7.H.1 — Surface area of prisms.
A right rectangular prism (box)
length 8 in, width 5 in, height 3 in
Step 01 of 05
Surface areais the total area of every outer face. For a box, that's six rectangles arranged in three matching pairs: top & bottom, front & back, left & right.
Strategy: think of "unfolding" the box into a flat NET, compute each face, then add them up.
Step 02 of 05
The three pairs of faces.
| Top & bottom | →each =l⋅w=8⋅5=40 |
| Front & back | →each =l⋅h=8⋅3=24 |
| Left & right | →each =w⋅h=5⋅3=15 |
Step 03 of 05
Total surface area = double each pair, then sum.
SA=2(40)+2(24)+2(15)=80+48+30=158 sq in
The general formula for a box:
SA=2(lw+lh+wh)
Step 04 of 05
Triangular prism — same idea, different bases. Two triangular ends + three rectangular sides.
Triangular prism with base triangle of base 6, height 4, and prism length 10. Triangle area is 21(6)(4)=12 per end. Three rectangles wrap around the triangle's three sides — multiply each side of the triangle by the prism length 10.
Step 05 of 05
General prism surface area. Two congruent base shapes plus the "lateral surface" — a rectangle whose width is the perimeter of the base and whose height is the prism's length.
SA=2⋅(base area)+(base perimeter)⋅(prism height)
Key insight
Surface area = sum of every face. For boxes, three pairs of rectangles. For any prism, two bases + lateral surface (perimeter of base × prism height). Always in SQUARE units.
7.H.2 — Volume of prisms.
The universal prism volume formula
V=B⋅h
Step 01 of 05
Volume of any prism = area of the base times the height (perpendicular distance between the two bases). The shape of the base doesn't matter — rectangle, triangle, hexagon, anything — same formula.
Step 02 of 05
Rectangular prism (box).
V=l⋅w⋅h
The box from the previous skill: V=8⋅5⋅3=120 cubic inches. Note the units — VOLUME is always in CUBIC units, not square.
Step 03 of 05
Triangular prism. Compute the triangle's area first, then multiply by the prism's length.
V=(21⋅triangle base⋅triangle height)⋅prism length
Example: triangle with base 6 and height 4, prism length 10.
V=21(6)(4)(10)=12⋅10=120 cubic units
Step 04 of 05
Tip — identify the BASE first. The base of a prism is the shape that REPEATS at both ends. The "height" is then the perpendicular distance between those two ends, NOT necessarily the vertical direction.
A triangular prism resting on a rectangular side still has its triangles as bases (because they're the repeated shape). Its "height" is the distance between the two triangles.
Step 05 of 05
Word-problem application. A rectangular pool 20 ft long, 10 ft wide, 4 ft deep holds:
V=20⋅10⋅4=800 cubic feet of water
At about 7.48 gallons per cubic foot, that's roughly 6,000 gallons.
Key insight
Prism volume = base area × height. Identify the base as the shape that REPEATS, then multiply by the perpendicular distance between the two copies. Volume is in cubic units, surface area is in square units.