How to find the area of a triangle - ISEE Middle Level Quantitative Reasoning
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A triangle has base 80 inches and area 4,200 square inches. What is its height?
A triangle has base 80 inches and area 4,200 square inches. What is its height?
Use the area formula for a triangle, setting 
:
inches
Use the area formula for a triangle, setting
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The sum of the lengths of the legs of an isosceles right triangle is one meter. What is its area in square centimeters?
The sum of the lengths of the legs of an isosceles right triangle is one meter. What is its area in square centimeters?
The legs of an isosceles right triangle have equal length, so, if the sum of their lengths is one meter, which is equal to 100 centimeters, each leg measures half of this, or
centimeters.
The area of a triangle is half the product of its height and base; for a right triangle, the legs serve as height and base, so the area of the triangle is
square centimeters.
The legs of an isosceles right triangle have equal length, so, if the sum of their lengths is one meter, which is equal to 100 centimeters, each leg measures half of this, or
The area of a triangle is half the product of its height and base; for a right triangle, the legs serve as height and base, so the area of the triangle is
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Figure NOT drawn to scale
Square 
has area 1,600. 
; 
. Which of the following is the greater quantity?
(a) The area of 
(b) The area of 
Figure NOT drawn to scale
Square
(a) The area of
(b) The area of
Square 
has area 1,600, so the length of each side is 
.
Since 
,
Therefore, 
.
has as its area 
; 
has as its area 
.
Since 
and 
, it follows that
and
has greater area than 
.
Square
Since
Therefore,
Since
and
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The above figure depicts Square 
. 
, 
, and 
are the midpoints of 
, 
, and 
, respectively.
has area 
. What is the area of Square 
?
The above figure depicts Square
Since , , and are the midpoints of , , and , if we call 
the length of each side of the square, then
The area of 
is half the product of the lengths of its legs:
The area of the square is the square of the length of a side, which is 
. This is eight times the area of 
, so the correct choice is 
Since
The area of
The area of the square is the square of the length of a side, which is
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Which of the following is the greater quantity?
(a) The area of the above triangle
(b) 800
Which of the following is the greater quantity?
(a) The area of the above triangle
(b) 800
The area of a right triangle is half the product of the lengths of its legs, which here are 25 and 60. So
which is less than 800.
The area of a right triangle is half the product of the lengths of its legs, which here are 25 and 60. So
which is less than 800.
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The above figure gives the lengths of the three sides of the triangle in feet. Give its area in square inches.
The above figure gives the lengths of the three sides of the triangle in feet. Give its area in square inches.
The area of a right triangle is half the product of the lengths of its legs, which here are 
feet and 
feet.
Multiply each length by 12 to convert to inches - the lengths become 
and 
. The area in square inches is therefore
square inches.
The area of a right triangle is half the product of the lengths of its legs, which here are
Multiply each length by 12 to convert to inches - the lengths become
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Refer to the above figure. Which is the greater quantity?
(a) The perimeter of the triangle
(b) 3 feet
Refer to the above figure. Which is the greater quantity?
(a) The perimeter of the triangle
(b) 3 feet
The perimeter of the triangle - the sum of the lengths of its sides - is
inches.
3 feet are equivalent to 
inches, so this is the greater quantity.
The perimeter of the triangle - the sum of the lengths of its sides - is
3 feet are equivalent to
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Figure NOT drawn to scale.
In the above diagram, Square 
has area 400. Which is the greater quantity?
(a) The area of 
(b) The area of 
Figure NOT drawn to scale.
In the above diagram, Square
(a) The area of
(b) The area of
Square 
has area 400, so its common sidelength is the square root of 400, or 20. Therefore,
.
The area of a right triangle is half the product of the lengths of its legs.
has legs 
and 
, so its area is
.
has legs 
and 
, so its area is
.
has the greater area.
Square
The area of a right triangle is half the product of the lengths of its legs.
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Figure NOT drawn to scale
The above diagram depicts Parallelogram 
. Which is the greater quantity?
(a) The area of 
(b) The area of 
Figure NOT drawn to scale
The above diagram depicts Parallelogram
(a) The area of
(b) The area of
Opposite sides of a parallelogram have the same measure, so
Base 
of 
and base 
of 
have the same length; also, as can be seen below, both have the same height, which is the height of the parallelogram.
Therefore, the areas of 
and 
have the same area - 
.
Opposite sides of a parallelogram have the same measure, so
Base
Therefore, the areas of
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