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📝 Perimeter of triangle formula (14 MCQs)

📖 From Digital SAT Algebra • 3. Mathematical Models in Algebra • 14 questions available

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What is Perimeter of triangle formula?

Definition:
The perimeter of a triangle is the total distance around the outside of the triangle, calculated by summing the lengths of all three sides, with the formula $P = a + b + c$ , where $a, b,$ and $c$ are the lengths of the sides, and this formula applies to any triangle regardless of its type (scalene, isosceles, or equilateral).

Working:
To find the perimeter, simply measure or know the lengths of the three sides and add them together, ensuring all sides are in the same unit, and for special triangles like equilateral, where all sides are equal, the formula simplifies to $P = 3s$ , where $s$ is the side length, and for isosceles with two equal sides, $P = 2a + b$ .

Example:
If a triangle has side lengths of 5 cm, 7 cm, and 9 cm, its perimeter is $P = 5 + 7 + 9 = 21$ cm, and if it is an equilateral triangle with side 6 cm, the perimeter is $3 \times 6 = 18$ cm.

Reason:
The perimeter formula is widely used in real-life applications such as fencing a triangular field, framing a triangular structure, calculating the border length of a triangular garden, and in various construction and design projects.

Easy

Medium

Hard

📝 All Perimeter of triangle formula MCQs

Q1. A triangle has side lengths $7$ cm, $11$ cm, and $15$ cm. What is its perimeter?

A.28 cm

B.31 cm

C.33 cm ✅

D.35 cm

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: The perimeter of a triangle is found by adding the lengths of all three sides. Therefore, $7+11+15=33$ cm. The other choices result from arithmetic mistakes or incorrectly omitting or altering one of the side lengths.

Q2. Three sides of a triangular garden measure $9.5$ m, $12$ m, and $14.5$ m. Which expression correctly represents the perimeter before calculating its value?

9.5+12+14.5

✅

9.5\times12\times14.5

2(9.5+12+14.5)

9.5+12-14.5

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Perimeter represents the complete distance around the boundary, so every side is included exactly once. Adding $9.5$ , $12$ , and $14.5$ correctly models the boundary length. Multiplication, doubling, or subtraction does not represent this situation.

Q3. Two triangles have perimeters of $36$ cm. Triangle A has sides $10$ , $12$ , and $14$ cm. Triangle B has two sides of $9$ cm and $13$ cm. What must the third side of Triangle B be?

A.12 cm

B.13 cm

C.14 cm ✅

D.15 cm

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Since Triangle B has the same perimeter as Triangle A, its perimeter must be $36$ cm. Its known sides total $9+13=22$ cm, so the missing side must be $36-22=14$ cm. This also demonstrates that equal perimeters do not require equal side lengths.

Q4. A designer increases each side of a triangular sign by $2$ cm. The original perimeter is $27$ cm. What is the new perimeter?

A.29 cm

B.31 cm

C.33 cm ✅

D.35 cm

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Each of the three sides increases by $2$ cm, so the total perimeter increases by $3\times2=6$ cm. Adding this increase to the original $27$ cm gives $33$ cm. A common mistake is adding only $2$ cm once.

Q5. A triangular path has sides $x$ m, $x+4$ m, and $2x-1$ m. If its perimeter is $35$ m, what is $x$ ?

A.6

B.7

C.8 ✅

D.9

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Adding the three sides gives $x+(x+4)+(2x-1)=35$ . Combining like terms produces $4x+3=35$ , so $4x=32$ and $x=8$ . Checking gives side lengths $8$ , $12$ , and $15$ , whose sum is $35$ m.

Q6. A triangular frame has two equal sides of $13$ cm and a base of $10$ cm. A student says its perimeter is $13+10=23$ cm because the equal sides represent one measurement. What is the correct conclusion?

A.23 cm because equal sides are counted once

B.26 cm because only the equal sides matter

C.36 cm because all three physical sides must be counted ✅

D.46 cm because the two equal sides are doubled

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Even when two sides have equal lengths, they are still two separate sides of the boundary. Therefore, both $13$ cm sides and the $10$ cm base must be included: $13+13+10=36$ cm. Equality of lengths does not remove a side.

Q7. A triangular piece of land has sides $18$ m, $24$ m, and $30$ m. A contractor charges $15 per meter to place fencing around it. What is the total fencing cost?

A.720

B.960

C.1080 ✅

D.1200

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: First calculate the complete boundary: $18+24+30=72$ m. The contractor charges $' in math mode at position 41: … total cost is \̲(̲72\times15=\" style="color:#cc0000">15 for each meter, so the total cost is \(72\times15=\$ 1,080. The key modelling step is finding the perimeter before applying the cost per meter.

Q8. A triangular banner has perimeter $48$ cm. Two sides are $17$ cm and $14$ cm. The manufacturer can use only a third side that is a whole number. Which value is possible?

A.16 cm

B.17 cm ✅

C.18 cm

D.19 cm

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The missing side is $48-17-14=17$ cm. It also forms a valid triangle because the sum of the two shorter sides, $17+14=31$ , exceeds the third side. Thus $17$ cm satisfies both the perimeter requirement and the triangle condition.

Q9. A student calculates the perimeter of a triangle with sides $8$ cm, $10$ cm, and $12$ cm as $8+10=18$ cm, then says the $12$ -cm side is not needed because it is the longest side. What is wrong with the reasoning?

A.The longest side should be doubled

B.Only the shortest side should be included

C.Perimeter requires all three boundary sides ✅

D.The longest side is always an interior measurement

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Perimeter measures the entire outside boundary, regardless of which side is longest or shortest. All three sides contribute once, so the correct calculation is $8+10+12=30$ cm. The student's approach measures only part of the boundary and therefore cannot represent the perimeter.

Q10. A triangular plot is shown on a coordinate grid with vertices $A(1,1)$ , $B(7,1)$ , and $C(4,5)$ . The horizontal side $AB$ is $6$ units, while each sloping side is $5$ units. What is the perimeter?

A.14 units

B.15 units

C.16 units ✅

D.18 units

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: From the grid information, $AB=6$ units and the two sloping sides are each $5$ units. Adding the three boundary lengths gives $6+5+5=16$ units. The important step is interpreting the graph's side lengths rather than counting only horizontal movement.

Q11. A triangular walking route has side lengths $2x+3$ , $x+5$ , and $3x-2$ meters. Another route has a perimeter of $54$ meters. What value of $x$ makes the first route equal in perimeter to the second?

A.6

B.7

C.8 ✅

D.9

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Set the first route's perimeter equal to 54: $(2x+3)+(x+5)+(3x-2)=54$ . Combining terms gives $6x+6=54$ , so $6x=48$ and $x=8$ . The resulting sides are $19$ , $13$ , and $22$ meters, totaling $54$ .

Q12. A triangular enclosure has perimeter $40$ m. One side is $4$ m longer than another, and the third side is twice the shorter side. What are the three side lengths?

A.8 m, 12 m, 20 m

B.9 m, 13 m, 18 m ✅

C.10 m, 14 m, 16 m

D.11 m, 13 m, 16 m

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Let the shortest side be $x$ . The other sides are $x+4$ and $2x$ . Their sum is $4x+4=40$ , giving $x=9$ . Therefore, the sides are $9$ , $13$ , and $18$ m. These add to $40$ and satisfy the triangle inequality.

Q13. A triangular sign originally has sides $a$ , $b$ , and $c$ . A manufacturer cuts $3$ cm from side $a$ , adds $5$ cm to side $b$ , and leaves side $c$ unchanged. How does the perimeter change?

A.Decreases by 8 cm

B.Decreases by 2 cm

C.Increases by 2 cm ✅

D.Increases by 8 cm

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The change in total perimeter depends on the changes to all three sides, not on their original lengths. Side $a$ contributes $-3$ cm and side $b$ contributes $+5$ cm, while side $c$ contributes $0$ . The net change is $+2$ cm.

Q14. A triangle has perimeter $30$ cm and positive side lengths that are whole numbers. Which set of side lengths produces the greatest possible longest side while still forming a triangle?

8,10,12

✅

5,10,15

6,7,17

4,13,13

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: For a valid triangle, the sum of any two sides must exceed the third. Among the choices, $8,10,12$ is valid and has longest side 12. The set $5,10,15$ fails because $5+10$ is not greater than $15$ , while $6,7,17$ also fails.

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