ISEE Lower Level Mathematics Achievement Quiz: Triangle Area

What this quiz covers

This quiz focuses on Triangle Area, giving you a quick way to practice the rules, question types, and explanations that matter most for ISEE Lower Level Mathematics Achievement.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. Two times as large
2. Three times as large
3. Four times as large (correct answer)
4. Eight times as large

Explanation: Let the original area be A = (1/2) × b × h. The new triangle has a base of 2b and a height of 2h. Its area is A_new = (1/2) × (2b) × (2h) = (1/2) × 4 × b × h = 4 × [(1/2) × b × h]. So, the new area is 4 times the original area.

Question 2

1. 3 feet
2. 6 feet (correct answer)
3. 15 feet
4. 20 feet

Explanation: The formula for the area of a triangle is Area = (1/2) × base × height. We can rearrange this to find the height: height = (2 × Area) / base. Plugging in the given values: height = (2 × 30 square feet) / 10 feet = 60 / 10 = 6 feet.

Question 3

1. 18 square inches
2. 30 square inches (correct answer)
3. 34 square inches
4. 60 square inches

Explanation: To find the area of a triangle, you only need the length of a base and the corresponding height. In this problem, the base is 15 inches and the height is 4 inches. The lengths of the other two sides (10 inches and 9 inches) are extra information not needed for the area calculation. Area = (1/2) × base × height = (1/2) × 15 inches × 4 inches = 30 square inches.

Question 4

1. 30 square inches
2. 60 square inches (correct answer)
3. 90 square inches
4. 120 square inches

Explanation: First, find the area of one triangular piece: Area = (1/2) × base × height = (1/2) × 5 inches × 4 inches = 10 square inches. Since there are 6 identical pieces, the total area is 6 × 10 square inches = 60 square inches.

Question 5

1. 4 cm
2. 6 cm
3. 8 cm
4. 9 cm (correct answer)

Explanation: The area formula is (base × height) / 2 = 18. This means base × height = 36. We need to find pairs of whole number factors of 36 where the base is greater than the height. The pairs are (36, 1), (18, 2), (12, 3), and (9, 4). The possible values for the base are 36, 18, 12, and 9. Of the choices given, only 9 cm is a possible length for the base.

Question 6

1. 23 cm²
2. 63 cm² (correct answer)
3. 126 cm²
4. 252 cm²

Explanation: A diagonal divides a parallelogram into two identical triangles. First, find the area of the parallelogram: Area = base × height = 14 cm × 9 cm = 126 cm². The area of one of the triangles is half the area of the parallelogram. So, the area of one triangle is 126 cm² / 2 = 63 cm².

Question 7

1. 42 cm² (correct answer)
2. 48 cm²
3. 84 cm²
4. 90 cm²

Explanation: The area of any triangle is calculated with the formula Area = (1/2) × base × height. The lengths of the other sides (8 cm and 15 cm) are extra information and not needed. Using the given base and height: Area = (1/2) × 12 cm × 7 cm = 6 cm × 7 cm = 42 cm².

Question 8

1. 10 square inches
2. 20 square inches
3. 40 square inches (correct answer)
4. 80 square inches

Explanation: The area of a triangle is directly proportional to its base. If the base is doubled and the height remains the same, the area will also double. Therefore, the new area will be 2 × 20 square inches = 40 square inches.

Question 9

1. 50 square feet
2. 100 square feet
3. 300 square feet (correct answer)
4. 600 square feet

Explanation: The diagonal line divides the rectangle into two congruent right triangles. The area of the entire rectangle is 30 feet × 20 feet = 600 square feet. The area of one triangular section is half of the rectangle's area, which is 600 / 2 = 300 square feet. Alternatively, one triangle has a base of 30 feet and a height of 20 feet, so its area is (1/2) × 30 × 20 = 300 square feet.

Question 10

1. 68 square inches
2. 70 square inches
3. 74 square inches (correct answer)
4. 77 square inches

Explanation: This is a two-step problem. First, find the area of the original rectangular paper: Area = length × width = 10 inches × 8 inches = 80 square inches. Second, find the area of the triangle that was cut off: Area = (1/2) × base × height = (1/2) × 4 inches × 3 inches = 6 square inches. Finally, subtract the area of the triangle from the area of the rectangle to find the remaining area: 80 - 6 = 74 square inches.

Question 11

1. 24 cm² (correct answer)
2. 30 cm²
3. 40 cm²
4. 48 cm²

Explanation: In a right triangle, the two shorter sides that form the right angle are the base and the height. The longest side (the hypotenuse) is not used to calculate the area. Here, the base and height are 6 cm and 8 cm. Area = (1/2) × base × height = (1/2) × 6 cm × 8 cm = (1/2) × 48 cm² = 24 cm².

Question 12

1. 15,000 square yards (correct answer)
2. 20,000 square yards
3. 30,000 square yards
4. 400 square yards

Explanation: To find the area of the triangle, use the formula Area = (1/2) × base × height. Substitute the given values: Area = (1/2) × 300 yards × 100 yards = (1/2) × 30,000 square yards = 15,000 square yards.

Question 13

1. 25 square inches
2. 120 square inches
3. 60 square inches
4. 30 square inches (correct answer)

Explanation: When you encounter problems involving isosceles triangles split by their height, remember that this height line creates two congruent right triangles and acts as a line of symmetry. Let's work through this step-by-step. The original isosceles triangle has a base of 10 inches and height of 12 inches. When the height is drawn from the top vertex to the base, it splits the triangle into two identical right triangles. Each right triangle has a base of 5 inches (half of the original 10-inch base) and the same height of 12 inches. To find the area of one right triangle, use the formula: Area = $\frac{1}{2} \times \text{base} \times \text{height}$ Area = $\frac{1}{2} \times 5 \times 12 = 30$ square inches Looking at the wrong answers: Choice A (25 square inches) likely comes from incorrectly using 5 × 5 instead of 5 × 12, perhaps confusing the base measurement. Choice B (120 square inches) results from calculating 10 × 12 without applying the $\frac{1}{2}$ factor—this would be the area if the shape were a rectangle. Choice C (60 square inches) is the area of the entire original triangle ($\frac{1}{2} \times 10 \times 12$), not just one of the smaller triangles. Remember this key insight: when an isosceles triangle's height splits it into two right triangles, each small triangle has half the base of the original triangle but the same height. Always double-check whether you're finding the area of one piece or the whole figure.

Question 14

1. 14.2 m²
2. 46 m²
3. 28.4 m²
4. 23 m² (correct answer)

Explanation: When you encounter a triangle area problem, you need to apply the fundamental triangle area formula: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$. Let's work through this step-by-step. You have a base of 9.2 meters and a height of 5 meters. Substituting into the formula: $\text{Area} = \frac{1}{2} \times 9.2 \times 5 = \frac{1}{2} \times 46 = 23 \text{ m}^2$ Now let's examine why the other answers are incorrect. Choice A (14.2 m²) represents a common error where students add the base and height instead of multiplying them. Choice B (46 m²) shows what happens when you forget the crucial $\frac{1}{2}$ in the triangle formula—you're essentially calculating the area of a rectangle with the same base and height. Choice C (28.4 m²) appears to come from incorrectly doubling one of the measurements before applying the formula, possibly calculating $\frac{1}{2} \times 9.2 \times 5 \times 2$. The correct answer is D) 23 m². Remember this key strategy: triangles always have exactly half the area of a rectangle with the same base and height. If you ever forget the triangle formula, just think "rectangle area divided by 2." Also, always double-check that your final answer includes the proper units (m² for area) and seems reasonable compared to the given measurements.

Question 15

1. 20 cm²
2. 25 cm²
3. 50 cm² (correct answer)
4. 100 cm²

Explanation: First, find the area of the large square: Area = side × side = 10 cm × 10 cm = 100 cm². The two diagonals divide the square into four identical triangles, so the area of the square is split evenly among them. The area of one small triangle is 100 cm² / 4 = 25 cm². The combined area of two of these triangles is 2 × 25 cm² = 50 cm².

Question 16

1. 18 square inches
2. 36 square inches
3. 216 square inches (correct answer)
4. 432 square inches

Explanation: First, make sure both measurements are in the same unit. Since the answer should be in square inches, convert the base from feet to inches: 2 feet × 12 inches/foot = 24 inches. Now, calculate the area with base = 24 inches and height = 18 inches. Area = (1/2) × 24 inches × 18 inches = 12 × 18 = 216 square inches.

Question 17

1. 5.5 inches
2. 88 inches
3. 22 inches
4. 11 inches (correct answer)

Explanation: When you see a question about finding the height of a triangle given its area and base, you're working with the triangle area formula: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$. To find the height, you need to rearrange this formula. Since you know the area is 44 square inches and the base is 8 inches, substitute these values: $44 = \frac{1}{2} \times 8 \times \text{height}$. Simplifying the right side: $44 = 4 \times \text{height}$. To isolate the height, divide both sides by 4: $\text{height} = \frac{44}{4} = 11$ inches. Let's examine why the other answers are incorrect. Choice A (5.5 inches) would give you an area of $\frac{1}{2} \times 8 \times 5.5 = 22$ square inches—exactly half the correct area. This suggests you might have forgotten to multiply by the full base or made an arithmetic error. Choice B (88 inches) is double the given area, which could result from forgetting the $\frac{1}{2}$ in the formula and calculating $\text{height} = \frac{44}{8} \times 2$. Choice C (22 inches) would give an area of $\frac{1}{2} \times 8 \times 22 = 88$ square inches, suggesting you might have confused the area value somewhere in your calculation. Remember this key strategy: when working with triangle area problems, always write out the formula first, then carefully substitute the known values and solve for the unknown. Double-check your arithmetic by plugging your answer back into the original formula.

Question 18

1. 20 square meters
2. 40 square meters (correct answer)
3. 80 square meters
4. 120 square meters

Explanation: First, find the area of the larger garden: Area = (1/2) × 16 m × 10 m = 80 m². Next, find the area of the smaller garden: Area = (1/2) × 10 m × 8 m = 40 m². Finally, find the difference between the two areas: 80 m² - 40 m² = 40 m².

Question 19

1. $150
2. $300 (correct answer)
3. $450
4. $600

Explanation: This is a two-step problem. First, calculate the area of the triangular section: Area = (1/2) × base × height = (1/2) × 20 feet × 15 feet = 10 × 15 = 150 square feet. Second, calculate the total cost by multiplying the area by the cost per square foot: Cost = 150 square feet × $2/square foot =$300.

Question 20

1. The area of Triangle A is greater than the area of Triangle B.
2. The area of Triangle B is greater than the area of Triangle A.
3. The areas of Triangle A and Triangle B are equal. (correct answer)
4. The area of Triangle B is 2 square meters greater than Triangle A.

Explanation: First, calculate the area of Triangle A: Area = (1/2) × 10 m × 6 m = 30 m². Next, calculate the area of Triangle B: Area = (1/2) × 12 m × 5 m = 30 m². Since both triangles have an area of 30 square meters, their areas are equal.