
Rectangular prism with two openingsAn outer rectangular prism measuring 30 by 16 by 8 units has two through-openings, each measuring 6 by 5 across the top and extending through the full height; remaining volume is outer volume minus both opening volumes.

Figure not drawn to scale.

Remaining material
\[V_{\text{remaining}}=V_{\text{outer}}-\sum V_{\text{removed}}\]

Every removed region must use the same volume units as the outer solid.

Example removed-volume audit
PartCalculationSigned contribution
Outer block\(30\cdot16\cdot8=3840\)\(+3840\text{ units}^3\)
Opening 1\(6\cdot5\cdot8=240\)\(-240\text{ units}^3\)
Opening 2\(6\cdot5\cdot8=240\)\(-240\text{ units}^3\)
Remaining\(3840-240-240\)\(3360\text{ units}^3\)

Density and cubic-unit conversion

Density relationships
\[D=\frac{m}{V}\qquad m=DV\qquad V=\frac{m}{D}\]

Match the volume unit in the solid calculation to the volume unit in the density before operating.

Dimension and conversion check
MeasurementExample conversionExponent
Length\(1\text{ ft}=12\text{ in}\)first power
Area\(1\text{ ft}^2=144\text{ in}^2\)square the factor
Volume\(1\text{ ft}^3=1728\text{ in}^3\)cube the factor

Open-top boxes

Open-top box from a rectangular sheetA 24-by-18 rectangular sheet has four corner squares of side x removed. Folding creates an open-top box of height x with base dimensions 24 minus 2x by 18 minus 2x.
Open-top box from a rectangular sheetA 24-by-18 rectangular sheet has four corner squares of side x removed. Folding creates an open-top box of height x with base dimensions 24 minus 2x by 18 minus 2x.Flat sheetx2418Folded open-top boxheight xbase: (L−2x) by (W−2x)Figure not drawn to scale.
Open-top box from a rectangular sheetA 24-by-18 rectangular sheet has four corner squares of side x removed. Folding creates an open-top box of height x with base dimensions 24 minus 2x by 18 minus 2x.

Figure not drawn to scale.

Open-top box volume
\[V=x(L-2x)(W-2x)\]

Cut size x becomes the box height; two cuts reduce each affected sheet dimension.

Worked example

Fold a box from a sheet

Squares of side 3 in are removed from a 24-by-18-inch sheet. Find the resulting box volume.

  1. New base

    Length \(24-2(3)=18\); width \(18-2(3)=12\).

  2. Height and volume

    Height is 3, so \(V=(18)(12)(3)=648\text{ in}^3\).

The open-top box holds \(648\text{ in}^3\).

Water level in a rectangular tank

Water depth
\[h_{\text{water}}=\frac{V_{\text{water}}}{A_{\text{tank base}}}\]

If total tank height is H, the unfilled distance is \(H-h_{\text{water}}\).

Mini check

Prism check

A tank has base area \(120\text{ cm}^2\), total height 15 cm, and contains \(1080\text{ cm}^3\) of water. How far is the surface below the top?

  1. 6 cm
  2. 9 cm
  3. 15 cm
  4. 90 cm
Show answer and explanation

Answer: 6 cm.

Water depth is \(1080/120=9\) cm, so the remaining distance is \(15-9=6\) cm.

Key takeaways

Prism essentials

  • Choose one base; use its area B for volume and its perimeter P for lateral area.
  • Prism height is perpendicular distance between the congruent bases.
  • Total prism area is \(Ph+2B\); volume is \(Bh\).
  • Subtract holes, match density units, and cube linear conversion factors for volume.
  • An open-top box cut by x has height x and base dimensions reduced by 2x.
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Put these notes into practice

Apply the ideas with SAT-style questions, then reinforce key details with flashcards.