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The base of a regular pyramid is a square and each of the other four sides is an equilateral triangle, length of each side being 20 cm. The vertical height of the pyramid, in cm, is
- a)12
- b)10√2
- c)8√3
- d)5√5
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
The base of a regular pyramid is a square and each of the other four s...
It is given that the base of the pyramid is square and each of the four sides are equilateral triangles.
Length of each side of the equilateral triangle = 20cm
Since the side of the triangle will be common to the square as well, the side of the square = 20cm
Let h be the vertical height of the pyramid ie OA
OB = 10 since it is half the side of the square
AB is the height of the equilateral triangle i.e 10√3
AOB is a right angle, so applying the Pythagorean formula, we get
OA2 + OB2 = AB2
h2 + 100 = 300
h = 10√2
Length of each side of the equilateral triangle = 20cm
Since the side of the triangle will be common to the square as well, the side of the square = 20cm
Let h be the vertical height of the pyramid ie OA
OB = 10 since it is half the side of the square
AB is the height of the equilateral triangle i.e 10√3
AOB is a right angle, so applying the Pythagorean formula, we get
OA2 + OB2 = AB2
h2 + 100 = 300
h = 10√2
Most Upvoted Answer
The base of a regular pyramid is a square and each of the other four s...
Let ABCD be the bottom face. let O be the top vertex of the pyramid. Let P be the mid point of a side, say AB. also let X be the projection of O on ABCD.
In triangle OAB, OP=20√3/2= 10√3
Now in triangle OPX, PX=10 and OP=10√3 and angle OXP=π/2. therefore, we get OX=10√2.
In triangle OAB, OP=20√3/2= 10√3
Now in triangle OPX, PX=10 and OP=10√3 and angle OXP=π/2. therefore, we get OX=10√2.
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Community Answer
The base of a regular pyramid is a square and each of the other four s...
To find the vertical height of the pyramid, we can use the Pythagorean theorem.
Let's call the height of each equilateral triangle side "h". The height of the pyramid is the height of the equilateral triangle plus the height of the square base.
By drawing an altitude in the equilateral triangle, we can see that it creates a right triangle with one leg being half of the side length of the equilateral triangle (10 cm) and the hypotenuse being the height of the equilateral triangle (h).
Using the Pythagorean theorem, we can solve for "h":
h^2 = (10 cm)^2 + (h/2)^2
h^2 = 100 cm^2 + (h^2)/4
4h^2 = 400 cm^2 + h^2
3h^2 = 400 cm^2
h^2 = 400 cm^2 / 3
h^2 = 133.33 cm^2
Taking the square root of both sides, we find:
h = √(133.33 cm^2)
h ≈ 11.55 cm
The height of the pyramid is the height of the equilateral triangle plus the height of the square base:
Height = 11.55 cm + 10 cm
Height ≈ 21.55 cm
Since none of the given options match exactly, the closest answer is 21.55 cm, which is approximately 22 cm. Therefore, the correct answer is not provided.
Let's call the height of each equilateral triangle side "h". The height of the pyramid is the height of the equilateral triangle plus the height of the square base.
By drawing an altitude in the equilateral triangle, we can see that it creates a right triangle with one leg being half of the side length of the equilateral triangle (10 cm) and the hypotenuse being the height of the equilateral triangle (h).
Using the Pythagorean theorem, we can solve for "h":
h^2 = (10 cm)^2 + (h/2)^2
h^2 = 100 cm^2 + (h^2)/4
4h^2 = 400 cm^2 + h^2
3h^2 = 400 cm^2
h^2 = 400 cm^2 / 3
h^2 = 133.33 cm^2
Taking the square root of both sides, we find:
h = √(133.33 cm^2)
h ≈ 11.55 cm
The height of the pyramid is the height of the equilateral triangle plus the height of the square base:
Height = 11.55 cm + 10 cm
Height ≈ 21.55 cm
Since none of the given options match exactly, the closest answer is 21.55 cm, which is approximately 22 cm. Therefore, the correct answer is not provided.
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