Digital SAT · Math · Unit 4 · Geometry & Trigonometry

Lesson 4.5 · Areas & Volumes

On the test: 2–3 questions a test — a volume from the reference sheet, a surface area, and very often a SCALE question: a model, similar solids, or a side changed by a percent

Big idea. Every formula on the reference sheet is one of three ideas: an area is base × height (a triangle half of it), a prism-like solid is base area × height (a pointed one a third of it), and a sphere is its own rule. Then one master rule for similar shapes: lengths × k, areas × k², volumes × k³ — and to go backwards, take the square or cube root.

🎛 Move it · why the area formulas work

Pick a shape. Tick “show why”: the parallelogram turns into a rectangle, two trapezoids make a parallelogram.

base10height6lean5

show why

Cut the red triangle off the left and slide it to the right: the parallelogram becomes a rectangle b × h. (Not b × the slanted side!)

🧑‍🏫 Rule 1 · Rectangle: area length × width, perimeter 2(l + w)step 1 of 10

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📋 The question · SAT · March US 2026 (A) · Module 2 Q9
A rectangle has a length of 10 centimeters and a width of 6,000 centimeters. What is the area, in square centimeters, of the rectangle?
1. A)10
2. B)6,010
3. C)60,000
4. D)66,000
multiply
$10 \times 6000 = 60, 000$
March US 2026 (A): C. (B added them.)

🧑‍🏫 Rule 2 · A square from its diagonalstep 1 of 4

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two ways
$s = \frac{d}{2} or A = \frac{d ^{2}}{2}$
The diagonal makes two 45-45-90 triangles.

🧑‍🏫 Rule 3 · Triangle, parallelogram, trapezoidstep 1 of 5

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triangle
$A = \frac{1}{2} bh$
Half the rectangle around it.

🧑‍🏫 Rule 4 · Circles and composite figuresstep 1 of 2

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📋 The question · SAT · October US 2024 · Math 800 Book Q41
$figure from the test$
The figure shown consists of a rectangle and a semicircle, where the length of $\overline{W X}$ is 20 units and the length of $\overline{W Z}$ is 5 units. The diameter of the semicircle is $\overline{W X}$ . Which of the following expressions represents the area, in square units, of the figure?
1. A) $20 + 5 + 10 π$
2. B) $20 \times 5 + 10 π r$
3. C) $20 \times 5 + 50 π$
4. D) $20 \times 5 + 100 π$
split it
$20 \times 5 + \frac{1}{2} π \cdot 1 0^{2}$
October US 2024: rectangle + semicircle on the diameter 20 (radius 10).

🎛 Move it · the solids

Pick a solid and change it. For the cone and pyramid, tick the box to see the cylinder/prism it fills a third of.

radius r6height h8

show the cylinder it fills one third of

A cone is exactly ⅓ of the cylinder around it (dashed). The slant height ℓ is the hypotenuse of the right triangle radius–height–slant — Pythagoras.

Rule · On the reference sheet

V = lwh · V = πr²h · V = ⅓πr²h · V = ⅓lwh · V = ⁴⁄₃πr³. You must know WHICH to use and what each letter is — the sheet will not tell you that 14 is a diameter.

🧑‍🏫 Rule 5 · Prisms: base area × heightstep 1 of 4

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📋 The question · SAT · June 2023 · Math 800 Book Q5
The volume of a right rectangular prism with a square base is 2,448 cubic centimeters. If the area of the square base is 144 square centimeters, what is the area, in square centimeters, of one of the four lateral faces of the prism?
1. A)17
2. B)204
3. C)540
4. D)816
height
$h = \frac{2448}{144} = 17$
June 2023.

🧑‍🏫 Rule 6 · Cylinder: πr²hstep 1 of 5

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📋 The question · SAT · March US 2023 · Math 800 Book Q1
A right circular cylinder has a height of 6 inches. The radius of the base of the cylinder is 5 inches. What is the volume, in cubic inches, of the cylinder?
1. A) $10 π$
2. B) $30 π$
3. C) $50 π$
4. D) $150 π$
straight in
$π \cdot 25 \cdot 6 = 150 π$
March US 2023: D.

🧑‍🏫 Rule 7 · Pyramid and cone: one thirdstep 1 of 4

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why ⅓
$V_{cone} = \frac{1}{3} V_{cylinder}, V_{pyramid} = \frac{1}{3} V_{prism}$
Same base, same height: three cones fill the cylinder.

🧑‍🏫 Rule 8 · Sphere and hemispherestep 1 of 6

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the formulas
$V_{sphere} = \frac{4}{3} π r^{3}, V_{hemi} = \frac{2}{3} π r^{3}$
Cube the radius.

🧑‍🏫 Rule 9 · Combined solids and the space leftstep 1 of 3

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📋 The question · SAT · March US 2023 · Math 800 Book Q2
$figure from the test$
A hemisphere and a cone have circular bases of equal circumference and have been connected at their bases, as shown in the figure. The diameter of the base of the hemisphere is 6 centimeters (cm) and the total height of the figure is 9 cm. What is the total volume, in cm³, of the cone and hemisphere?
1. A) $18 π$
2. B) $36 π$
3. C) $45 π$
4. D) $162 π$
split it
$r = 3; h_{cone} = 9 - 3 = 6$
March US 2023: the dome takes 3 of the 9.

🎛 Move it · unfold the box

Change the box; take the lid off.

length6width4height3

it has a lid

Surface area is the paper you need to wrap it: unfold the box and add the faces. A box has three PAIRS of equal rectangles: SA = 2(lw + lh + wh). No lid → one l·w face fewer (an open cube: 5s², not 6s²).

🧑‍🏫 Rule 10 · Box and cubestep 1 of 3

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📋 The question · SAT · August 2024 · Math 800 Book Q32
A right rectangular prism has a length of 26 centimeters (cm), a width of 16 cm, and a height of 14 cm. What is the surface area, in cm², of the right rectangular prism?
(type the answer)
three pairs
$2 (26 \cdot 16 + 26 \cdot 14 + 16 \cdot 14) = 2008$
August 2024.

🧑‍🏫 Rule 11 · Cylinder and spherestep 1 of 3

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📋 The question · Gen Z Study · cylinder surface
A closed cylinder has radius 3 and height 5. Its total surface area is $k π$ . What is $k$ ?
(type the answer)
cylinder
$2 π r^{2} + 2 π r h = 18 π + 30 π$
Two lids + the unrolled label: 48.

🧑‍🏫 Rule 12 · Pyramids and cones: the slant heightstep 1 of 10

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the right triangle
$ℓ^{2} = h^{2} + (half the base side)^{2}$
For a cone: $ℓ^{2} = h^{2} + r^{2}$ .

🎛 Move it · the ratio rules

Grow k and count the squares and cubes. Then try “backwards”, “stretch one dimension” and “a % change”.

scale factor k2

Similar figures, scale factor k: every length (sides, perimeter, radius, height, slant) × k; every area (faces, surface area) × k²; every volume × k³. Count them: a 2× square holds 4 unit squares, a 2× cube holds 8 unit cubes. A model at 1/15 scale has 1/225 of the area.

🧑‍🏫 Rule 13 · Lengths × k, areas × k², volumes × k³step 1 of 10

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📋 The question · SAT · March 2024 · Math 800 Book Q15
Each side of square A has a length of 11 inches. Each side of square A is multiplied by a scale factor of 3 to create square B. What is the length, in inches, of each side of square B?
1. A)8
2. B)11
3. C)14
4. D)33
lengths
$11 \times 3 = 33$
March 2024: D.

🧑‍🏫 Rule 14 · Backwards: from an area or volume ratio to kstep 1 of 11

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the move
$k = area ratio = 3 volume ratio$
Always go back to k first.

🧑‍🏫 Rule 15 · Only some dimensions changestep 1 of 2

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📋 The question · SAT · August US 2024 · Math 800 Book Q38
$figure from the test$
The figure shown is a right circular cylinder with a radius $r$ and a height of $h$ . A second right circular cylinder (not shown) has a volume that is 576 times as large as the volume of the cylinder shown. Which of the following could represent the radius $R$ , in terms of $r$ , and the height $H$ , in terms of $h$ , of the second cylinder?
1. A) $R = 9 r$ and $H = 8 h$
2. B) $R = 9 r$ and $H = 64 h$
3. C) $R = 8 r$ and $H = 9 h$
4. D) $R = 64 r$ and $H = 9 h$
where each letter sits
$π (a r)^{2} (bh) = a^{2} b \cdot π r^{2} h$
August US 2024: the radius is squared, the height is not.

🧑‍🏫 Rule 16 · A percent change on every sidestep 1 of 2

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📋 The question · SAT · June 2024 · Math 800 Book Q28
A rectangular banner has an area of 2,500 square inches. A copy of the banner is made in which the length and width of the original banner are each increased by 40%. What is the area of the copy, in square inches?
1. A)2,540
2. B)2,580
3. C)3,500
4. D)4,900
multiply the multipliers
$1.4 \times 1.4 = 1.96 \Rightarrow 2500 \times 1.96 = 4900$
June 2024: D.

🎛 Move it · density and units

Change the cube’s edge and mass.

edge (cm)2mass (g)2.72

Density = mass ÷ volume (so mass = density × volume). Find the VOLUME first — a cube’s is the edge CUBED, not × 3. Converting areas and volumes: square or cube the length conversion (cm² → m²: ÷ 100², not ÷ 100).

🧑‍🏫 Rule 17 · Density = mass ÷ volumestep 1 of 6

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📋 The question · SAT · March 2024 · Math 800 Book Q10
To study fluctuations in leaf water potential, samples of wood were taken from 25 trees and cut in the shape of a cube. The length of the edge of one of these cubes is 2.00 centimeters. If this cube has a mass of 2.72 grams, what is the density of this cube, in grams per cubic centimeter?
(type the answer)
cube first
$V = 2^{3} = 8; \frac{2.72}{8} = 0.34$
March 2024.

🧑‍🏫 Rule 18 · Square and cubic unitsstep 1 of 4

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the rule
$1 yd = 3 ft \Rightarrow 1 yd^{2} = 9 ft^{2}, 1 yd^{3} = 27 ft^{3}$
Square or cube the length conversion.

🧑‍🏫 Rule 19 · The book’s other questions in this chapterstep 1 of 3

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📋 The question · SAT · June 2024 · Math 800 Book Q27
A chemist combines water and isopropanol to make a mixture with a volume of 52 milliliters (mL). The volume of isopropanol in the mixture is 10 mL. What is the volume of water, in mL, in the mixture? (Assume that the volume of the mixture is the sum of the volumes of water and isopropanol before they were mixed.)
(type the answer)
volumes add
$52 - 10 = 42$
June 2024.

Real question 1SAT · August US 2024 · Math 800 Book Q38

$figure from the test$

The figure shown is a right circular cylinder with a radius $r$ and a height of $h$ . A second right circular cylinder (not shown) has a volume that is 576 times as large as the volume of the cylinder shown. Which of the following could represent the radius $R$ , in terms of $r$ , and the height $H$ , in terms of $h$ , of the second cylinder?

Take with you

▸Rectangle lw, perimeter 2(l + w); square from its diagonal: d²/2. Triangle ½bh, parallelogram bh, trapezoid ½(b₁ + b₂)h, circle πr², composites = add the pieces.
▸Prism and cylinder: base × height. Pyramid and cone: ⅓ of that. Sphere ⁴⁄₃πr³, hemisphere ⅔πr³. Diameter → halve it first.
▸Surface area = the faces unfolded: box 2(lw + lh + wh), cube 6s² (5s² with no lid), cylinder 2πr² + 2πrh, sphere 4πr²; slant height from Pythagoras.
▸Similar shapes: lengths × k, areas × k², volumes × k³. Backwards: k = √(area ratio) = ∛(volume ratio).
▸Different factors: put each where it sits in the formula (πR²H → a²b). Every side +40% → area × 1.96.
▸Density = mass ÷ volume. Square / cube a unit conversion for areas / volumes (1 yd³ = 27 ft³).

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