How to find the area of a figure - HSPT Math

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Question

What is the area of a square with a side length of 4?

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Answer

The area of a square is represented by the equation $\dpi{100} Area = side^{2}$ .

Therefore the area of this square is $\dpi{100} 4^{2}=16$ .

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Question

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

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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

The area of the square is 81. What is the sum of the lengths of three sides of the square?

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Answer

A square that has an area of 81 has sides that are the square root of 81 (side2 = area for a square). Thus each of the four sides is 9. The sum of three of these sides is .

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Question

What is the area of a triangle with a base of 5 and a height of 20?

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Answer

When searching for the area of a triangle we are looking for the amount of the space enclosed by the triangle.

The equation for area of a triangle is $\frac{1}{2}(base)(height)= Area: of: Triangle$

Plug in the numbers for base and height into the equation yielding $\frac{1}{2}(5)(20)= Area: of: Triangle$

Then multiply the numbers together to arrive at the answer which is .

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Question

What is the area of a rectangle with length $14\ cm$ and width $12\ cm$ ?

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Answer

The formula for the area, , of a rectangle when we are given its length, , and width, , is .

To calculate this area, just multiply the two terms.

$A = 14\ cm * 12\ cm = 168; cm^{2}$

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Question

A trapezoid has height 32 inches and bases 25 inches and 55 inches. What is its area?

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Answer

Use the following formula, with :

$A = \frac{1}{2} (B+b)h = \frac{1}{2} (55+25) \cdot 32 = 1,280$

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Question

Square A has sides measuring 5 meters. A second square, Square B, has sides that are 2 meters longer than the sides of Square A. What is the difference in area of Square A and Square B?

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Answer

The area of Square A is 5 * 5, or 25 m2.

Since each of Square B's sides is 2 meters longer, the sides measure 7 meters. Therefore, the area of square B is 49 m2.

Subtract to find the difference in areas: $49-25=24\ m^2$

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Question

A trapezoid has a height of inches and bases measuring inches and inches. What is its area?

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Answer

Use the following formula, with :

$A = \frac{1}{2} (B+b)h = \frac{1}{2} (36+24) \cdot 25 = 750$

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Question

What is the area of a square with perimeter 64 inches?

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Answer

The perimeter of a square is four times its sidelength, so a square with perimeter 64 inches has sides with length 16 inches. Use the area formula:

$A = s^{2} = 16^{2} = 256$

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Question

What is the area of the trapezoid?

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Answer

To find the area of a trapezoid, multiply the sum of the bases (the parallel sides) by the height (the perpendicular distance between the bases), and then divide by 2.

$A = \frac{1}{2} \cdot (7 + 15) \cdot 9 = \frac{1}{2} \cdot 22 \cdot 9 = 99 \textrm{ m}^2$

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Question

What is the area of the above trapezoid?

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Answer

To find the area of a trapezoid, multiply one half (or 0.5, since we are working with decimals) by the sum of the lengths of its bases (the parallel sides) by its height (the perpendicular distance between the bases). This quantity is

$A = 0.5 \cdot (7.2 + 13.4) \cdot 10.6 =0.5 \cdot 20.6 \cdot 10.6 = 109.18\textrm{ m}^{2}$

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Question

What is the area of a triangle with a base of 22 cm and a height of 9 cm?

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Answer

Use the area of a triangle formula

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

Plug in the base and height. This gives you $99\ cm^{2}$ .

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Question

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

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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?

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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 ?

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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

Order the following from least area to greatest area:

Figure A: A rectangle with length 10 inches and width 14 inches.

Figure B: A square with side length 1 foot.

Figure C: A triangle with base 16 inches and height 20 inches.

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Answer

Figure A has area $10 \times 14 = 140$ square inches.

Figure B has area $12 \times 12 = 144$ square inches, 1 foot being equal to 12 inches.

Figure C has area $\frac{1}{2} \times 16 \times 20 = 160$ square inches.

The figures, arranged from least area to greatest, are A, B, C.

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Question

The area of the following rectangle is . Solve for .

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Answer

The area of a rectangle can be found by multiplying the length by the width.

$A=l\times w$

$48=3x\times x=3x^2$

$\frac{48}{3}=\frac{3x^2}{3}$

$\sqrt{16}=\sqrt{x^2}$

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Question

Find the area of the following parallelogram:

Note: The formula for the area of a parallelogram is $A=b\times h$ .

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Answer

The base of the parallelogram is 10, while the height is 5.

$A=b\times h$

$A=10\times5=50: in^2$

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Question

Give the surface area of the above box in square inches.

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Answer

Use the surface area formula, substituting :

$A = 2 \cdot 22 \cdot 12 + 2 \cdot 22 \cdot 12 + 2\cdot 12\cdot 12$

square inches

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Question

The above diagram depicts a rectangle with isosceles triangle $\Delta ECM$ . If is the midpoint of $\overline{CT}$ , and the area of the orange region is , then what is the length of one leg of $\Delta ECM$ ?

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Answer

The length of a leg of $\Delta ECM$ is equal to the height of the orange region, which is a trapezoid. Call this length/height .

Since the triangle is isosceles, then , and since is the midpoint of $\overline{CT}$ , . Also, since opposite sides of a rectangle are congruent,

Therefore, the orange region is a trapezoid with bases and and height . Its area is 72, so we can set up and solve this equation using the area formula for a trapezoid:

$\frac{1}{2} (B + b)h = 72$

$\frac{1}{2} (2h + h)h = 72$

$\frac{1}{2} (3h )h = 72$

$\frac{3}{2}h^{2} = 72$

$\frac{3}{2}h^{2} \cdot \frac{2}{3} = 72 \cdot \frac{2}{3}$

$h^{2}= 48$

$h = \sqrt {48}$

This is the length of one leg of the triangle.

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