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📝 Area of triangle formula (15 MCQs)

📖 From Digital SAT Algebra • 3. Mathematical Models in Algebra • 15 questions available

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What is Area of triangle formula?

Definition:
The area of a triangle is the measure of the region enclosed by the triangle, calculated using the formula $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$ , where the base is any side of the triangle, and the height is the perpendicular distance from the opposite vertex to that base, representing the space inside the triangle in square units.

Working:
To find the area, identify a base (any side) and measure the corresponding height (the perpendicular distance to the base from the opposite vertex), then multiply the base by the height and divide by 2, so if $b$ is the base and $h$ is the height, the area is $A = \frac{1}{2}bh$ , and for right triangles, the two legs can serve as base and height.

Example:
If a triangle has a base of 10 cm and a height of 6 cm, its area is $A = \frac{1}{2} \times 10 \times 6 = \frac{1}{2} \times 60 = 30$ cm $^2$ , meaning the triangle occupies 30 square centimeters of space.

Reason:
The area formula is fundamental in geometry and is extensively used in architecture, construction, land measurement, and design, helping calculate materials needed for floors, roofs, and various triangular surfaces.

Easy

Medium

Hard

📝 All Area of triangle formula MCQs

Q1. A triangle has a base of $14$ cm and a perpendicular height of $9$ cm. What is its area?

A.54 cm²

B.63 cm² ✅

C.126 cm²

D.252 cm²

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The area is found by multiplying the base and perpendicular height and then taking half of the product. Thus $A=\frac{1}{2}(14)(9)=63$ cm². The height must be perpendicular to the selected base.

Q2. Which change always doubles the area of a triangle while keeping the other dimension unchanged?

A.Double the base ✅

B.Double both base and height

C.Increase the base by 50%

D.Halve the height

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Since area depends on one-half the product of base and height, doubling either one while keeping the other fixed doubles the product. Therefore, doubling the base alone doubles the triangle's area.

Q3. Two triangles have the same base length. Triangle P has height $8$ cm, while Triangle Q has height $12$ cm. How should their areas be compared?

A.P has twice Q's area

B.Q has

1.5

times P's area ✅

C.Q has twice P's area

D.Their areas are equal

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: With equal bases, triangle areas are directly proportional to their perpendicular heights. The ratio is $12:8=3:2$ , so Triangle Q has $1.5$ times the area of Triangle P.

Q4. A student says that a triangle with base $10$ cm and height $6$ cm has area $60$ cm² because $10\times6=60$ . What is the best evaluation?

A.Correct because triangles use the full base-height product

B.Incorrect because the product must be doubled

C.Incorrect because the product must be halved ✅

D.Correct only for equilateral triangles

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The student's multiplication gives the area of a corresponding rectangle, not the triangle. A triangle with the same base and perpendicular height occupies half that rectangle, so the correct area is $\frac{1}{2}(10)(6)=30$ cm².

Q5. A triangular garden has area $96$ m² and a base of $16$ m. A landscaper wants to determine the perpendicular height before ordering materials. What height should be used?

A.6 m

B.8 m

C.10 m

D.12 m ✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: Starting with $A=\frac{1}{2}bh$ , substitute $96$ for area and $16$ for base. Then $96=8h$ , giving $h=12$ m. The calculation shows why the perpendicular height, rather than a slanted side, is required.

Q6. A designer keeps the area of a triangular sign fixed at $120$ cm². If the base changes from $20$ cm to $24$ cm, what happens to the required height?

A.It increases from 12 cm to 14.4 cm

B.It decreases from 12 cm to 10 cm ✅

C.It remains 12 cm

D.It decreases from 12 cm to 10.0 cm only if the triangle is right-angled

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: For fixed area, the product $bh$ must remain constant. Initially $120=\frac12(20)h$ , so $h=12$ cm. With base $24$ cm, $120=\frac12(24)h$ , giving $h=10$ cm. Increasing the base therefore reduces the required height.

Q7. A triangular plot has a base of $30$ m and a perpendicular height of $18$ m. A path divides it into two triangles with areas $90$ m² and $180$ m². What percentage of the original plot is the smaller region?

A.0.16699999999999998

B.0.25

C.0.33299999999999996 ✅

D.0.5

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The entire plot has area $\frac12(30)(18)=270$ m². The smaller region is $90$ m², so its percentage is $\frac{90}{270}\times100=33.3\%$ . Therefore, the correct choice is 33.3%, not 25%.

Q8. A student calculates the area of a triangle with base $15$ cm and height $8$ cm as $120$ cm². Another student calculates $60$ cm². Which reasoning is correct, and why?

A.The first student is correct because triangles use

bh

B.The second student is correct because a triangle is half of a corresponding rectangle ✅

C.Both are correct under different units

D.Neither is correct because height must equal base

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The base-height product is $15\times8=120$ cm², but the triangle occupies half the corresponding rectangle. Therefore, the correct area is $60$ cm². The second student's reasoning properly accounts for the triangular shape.

Q9. A triangular frame has a fixed base of $18$ cm. Its height is increased from $7$ cm to $11$ cm. By how much does its area increase?

A.18 cm²

B.36 cm² ✅

C.54 cm²

D.72 cm²

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The original area is $\frac12(18)(7)=63$ cm², while the new area is $\frac12(18)(11)=99$ cm². The increase is $99-63=36$ cm², so the correct answer is 36 cm².

Q10. Consider a graph showing three triangles that all have base endpoints on the same horizontal line from $x=0$ to $x=10$ , while their third vertices have heights $4$ , $7$ , and $9$ units above the line. Which triangle has the greatest area?

A.The triangle with height 4

B.The triangle with height 7

C.The triangle with height 9 ✅

D.All have the same area

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: All three triangles have the same base length of $10$ units, so their areas depend directly on their heights. The triangle reaching $9$ units has the largest base-height product and therefore the greatest area.

Q11. A coordinate graph shows triangle A with base $12$ units and perpendicular height $5$ units, and triangle B with base $8$ units and perpendicular height $9$ units. Which conclusion is correct?

A.Triangle A has greater area

B.Triangle B has greater area ✅

C.Both have the same area

D.The areas cannot be compared without side lengths

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Triangle A has area $\frac12(12)(5)=30$ square units. Triangle B has area $\frac12(8)(9)=36$ square units. Therefore, Triangle B is larger in area even though its base is shorter, because its height is sufficiently greater.

Q12. A triangular banner must have area $150$ cm². A supplier offers material with a base of $25$ cm. The designer first calculates a height of $6$ cm but then realizes the triangle must use the full rectangular material area. What is the correct perpendicular height for the triangle?

A.6 cm

B.8 cm

C.10 cm

D.12 cm ✅

💡 Difficulty: hard | ✅ Correct: D

📖 Explanation: Using $150=\frac12(25)h$ , multiply both sides by $2$ to obtain $300=25h$ . Therefore $h=12$ cm, not 10 cm. The full rectangular material area should not be confused with the triangular region's area.

Q13. A triangular field and a parallelogram-shaped field have the same base $20$ m and the same perpendicular height $15$ m. How does the triangle's area compare with the parallelogram's area?

A.The triangle is equal to it

B.The triangle is half of it ✅

C.The triangle is twice it

D.The triangle is one-fourth of it

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The triangular area is $\frac12(20)(15)=150$ m², while the parallelogram area is $20\times15=300$ m². Thus, with identical base and perpendicular height, the triangle occupies exactly half the area.

Q14. A triangular logo has area $72$ cm². Its designer considers changing the base and height simultaneously. Which pair of new dimensions preserves the same area?

A.Base

12

cm and height

12

B.Base

18

cm and height

8

cm ✅

C.Base

24

cm and height

8

D.Base

16

cm and height

10

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: For the area to remain $72$ cm², the new base-height product must equal $144$ . For $18$ cm and $8$ cm, the product is $144$ , giving $\frac12(18)(8)=72$ cm². The other pairs produce different areas.

Q15. A triangular piece of land has a fixed perimeter of $30$ m. One possible triangle has base $10$ m and perpendicular height $6$ m. A student claims that any triangle with the same perimeter must have the same area. Which response is best?

A.The claim is always true

B.The claim is true only for right triangles

C.The claim is false because area also depends on the triangle's shape and corresponding height ✅

D.The claim is true whenever the base is unchanged

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: Equal perimeter does not guarantee equal area because different side arrangements can produce different heights and shapes. Since area depends on the product of a chosen base and its perpendicular height, triangles with equal perimeter may have different areas.

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