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📝 Solve geometry formulas (10 MCQs)

📖 From Digital SAT Algebra • 2. Linear Equations And Inequalities • 10 questions available

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What is Solve geometry formulas?

Definition:
Solving geometry formulas means rearranging standard geometric equations (area, perimeter, volume, etc.) to find an unknown dimension. This involves algebraic manipulation to isolate the desired variable, using the given formula and known values.

Working:
For the perimeter of a rectangle $P = 2l + 2w$ , to find $l$ , subtract $2w$ : $P - 2w = 2l$ , divide by 2: $l = (P - 2w)/2$ . For the area of a circle $A = \pi r^2$ , solve for $r$ : $r = \sqrt{A/\pi}$ .

Example:
A rectangle has area 24 sq units and length 6 units. Find width. Using $A = lw$ , $24 = 6w$ , divide: $w = 4$ units.

Reason:
Geometry formulas are practical; solving them helps find missing dimensions in construction, design, and everyday measurements.

Easy

Medium

Hard

📝 All Solve geometry formulas MCQs

Q1. A triangle has area 84 square centimeters and base 14 centimeters. Which equation correctly isolates the height $h$ from $A=\frac{1}{2}bh$ , and what is the height?

h=\frac{A}{2b}=3

h=\frac{2A}{b}=12

cm ✅

h=\frac{b}{2A}=\frac{1}{12}

h=2Ab=2352

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: To isolate $h$ , first multiply both sides by 2, giving $2A=bh$ . Dividing by $b$ produces $h=\frac{2A}{b}$ . Substituting $A=84$ and $b=14$ gives $h=12$ centimeters.

Q2. A triangular garden has a fixed area of 120 square meters. A designer considers bases of 15, 20, and 30 meters. Which base requires the greatest height?

A.15 m ✅

B.20 m

C.30 m

D.All require the same height

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Using $h=\frac{2A}{b}$ , the required height decreases as the base increases. The heights are 16, 12, and 8 meters respectively. Therefore, the smallest base of 15 meters requires the greatest height.

Q3. A triangular sign has base 18 inches and height 10 inches. A manufacturer increases the base by 20% but wants the area unchanged. What should the new height be?

A.8 inches

B.8 1/3 inches ✅

C.10 inches

D.12 inches

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The original area is $\frac12(18)(10)=90$ square inches. The new base is 21.6 inches. Setting $\frac12(21.6)h=90$ gives $h=\frac{180}{21.6}=8\frac13$ inches, preserving the original area.

Q4. A student rearranges $P=a+b+c$ to solve for $c$ and writes $c=P+a+b$ . What is the most important error in this rearrangement?

A.The student multiplied instead of divided

B.The terms

a

and

b

should be subtracted from

P

✅

C.The perimeter formula cannot be rearranged

D.The variable

c

must be divided by

P

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Since $a$ and $b$ are added to $c$ , they must be removed by subtraction. Starting with $P=a+b+c$ , subtracting $a$ and $b$ from both sides gives $c=P-a-b$ .

Q5. A triangular parcel has perimeter 74 meters. Two sides measure 21 meters and 27 meters. The owner wants to determine the third side before calculating the fencing cost. Which value is correct?

A.22 m

B.24 m

C.26 m ✅

D.28 m

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: The perimeter is the sum of all three side lengths. Therefore, $74=21+27+c$ . Combining the known sides gives 48, so $c=74-48=26$ . The correct third side is therefore 26 meters.

Q6. A graph plots perimeter $P$ on the vertical axis against an unknown side $c$ on the horizontal axis for fixed sides $a=12$ and $b=18$ . Which description matches the graph?

A.A line with slope 1 and vertical intercept 30 ✅

B.A line with slope 30 and vertical intercept 1

C.A horizontal line at

P=30

D.A line with slope -1 and intercept 30

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: From $P=a+b+c$ , substituting the fixed sides gives $P=30+c$ . This is a linear relationship with slope 1, meaning every one-unit increase in $c$ increases the perimeter by one unit, and its vertical intercept is 30.

Q7. An investment uses $I=Prt$ , where $I$ is interest, $P$ is principal, $r$ is the annual rate, and $t$ is time. If $I$ , $P$ , and $t$ are known, which expression isolates $r$ ?

r=\frac{I}{Pt}

✅

r=\frac{IP}{t}

r=\frac{Pt}{I}

r=I-P-t

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Starting with $I=Prt$ , divide both sides by $Pt$ . This gives $r=\frac{I}{Pt}$ . The rearrangement works because division by the factors multiplying $r$ isolates the desired variable.

Q8. A student calculates simple interest using $I=Prt$ with $P=5000$ , $r=0.06$ , and $t=2$ , obtaining 600. Another student obtains 60000 because the rate was entered as 6. Which reasoning is correct?

A.The second student is correct because percentages should always be entered as whole numbers

B.The first student is correct because

6\%=0.06

in decimal form ✅

C.Both are correct because rates can use either form without adjustment

D.Neither is correct because time must be converted to months

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: A percentage rate must be converted to decimal form when used directly in $I=Prt$ . Since $6\%=0.06$ , the calculation is $5000(0.06)(2)=600$ . Using 6 incorrectly makes the result one hundred times too large.

Q9. A triangle's area is modeled by $A=\frac12bh$ . A graph shows that when $b=8$ , the area is 32, and when $b=12$ , the area is 48, while height remains constant. What is the height?

A.4

B.6

C.8 ✅

D.12

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Because $h$ is constant, the relationship between area and base is linear. Using $A=\frac12bh$ with $A=32$ and $b=8$ , we obtain $32=4h$ , so $h=8$ . The second point confirms this because $\frac12(12)(8)=48$ .

Q10. A savings account earns $I=Prt$ . One account has $P=4000$ , $r=5\%$ , and $t=3$ years. Another has $P=6000$ , $r=4\%$ , and $t=2$ years. Which account earns more simple interest, and by how much?

A.First account by $120 ✅

B.First account by $240

C.Second account by $120

D.Second account by $240

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: For the first account, $I=4000(0.05)(3)=600$ . For the second, $I=6000(0.04)(2)=480$ . Comparing the two results shows that the first account earns $120 more, not$ 240. Therefore, option A is mathematically correct.

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