Calculating the surface area of a tetrahedron - GMAT Quantitative
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The slant height of a pyramid is one and one-half times the perimeter of its square base. The base has sides of length 15 inches. What is the surface area of the pyramid?
The slant height of a pyramid is one and one-half times the perimeter of its square base. The base has sides of length 15 inches. What is the surface area of the pyramid?
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The square base of the pyramid has four sides with length 15 inches, making its perimeter four times that, or 60 inches. The slant height is
inches.
Therefore, the area of the base is 
square inches.
Each of the four lateral triangles has area 
square inches.
The total surface area is 
square inches.
The square base of the pyramid has four sides with length 15 inches, making its perimeter four times that, or 60 inches. The slant height is
Therefore, the area of the base is
Each of the four lateral triangles has area
The total surface area is
In three-dimensional space, the four vertices of a tetrahedron - a solid with four faces - have Cartesian coordinates 
.
Give the surface area of the tetrahedron.
In three-dimensional space, the four vertices of a tetrahedron - a solid with four faces - have Cartesian coordinates
Give the surface area of the tetrahedron.
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The tetrahedron looks like this:
is the origin and 
are the other three points, which are 60 units away from the origin on each of the three (perpendicular) axes.
The bottom, front, and left faces are each right triangles whose legs each measure 60. Each face has area
.
The remaining face has three edges each a hypotenuse of one of three congruent right triangles, so its sides are congruent, and it is an equilateral triangle. Its sidelength can be found via the 45-45-90 Theorem to be 
, so its area is
The total area is
The tetrahedron looks like this:
The bottom, front, and left faces are each right triangles whose legs each measure 60. Each face has area
The remaining face has three edges each a hypotenuse of one of three congruent right triangles, so its sides are congruent, and it is an equilateral triangle. Its sidelength can be found via the 45-45-90 Theorem to be
The total area is
A regular tetrahedron comprises four faces, each of which is an equilateral triangle. If the sum of the lengths of its edges is 120, what is its surface area?
A regular tetrahedron comprises four faces, each of which is an equilateral triangle. If the sum of the lengths of its edges is 120, what is its surface area?
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As shown in the diagram below, a regular tetrahedron has six congruent edges, so each has length 
:
The area of one face is the area of an equilateral triangle with sidelength 20, which is
The total surface area is four times this, or 
.
As shown in the diagram below, a regular tetrahedron has six congruent edges, so each has length
The area of one face is the area of an equilateral triangle with sidelength 20, which is
The total surface area is four times this, or
Refer to the above diagram, which shows a tetrahedron.
, and 
. Give the surface area of the tetrahedron.
Refer to the above diagram, which shows a tetrahedron.
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Three of the surfaces of the tetrahedron - 
, 
, and 
- are isosceles right triangles with hypotenuse 30, so by the 45-45-90 Theorem, each leg measures this length divided by 
, or
.
The area of each of these triangles is half the product of its legs, so each area is
Also, the legs are of the same measure among the triangles, the hypotenuses are as well, so the fourth surface 
is an equilateral triangle. Its sidelength is 
, so we use the equilateral triangle area formula to calculate its area:
Add the areas of the faces:
Three of the surfaces of the tetrahedron -
The area of each of these triangles is half the product of its legs, so each area is
Also, the legs are of the same measure among the triangles, the hypotenuses are as well, so the fourth surface
Add the areas of the faces:
A regular tetrahedron is a solid with four faces, each of which is an equilateral triangle.
If the lengths of all of the edges of a regular tetrahedron are added, the total length is 120. What is the surface area of the tetrahedron?
A regular tetrahedron is a solid with four faces, each of which is an equilateral triangle.
If the lengths of all of the edges of a regular tetrahedron are added, the total length is 120. What is the surface area of the tetrahedron?
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A tetrahedron looks like this:
The tetrahedron has six edges, and in a regular tetrahedron, they are congruent, so each edge has length 
.
The area of one face of the tetrahedron, it being an equilateral triangle, can be calulated using the formula
There are four congruent faces, so the total surface area is
Since 
, the surface area of the tetrahedron is
A tetrahedron looks like this:
The tetrahedron has six edges, and in a regular tetrahedron, they are congruent, so each edge has length
The area of one face of the tetrahedron, it being an equilateral triangle, can be calulated using the formula
There are four congruent faces, so the total surface area is
Since
A regular tetrahedron is a solid with four faces, each of which is an equilateral triangle.
Each edge of a regular tetrahedron has length 
. What is the surface area of the tetrahedron?
A regular tetrahedron is a solid with four faces, each of which is an equilateral triangle.
Each edge of a regular tetrahedron has length
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The area of one face of the tetrahedron, it being an equilateral triangle, can be calulated using the formula
There are four congruent faces, so the total surface area is
Since 
, the surface area of the tetrahedron is
The area of one face of the tetrahedron, it being an equilateral triangle, can be calulated using the formula
There are four congruent faces, so the total surface area is
Since
Evaluate the surface area of the above tetrahedron.
Evaluate the surface area of the above tetrahedron.
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Three of the faces of the tetrahedron are isosceles right triangles with legs of length 8, and, subsequently, by the 45-45-90 Theorem, hypotenuses of length 
. The fourth face, consequently, is an equilateral triangle with three sides of length 
.
Each of the three right triangle faces has area equal to half the product of its legs , which is
.
The equilateral face has as its area
The sum of the areas of the faces is
Three of the faces of the tetrahedron are isosceles right triangles with legs of length 8, and, subsequently, by the 45-45-90 Theorem, hypotenuses of length
Each of the three right triangle faces has area equal to half the product of its legs , which is
The equilateral face has as its area
The sum of the areas of the faces is
The cube in the above figure has surface area 384. Give the surface area of the tetrahedron with vertices 
, shown in red.
The cube in the above figure has surface area 384. Give the surface area of the tetrahedron with vertices
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The surface area formula can be used to find the length of each edge of the cube:
Three faces of the tetrahedron - 
, 
, 
- are right triangles with legs of length 8, so the area of each is half the product of the lengths of their legs:
.
Each triangle is isosceles, so, by the 45-45-90 Theorem, each of their hypotentuses measures 
times a leg, or 
. is therefore an equilateral triangle with sidelenghth 
. Its area can be found as follows:
The total surface area is
The surface area formula can be used to find the length of each edge of the cube:
Three faces of the tetrahedron -
Each triangle is isosceles, so, by the 45-45-90 Theorem, each of their hypotentuses measures
The total surface area is
The above diagram shows a regular right triangular pyramid. Its base 
is an equilateral triangle; the other three faces are congruent isosceles triangles, with 
an altitude of 
. Give the surface area of the pyramid.
The above diagram shows a regular right triangular pyramid. Its base
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The base is an equilateral triangle with sidelength 12, so its area can be calculated as follows:
.
Each of the three other faces is congruent, with base 12. The area of each is the product of its base and its height. To find the common height, we examine 
, which, since 
is an altitude of isosceles 
, is a right triangle with hypotenuse of length 18 and one leg of length 
. We can find 
using the Pythagorean Theorem:
The area of 
is half the product of this height and the base:
All three lateral faces have this area.
Now add the areas of the four faces:
The base is an equilateral triangle with sidelength 12, so its area can be calculated as follows:
Each of the three other faces is congruent, with base 12. The area of each is the product of its base and its height. To find the common height, we examine
The area of
All three lateral faces have this area.
Now add the areas of the four faces:
In three-dimensional space, the four vertices of a tetrahedron - a solid with four faces - have Cartesian coordinates .
Give the surface area of the tetrahedron.
In three-dimensional space, the four vertices of a tetrahedron - a solid with four faces - have Cartesian coordinates
Give the surface area of the tetrahedron.
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The tetrahedron looks like this:
is the origin and are the other three points, which are 60 units away from the origin on each of the three (perpendicular) axes.
The bottom, front, and left faces are each right triangles whose legs each measure 60. Each face has area
.
The remaining face has three edges each a hypotenuse of one of three congruent right triangles, so its sides are congruent, and it is an equilateral triangle. Its sidelength can be found via the 45-45-90 Theorem to be , so its area is
The total area is
The tetrahedron looks like this:
The bottom, front, and left faces are each right triangles whose legs each measure 60. Each face has area
The remaining face has three edges each a hypotenuse of one of three congruent right triangles, so its sides are congruent, and it is an equilateral triangle. Its sidelength can be found via the 45-45-90 Theorem to be
The total area is
The slant height of a pyramid is one and one-half times the perimeter of its square base. The base has sides of length 15 inches. What is the surface area of the pyramid?
The slant height of a pyramid is one and one-half times the perimeter of its square base. The base has sides of length 15 inches. What is the surface area of the pyramid?
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The square base of the pyramid has four sides with length 15 inches, making its perimeter four times that, or 60 inches. The slant height is
inches.
Therefore, the area of the base is square inches.
Each of the four lateral triangles has area square inches.
The total surface area is square inches.
The square base of the pyramid has four sides with length 15 inches, making its perimeter four times that, or 60 inches. The slant height is
Therefore, the area of the base is
Each of the four lateral triangles has area
The total surface area is
A regular tetrahedron comprises four faces, each of which is an equilateral triangle. If the sum of the lengths of its edges is 120, what is its surface area?
A regular tetrahedron comprises four faces, each of which is an equilateral triangle. If the sum of the lengths of its edges is 120, what is its surface area?
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As shown in the diagram below, a regular tetrahedron has six congruent edges, so each has length :
The area of one face is the area of an equilateral triangle with sidelength 20, which is
The total surface area is four times this, or .
As shown in the diagram below, a regular tetrahedron has six congruent edges, so each has length
The area of one face is the area of an equilateral triangle with sidelength 20, which is
The total surface area is four times this, or
Refer to the above diagram, which shows a tetrahedron.
, and . Give the surface area of the tetrahedron.
Refer to the above diagram, which shows a tetrahedron.
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Three of the surfaces of the tetrahedron - , , and - are isosceles right triangles with hypotenuse 30, so by the 45-45-90 Theorem, each leg measures this length divided by , or
.
The area of each of these triangles is half the product of its legs, so each area is
Also, the legs are of the same measure among the triangles, the hypotenuses are as well, so the fourth surface is an equilateral triangle. Its sidelength is , so we use the equilateral triangle area formula to calculate its area:
Add the areas of the faces:
Three of the surfaces of the tetrahedron -
The area of each of these triangles is half the product of its legs, so each area is
Also, the legs are of the same measure among the triangles, the hypotenuses are as well, so the fourth surface
Add the areas of the faces:
A regular tetrahedron is a solid with four faces, each of which is an equilateral triangle.
If the lengths of all of the edges of a regular tetrahedron are added, the total length is 120. What is the surface area of the tetrahedron?
A regular tetrahedron is a solid with four faces, each of which is an equilateral triangle.
If the lengths of all of the edges of a regular tetrahedron are added, the total length is 120. What is the surface area of the tetrahedron?
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A tetrahedron looks like this:
The tetrahedron has six edges, and in a regular tetrahedron, they are congruent, so each edge has length .
The area of one face of the tetrahedron, it being an equilateral triangle, can be calulated using the formula
There are four congruent faces, so the total surface area is
Since , the surface area of the tetrahedron is
A tetrahedron looks like this:
The tetrahedron has six edges, and in a regular tetrahedron, they are congruent, so each edge has length
The area of one face of the tetrahedron, it being an equilateral triangle, can be calulated using the formula
There are four congruent faces, so the total surface area is
Since
A regular tetrahedron is a solid with four faces, each of which is an equilateral triangle.
Each edge of a regular tetrahedron has length . What is the surface area of the tetrahedron?
A regular tetrahedron is a solid with four faces, each of which is an equilateral triangle.
Each edge of a regular tetrahedron has length
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The area of one face of the tetrahedron, it being an equilateral triangle, can be calulated using the formula
There are four congruent faces, so the total surface area is
Since , the surface area of the tetrahedron is
The area of one face of the tetrahedron, it being an equilateral triangle, can be calulated using the formula
There are four congruent faces, so the total surface area is
Since
Evaluate the surface area of the above tetrahedron.
Evaluate the surface area of the above tetrahedron.
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Three of the faces of the tetrahedron are isosceles right triangles with legs of length 8, and, subsequently, by the 45-45-90 Theorem, hypotenuses of length . The fourth face, consequently, is an equilateral triangle with three sides of length .
Each of the three right triangle faces has area equal to half the product of its legs , which is
.
The equilateral face has as its area
The sum of the areas of the faces is
Three of the faces of the tetrahedron are isosceles right triangles with legs of length 8, and, subsequently, by the 45-45-90 Theorem, hypotenuses of length
Each of the three right triangle faces has area equal to half the product of its legs , which is
The equilateral face has as its area
The sum of the areas of the faces is
The cube in the above figure has surface area 384. Give the surface area of the tetrahedron with vertices , shown in red.
The cube in the above figure has surface area 384. Give the surface area of the tetrahedron with vertices
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The surface area formula can be used to find the length of each edge of the cube:
Three faces of the tetrahedron - , , - are right triangles with legs of length 8, so the area of each is half the product of the lengths of their legs:
.
Each triangle is isosceles, so, by the 45-45-90 Theorem, each of their hypotentuses measures times a leg, or . is therefore an equilateral triangle with sidelenghth . Its area can be found as follows:
The total surface area is
The surface area formula can be used to find the length of each edge of the cube:
Three faces of the tetrahedron -
Each triangle is isosceles, so, by the 45-45-90 Theorem, each of their hypotentuses measures
The total surface area is
The above diagram shows a regular right triangular pyramid. Its base is an equilateral triangle; the other three faces are congruent isosceles triangles, with an altitude of . Give the surface area of the pyramid.
The above diagram shows a regular right triangular pyramid. Its base
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The base is an equilateral triangle with sidelength 12, so its area can be calculated as follows:
.
Each of the three other faces is congruent, with base 12. The area of each is the product of its base and its height. To find the common height, we examine , which, since is an altitude of isosceles , is a right triangle with hypotenuse of length 18 and one leg of length . We can find using the Pythagorean Theorem:
The area of is half the product of this height and the base:
All three lateral faces have this area.
Now add the areas of the four faces:
The base is an equilateral triangle with sidelength 12, so its area can be calculated as follows:
Each of the three other faces is congruent, with base 12. The area of each is the product of its base and its height. To find the common height, we examine
The area of
All three lateral faces have this area.
Now add the areas of the four faces:
The slant height of a pyramid is one and one-half times the perimeter of its square base. The base has sides of length 15 inches. What is the surface area of the pyramid?
The slant height of a pyramid is one and one-half times the perimeter of its square base. The base has sides of length 15 inches. What is the surface area of the pyramid?
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The square base of the pyramid has four sides with length 15 inches, making its perimeter four times that, or 60 inches. The slant height is
inches.
Therefore, the area of the base is square inches.
Each of the four lateral triangles has area square inches.
The total surface area is square inches.
The square base of the pyramid has four sides with length 15 inches, making its perimeter four times that, or 60 inches. The slant height is
Therefore, the area of the base is
Each of the four lateral triangles has area
The total surface area is
In three-dimensional space, the four vertices of a tetrahedron - a solid with four faces - have Cartesian coordinates .
Give the surface area of the tetrahedron.
In three-dimensional space, the four vertices of a tetrahedron - a solid with four faces - have Cartesian coordinates
Give the surface area of the tetrahedron.
Tap to see back →
The tetrahedron looks like this:
is the origin and are the other three points, which are 60 units away from the origin on each of the three (perpendicular) axes.
The bottom, front, and left faces are each right triangles whose legs each measure 60. Each face has area
.
The remaining face has three edges each a hypotenuse of one of three congruent right triangles, so its sides are congruent, and it is an equilateral triangle. Its sidelength can be found via the 45-45-90 Theorem to be , so its area is
The total area is
The tetrahedron looks like this:
The bottom, front, and left faces are each right triangles whose legs each measure 60. Each face has area
The remaining face has three edges each a hypotenuse of one of three congruent right triangles, so its sides are congruent, and it is an equilateral triangle. Its sidelength can be found via the 45-45-90 Theorem to be
The total area is