How to find the area of a triangle - ISEE Middle Level Math

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Question

What is the area of a triangle with a base of and a height of ?

Answer

The formula for the area of a triangle is $\dpi{100} Area=\frac{1}{2}\times base\times height$ .

Plug the given values into the formula to solve:

$\dpi{100} Area=\frac{1}{2}\times 12\times 3$

$\dpi{100} Area=\frac{1}{2}\times 36$

$\dpi{100} Area=18$

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Question

A triangle has a base of $\small 43$ and an area of $\small 2042.5$ . What is the height?

Answer

The area of a triangle is found by multiplying the base by the height and dividing by two:

$\small \frac{b\cdot h}{2}$

In this problem we are given the base, which is $\small 43$ , and the area, which is $\small 2042.5$ . First we write an equation using $\small h$ as our variable.

$\small \frac{43\cdot h}{2}=2042.5$

To solve this equation, first multply both sides by $\small 2$ , becuase multiplication is the opposite of division and therefore allows us to eliminate the $\small 2$ .

The left-hand side simplifies to:

$\small \frac{43\cdot h}{2}\cdot 2=43\cdot h$

The right-hand side simplifies to:

$\small 2042.5\cdot 2=4085$

So our equation is now:

$\small \small 43\cdot h=4085$

Next we divide both sides by $\small \small 43$ , because division is the opposite of multiplication, so it allows us to isolate the variable by eliminating $\small \small 43$ .

$\small \frac{43\cdot h}{43}= h$

$\small \frac{4085}{43}=95$

$\small h=95$

So the height of the triangle is $\small 95$ .

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Question

Bill paints a triangle on his wall that has a base parallel to the ground that runs from one end of the wall to the other. If the base of the wall is 8 feet, and the triangle covers 40 square feet of wall, what is the height of the triangle?

Answer

In order to find the area of a triangle, we multiply the base by the height, and then divide by 2.

$\small \frac{b\cdot h}{2}$

In this problem we are given the base and the area, which allows us to write an equation using $\small h$ as our variable.

$\small \frac{8\cdot h}{2}=40$

Multiply both sides by two, which allows us to eliminate the two from the left side of our fraction.

The left-hand side simplifies to:

$\small \frac{8\cdot h}{2}\cdot 2=8\cdot h$

The right-hand side simplifies to:

$\small 40\cdot 2=80$

Now our equation can be rewritten as:

$\small 8\cdot h=80$

Next we divide by 8 on both sides to isolate the variable:

$\small \frac{8\cdot h}{8}=h$

$\small \frac{80}{8}=10$

$\small h=10$

Therefore, the height of the triangle is $\small 10 : feet$ .

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Question

Note: Figure NOT drawn to scale.

The above triangle has area 36 square inches. If $x = 4.5 \textrm{ in}$ , then what is ?

Answer

The area of a triangle is one half the product of its base and its height - in the above diagram, that means

$A = \frac{1}{2}xy$ .

Substitute , and solve for .

$\frac{1}{2} \cdot 4.5 \cdot y = 36$

$2.25 y \div 2.25= 36\div 2.25$

$y = 16 \textrm{ in}$

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Question

Please use the following shape for the question.

What is the area of this shape?

Answer

From this shape we are able to see that we have a square and a triangle, so lets split it into the two shapes to solve the problem. We know we have a square based on the 90 degree angles placed in the four corners of our quadrilateral.

Since we know the first part of our shape is a square, to find the area of the square we just need to take the length and multiply it by the width. Squares have equilateral sides so we just take 5 times 5, which gives us 25 inches squared.

We now know the area of the square portion of our shape. Next we need to find the area of our right triangle. Since we know that the shape below the triangle is square, we are able to know the base of the triangle as being 5 inches, because that base is a part of the square's side.

To find the area of the triangle we must take the base, which in this case is 5 inches, and multipy it by the height, then divide by 2. The height is 3 inches, so 5 times 3 is 15. Then, 15 divided by 2 is 7.5.

We now know both the area of the square and the triangle portions of our shape. The square is 25 inches squared and the triangle is 7.5 inches squared. All that is remaining is to added the areas to find the total area. Doing this gives us 32.5 inches squared.

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Question

The hypotenuse of a right triangle is 25 inches; it has one leg 15 inches long. Give its area in square feet.

Answer

The area of a right triangle is half the product of the lengths of its legs, so we need to use the Pythagorean Theorem to find the length of the other leg. Set :

$a^{2} = c^{2} - b^{2}$

$a^{2} = 25^{2} - 15^{2}$

$a^{2} = 625-225$

$a^{2} =400$

$a = \sqrt{400} = 20$

The legs are 15 and 20 inches long. Divide both dimensions by 12 to convert from inches to feet:

$20 \div 12 = \frac{5}{3}$ feet

$15 \div 12 = \frac{5}{4}$ feet

Now find half their product:

$A = \frac{1}{2} \times \frac{5}{3}\times \frac{5}{4} = \frac{25}{24} = 1 \frac{1}{24}$ square feet

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Question

The hypotenuse of a right triangle is feet; it has one leg feet long. Give its area in square inches.

Answer

The area of a right triangle is half the product of the lengths of its legs, so we need to use the Pythagorean Theorem to find the length of the other leg. Set :

$a^{2} = c^{2} - b^{2}$

$a^{2} = 39 ^{2} - 36^{2}$

$a^{2} = 1,521 - 1,296$

$a^{2} = 225$

$a = \sqrt{225} = 15$

The legs have length and feet; multiply both dimensions by to convert to inches:

$15 \times 12 = 180$ inches

$36 \times 12 = 432$ inches.

Now find half the product:

$A = \frac{1}{2} \cdot 180 \cdot 432 = 38,880\ in^{2}$

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Question

What is the area of the triangle?

Answer

Area of a triangle can be determined using the equation:

$\small A=\frac{1}{2}bh$

$\small A=\frac{1}{2}(14)(5)=35$

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Question

The three angles of a triangle are labeled , , and . If is $97^{\circ}$ , what is the value of ?

Answer

Given that the three angles of a triangle always add up to 180 degrees, the following equation can be used:

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Question

A triangle has a height of 9 inches and a base that is one third as long as the height. What is the area of the triangle, in square inches?

Answer

The area of a triangle is found by multiplying the base times the height, divided by 2.

$\text{Area} =\frac{\text{base}\times \text{height}}{2}$

Given that the height is 9 inches, and the base is one third of the height, the base will be 3 inches.

$b=h\div3=9\div3=3$

We now have both the base (3) and height (9) of the triangle. We can use the equation to solve for the area.

$\text{Area} =\frac{9\times 3}{2}$

$\text{Area} =\frac{27}{2}$

The fraction cannot be simplified.

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Question

What is the area (in square feet) of a triangle with a base of feet and a height of feet?

Answer

The area of a triangle is found by multiplying the base times the height, divided by .

$Area =\frac{base\cdot height}{2}$

$Area =\frac{4\cdot 7}{2}$

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Question

In an equilateral triangle, which of the following is NOT true?

Answer

In an equilateral triangle, all sides and angles are equal. All the angles equal 60 degrees, so there is a 60 degree angle.

Therefore, the answer choice, “There is a 90 degree angle” is not true and is the correct answer choice.

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Question

If the angles of a triangle are equal to , , and , what is the value of ?

Answer

Given that there are 180 degrees in a triangle,

Thus, 30 is the correct answer.

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Question

Note: Figure NOT drawn to scale.

What percent of the above figure is green?

Answer

The area of the entire rectangle is the product of its length and width, or

$120 \times 50 = 6,000$ .

The area of the right triangle is half the product of its legs, or

$\frac{1}{2} \times 40 \times 80 = 1,600$

The area of the green region is therefore the difference of the two, or

.

The green region is therefore

$\frac{4,400}{6,000} \times 100 = 73 \frac{1}{3} %$

of the rectangle.

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Question

Note: Figure NOT drawn to scale.

Refer to the above diagram. Give the ratio of the area of the green region to that of the white region.

Answer

The area of the entire rectangle is the product of its length and width, or

$120 \times 50 = 6,000$ .

The area of the right triangle is half the product of its legs, or

$\frac{1}{2} \times 40 \times 80 = 1,600$

The area of the green region is therefore the difference of the two, or

.

The ratio of the area of the green region to that of the white region is

$\frac{4,400}{1,600} = \frac{4,400 \div 400}{1,600\div 400} = \frac{11}{4}$

That is, 11 to 4.

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Question

You have two traingular gardens next to each other. They both have a base of and a height of . What is the total area?

Answer

The area of a triangle is

$\frac{1}{2}baseheight$

and since there are two identical traingles, them put together will just be

.

So your answer will just be .

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Question

The above figure depicts Square with perimeter 240. , , and are the midpoints of $\overline{AD}$ , $\overline{BC}$ , and $\overline{AB}$ , respectively.

Give the area of Polygon .

Answer

Square has perimeter 240, so the length of each side is one fourth of this, or

$\frac{1}{4} \cdot 240 = 60$ .

Segment $\overline{XY}$ , as seen below, divides Polygon into two figures:

One figure is $\bigtriangleup XZY$ , with base and height 60 and 30, respectively. Its area is half their product, or

$A_{1} = \frac{1}{2} \cdot 60 \cdot 30 = 900$

The other figure is Rect , whose length and width are 60 and 30, respectively. Its area is their product, or

$A_{2} = \cdot 60 \cdot 30 = 1,800$

Add these areas:

$A_{1} +A_{2} = 900 + 1,800 = 2,700$ .

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Question

Give the area of the above triangle.

Answer

The area of a right triangle is half the product of the lengths of its legs, which here are 25 and 60. So

$A = \frac{1}{2} \cdot 25 \cdot 60 = 750$

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Question

Find the area of a triangle with a base of 10cm and a height that is half the base.

Answer

To find the area of a triangle, we will use the following formula:

$\text{area of triangle} = \frac{1}{2} \cdot b \cdot h$

Now, we know the base has a length of 10cm. We also know the height is half the base. Therefore, the height is 5cm. Knowing this, we can substitute into the formula. We get

$\text{area of triangle} = \frac{1}{2} \cdot 10\text{cm} \cdot 5\text{cm}$

$\text{area of triangle} = \frac{1}{2} \cdot 50\text{cm}^2$

$\text{area of triangle} = 25\text{cm}^2$

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Question

The roof of a skyscraper forms a right triangle with equal arms of length 50 meters. Find the area of the roof of the skyscraper.

Answer

The roof of a skyscraper forms a right triangle with equal arms of length 50 meters. Find the area of the roof of the skyscraper.

To find the area of a triangle, use the following:

$A_{triangle}=\frac{1}{2}bh$

Where b and h are the base and height.

In this case, we are told that the base and height are both 50 meters, thus, the perimeter will be:

$A_{triangle}=\frac{1}{2}50m50m=\frac{1}{2}2500m^2=1250m^2$

So our answer is:

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