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Quizzes > High School Quizzes > Mathematics

Area of a Triangle Practice Quiz

Integrate grade 6 challenges and triangle word problems

Difficulty: Moderate
Grade: Grade 6
Study OutcomesCheat Sheet
Colorful paper art promoting Triangle Area Challenge, a geometry-based trivia quiz for middle schoolers.

What is the formula to calculate the area of a triangle?
A = 2 - base - height
A = 1/2 - base - height
A = base - height
A = base + height
The area of a triangle is given by one half of the product of its base and height. This formula is essential for solving other problems involving triangles.
A triangle has a base of 8 cm and a height of 5 cm. What is its area?
40 cm²
20 cm²
13 cm²
10 cm²
Using the formula 1/2 - base - height = 1/2 - 8 - 5, the area is 20 square centimeters. This straightforward problem reinforces the basic area formula.
If the base of a triangle is doubled while the height remains the same, what happens to the area?
It quadruples
It doubles
It remains the same
It halves
The area of a triangle is directly proportional to the base when the height is constant. Doubling the base doubles the area, confirming the direct relationship in the formula.
A triangle has an area of 12 square units and a height of 4 units. What is the length of its base?
6 units
8 units
3 units
12 units
Using the area formula A = 1/2 - base - height, we rearrange it to calculate the base as (2A)/height, which gives (2 - 12)/4 = 6 units. This reinforces solving for missing dimensions.
Which of the following is true about finding the area of a triangle?
Only the base is needed
Both the base and the height are required
Only the height is needed
Either the base or the height can be used
Both the base and the corresponding height are necessary to calculate a triangle's area using the formula 1/2 - base - height. This is fundamental to ensuring accurate area computation.
A triangle has a base of 10 cm and an area of 35 square cm. What is its height?
14 cm
5 cm
3.5 cm
7 cm
Rearrange the area formula (1/2 - base - height = area) to find height as (2 - area)/base. Substituting the values gives (2 - 35)/10 = 7 cm.
If two triangles have the same height and one has a base of 6 meters while the other has a base of 9 meters, how do their areas compare?
The triangle with a 9 m base has 1.5 times the area of the triangle with a 6 m base
The triangle with a 6 m base has 1.5 times the area
They have the same area
The triangle with a 9 m base has twice the area
Since the area is proportional to the base when the height is constant, increasing the base from 6 m to 9 m increases the area by a factor of 9/6, which is 1.5. This demonstrates proportional reasoning.
A right triangle has a base of 8 units and a height of 6 units. What is its area?
14 square units
24 square units
48 square units
28 square units
Using the area formula for a triangle, 1/2 - base - height, the area is 1/2 - 8 - 6 = 24 square units. This applies to all triangles regardless of being right-angled.
A triangle's base is reduced by 20% while its height is increased by 25%. What is the overall percentage change in its area?
Decrease by 5%
No change (0%)
Increase by 20%
Increase by 5%
The new base is 0.8 times the original and the new height is 1.25 times the original, and 0.8 - 1.25 equals 1.0, meaning the area remains unchanged. This tests understanding of percentage change products.
A triangle has sides of lengths 7, 8, and 9 units. Which formula is most appropriate for finding its area?
Pythagorean theorem
Law of Sines
Heron's formula
1/2 - base - height
Heron's formula is specifically designed to compute the area of a triangle when all three sides are known. This is more appropriate than the base-height formula when the height is not obvious.
Using Heron's formula, find the area of a triangle with side lengths 6, 8, and 10.
28 square units
24 square units
20 square units
30 square units
Calculate the semi-perimeter as (6+8+10)/2 = 12, then the area is √(12 - (12-6) - (12-8) - (12-10)) = √576 = 24 square units. This is a standard application of Heron's formula.
If the area of a triangle is 50 square cm and its height is 10 cm, what is the length of the base?
10 cm
5 cm
15 cm
20 cm
Rearrange the formula to base = (2 - area) / height. Substituting the values: (2 - 50)/10 = 10 cm, which is the correct base length.
A triangle is similar to another triangle whose area is 32 square units. If the corresponding side lengths of the first triangle are twice as long as those of the second, what is the area of the first triangle?
128 square units
64 square units
96 square units
256 square units
The area of similar figures scales by the square of the side-length ratio. Doubling the side length increases the area by 2² = 4; hence, 4 - 32 = 128 square units.
A triangle has a base of 15 cm and an area of 45 square cm. What is its height?
9 cm
6 cm
3 cm
12 cm
Using the formula, height = (2 - area) / base = (2 - 45)/15 = 6 cm. This problem reinforces rearranging the area formula to solve for the unknown height.
Which of the following is true about triangles with equal bases but different heights?
The triangle with the smaller height has the larger area
Both triangles have the same area
The triangle with the greater height has the larger area
The base determines the area regardless of height
When two triangles share the same base, their areas are directly proportional to their heights. Thus, the triangle with the greater height will have the larger area.
A triangle has vertices at (0, 0), (4, 0), and (4, 3). What is the area of the triangle?
12 square units
6 square units
None of the above
7 square units
The triangle formed by these vertices is a right triangle with a base of 4 and a height of 3. Applying the area formula yields 1/2 - 4 - 3 = 6 square units.
In triangle ABC, if side AB is 9 units, side AC is 12 units, and the included angle is 30 degrees, what is the area of the triangle?
36 square units
54 square units
18 square units
27 square units
The area can be calculated using the formula 1/2 - AB - AC - sin(included angle). Since sin(30°) is 0.5, the area is 1/2 - 9 - 12 - 0.5 = 27 square units.
A triangle is inscribed in a circle with a radius of 5 units such that one of its sides is the diameter of the circle. What is the area of the triangle if the third vertex is positioned to maximize the perpendicular height from the diameter?
25 square units
20 square units
50 square units
30 square units
By Thales' theorem, the triangle is right-angled with the diameter as its hypotenuse (length 10). The maximum perpendicular height from the diameter is equal to the radius, 5 units, yielding an area of 1/2 - 10 - 5 = 25 square units.
The area of a triangle can be expressed as A = (1/2)ab sin C. For a triangle with sides a = 7, b = 10, and an included angle of 45°, what is the approximate area?
25.50 square units
22.00 square units
24.75 square units
28.00 square units
Substitute a = 7, b = 10, and sin 45° ≈ 0.7071 into the formula to compute the area: 1/2 - 7 - 10 - 0.7071 ≈ 24.75 square units. This demonstrates the use of trigonometric functions in area calculations.
A triangle's area is doubled by a transformation that scales all side lengths uniformly. What is the scaling factor applied?
1/√2
2
1.5
√2
When all sides of a triangle are scaled by a factor k, the area scales by k². To double the area, k² must equal 2, so the scaling factor is √2.
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Study Outcomes

  1. Apply the triangle area formula to solve real-world problems.
  2. Analyze given dimensions to calculate the area of various triangles.
  3. Interpret geometric diagrams to identify relevant sides and heights.
  4. Validate solutions through step-by-step reasoning and checking calculations.

Area Triangle Worksheet Cheat Sheet

  1. Fundamental Area Formula - Every triangle's area can be found with the trusty Area = ½ × base × height formula. It's like the Swiss Army knife of triangle area methods - always ready to go! For example, a triangle with an 8 cm base and a 5 cm height has an area of 20 cm². Learn more
    2. GeeksforGeeks: Area of Triangle
  2. Heron's Formula - When you only know the three sides, Heron's Formula swoops in: Area = √[s(s - a)(s - b)(s - c)], where s = (a + b + c)/2. No height? No problem - Heron's got your back! Great for scalene triangles when measuring altitude feels like a scavenger hunt. Learn more
    3. Wikipedia: Heron's Formula
  3. Equilateral Triangle Area - With all sides equal, an equilateral triangle's area is Area = (√3 / 4) × side². It's geometry's version of a perfectly symmetrical ice cream cone! If each side is 6 cm, your area is about 15.59 cm² - sweet and precise. Learn more
    4. Wikipedia: Equilateral Triangle
  4. Triangle Inequality Theorem - To even form a triangle, any two sides' lengths must sum to more than the third side. Think of it like three friends deciding if they can all fit on one bench - two must be small enough to share with the third! This rule keeps your triangle legit. Learn more
    5. Wikipedia: Triangle Inequality
  5. Triangle Types Properties - Triangles come in three flavors: equilateral (all sides and angles equal), isosceles (two equal sides/angles) and scalene (all different). Knowing these traits helps you pick the fastest area method and nail tricky problems. It's like matching each triangle to its perfect formula. Learn more
    6. GeeksforGeeks: Types of Triangles
  6. Angle Sum Property - No matter the shape, a triangle's interior angles always add up to 180°. It's the unbreakable promise of triangles - like three best friends always summing up to a perfect group. Use this to find missing angles before jumping into area calculations. Learn more
    7. GeeksforGeeks: Properties of Triangle
  7. Exterior Angle Property - An exterior angle equals the sum of the two opposite interior angles. It's like sneaking a peek at your friend's test answers - calculated but perfectly allowed in geometry land! Great for proofs and bonus problem-solving. Learn more
    8. GeeksforGeeks: Properties of Triangle
  8. Shoelace Formula - Got coordinates for your triangle's vertices? Plug them into the Shoelace Formula: Area = ½ |x₝(y₂ - y₃) + x₂(y₃ - y₝) + x₃(y₝ - y₂)| and voilà - area by arithmetic magic! Perfect for coordinate geometry fanatics. Learn more
    9. Wikipedia: Shoelace Formula
  9. Practice Makes Perfect - Drill yourself with problems using base/height, three sides, or coordinates. The more you mix and match methods, the more you'll unlock your inner triangle ninja! Plus, you'll breeze through exam questions like a pro. Learn more
    10. GeeksforGeeks: Area Practice
  10. Don't Forget Your Units - Always label your result in square units (cm², m², etc.). It's like signing your artwork - you want everyone to know your answer is complete and correct! Units keep your math crystal clear. Learn more
    11. Math.net: Area of a Triangle
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