Triangle Area
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ISEE Lower Level: Mathematics Achievement › Triangle Area
A small triangle has a certain area. A new, larger triangle is created by doubling both the base and the height of the small triangle. The area of the new triangle will be how many times as large as the area of the small triangle?
Four times as large
Eight times as large
Two times as large
Three 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.
The area of a triangular sail is 30 square feet. If the base of the sail measures 10 feet, what is the height of the sail?
20 feet
6 feet
15 feet
3 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.
A triangle has a base of 15 inches. The height corresponding to that base is 4 inches. The other two sides of the triangle measure 10 inches and 9 inches. What is the area of the triangle?
30 square inches
34 square inches
18 square inches
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.
A quilt design includes 6 identical triangular pieces of fabric. Each triangle has a base of 5 inches and a height of 4 inches. What is the total area of fabric used for all 6 pieces?
90 square inches
60 square inches
120 square inches
30 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.
The area of a triangle is 18 cm². The base and height are both whole numbers, and the base is longer than the height. Which of the following could be the length of the base?
8 cm
4 cm
6 cm
9 cm
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.
A diagonal line is drawn across a parallelogram that has a base of 14 cm and a height of 9 cm. What is the area of one of the two triangles formed by the diagonal?
126 cm²
63 cm²
23 cm²
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.
An obtuse triangle is drawn with a base of 12 cm. The height of the triangle corresponding to this base is 7 cm. The other two sides of the triangle measure 8 cm and 15 cm. What is the area of the triangle?
84 cm²
48 cm²
42 cm²
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.
A certain triangle has an area of 20 square inches. If a new triangle is drawn with double the original base but the same height, what will be the area of the new triangle?
80 square inches
20 square inches
10 square inches
40 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.
A rectangular classroom is 30 feet long and 20 feet wide. The room is divided into two equal triangular sections by a diagonal line for different activities. What is the area of one of the triangular sections?
600 square feet
100 square feet
50 square feet
300 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.
A rectangular piece of paper is 10 inches long and 8 inches wide. A triangle is cut from one corner of the paper. The cut triangle has a base of 4 inches and a height of 3 inches. What is the area, in square inches, of the paper that remains?
68 square inches
74 square inches
70 square inches
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.