Mathematics: Find the lateral area of a square based pyramid that the slant height is 10 and the base edge 10

Find the lateral area of a square based pyramid, №12094252, 19.01.2021 12:19

Mathematics

What is the lateral area of this regular octagonal pyramid? (picture one) 114.8 cm² 162.4 cm² 229.7 cm² 281.3 cm² 2.what is the slant height x of the square pyramid? (picture 2) the figure shows a square pyramid. the slant height is shown as a dashed line perpendicular to the base edge and is labeled as x. the length of the lateral edge is 8 meters. the lateral edge makes a 60 degree angle with the base edge express your answer in radical form. 3.what is the surface area of this square pyramid? (picture 3) round your answer to the nearest tenth,

Step-by-step answer

P Answered by PhD

41

Q1)
the lateral area of the pyramid is the total area of all the lateral faces excluding the base.
In this regular octagonal pyramid, the lateral sides are triangles. As there are 8 triangles we need to find the area of all 8 sides.
Area of one lateral triangle face = 1/2 * base * slant height
slant height is the hypotenuse of the right angled triangle formed from the base of the pyramid with the perpendicular height.
slant height - l
l² = 7² + 7² = 49 *2
l² = 98
l = √98
l = 9.9
Area = 1/2 * 5.8 cm * 9.9 cm
= 28.71 cm²
There are 8 sides
total lateral area = 8 * 28.71 = 229.68 rounded off is 229.7 cm²
third option is correct - 229.7 cm²

Q2)
in the triangular face, the lateral edge makes a 60° angle with the base edge. Therefore 2 of the angles are 60° each, since the sum of the interior angles of a triangle is 180°, the third angle too is 60°. this makes the triangle an equilateral triangle with equal angles, hence equal sides.
since lateral edge is 8 cm,base edge too is 8 cm.
since this is an equilateral triangle, the perpendicular line cuts the base edge at its midpoint, bisecting the line forming 2 right angled triangles.
in the right angled triangle, height of triangle is x slant height ,
base = 8 /2 = 4 cm
hypotenuse = 8 cm
We need to find x, use Pythogoras' theorem
4² + x² = 8²
16 + x² = 64
x² = 62 - 16
x = √48
x = √4x√4x√3
= 2x2√3
= 4√3 cm

Q3)
surface area of the square pyramid
surface area of the base + surface area of triangular faces
square area = length x length
= 6.2 x 6.2
= 38.44 yd²
triangular face area = 1/2 * length * height
since the angle between lateral edge and base edge is 60°, its an equilateral triangle where all sides are equal. in this case each side is 6.2 yd.
to find the perpendicular height, use pythogoras' theorem
the perpendicular line(slant height ) cuts the base edge at its midpoint, therefore length of the right angled triangle is = 6.2/2 = 3.1 yd

slant height - l
l² + 3.1² = 6.2²
l² = 38.44 -9.61
l² = 28.83
l = 5.37
area = 1/2 *length *height
= 1/2 * 6.2 * 5.37
= 16.64 yd²
there are 4 triangles = 4 * 16.64 = 66.58 yd²
total area = 38.44 + 66.58 = 105 yd²
correct answer is 3rd option - 105 yd²

Mathematics

What is the lateral area of this regular octagonal pyramid? (picture one) 114.8 cm² 162.4 cm² 229.7 cm² 281.3 cm² 2.what is the slant height x of the square pyramid? (picture 2) the figure shows a square pyramid. the slant height is shown as a dashed line perpendicular to the base edge and is labeled as x. the length of the lateral edge is 8 meters. the lateral edge makes a 60 degree angle with the base edge express your answer in radical form. 3.what is the surface area of this square pyramid? (picture 3) round your answer to the nearest tenth,

Step-by-step answer

P Answered by PhD

41

Q1)
the lateral area of the pyramid is the total area of all the lateral faces excluding the base.
In this regular octagonal pyramid, the lateral sides are triangles. As there are 8 triangles we need to find the area of all 8 sides.
Area of one lateral triangle face = 1/2 * base * slant height
slant height is the hypotenuse of the right angled triangle formed from the base of the pyramid with the perpendicular height.
slant height - l
l² = 7² + 7² = 49 *2
l² = 98
l = √98
l = 9.9
Area = 1/2 * 5.8 cm * 9.9 cm
= 28.71 cm²
There are 8 sides
total lateral area = 8 * 28.71 = 229.68 rounded off is 229.7 cm²
third option is correct - 229.7 cm²

Q2)
in the triangular face, the lateral edge makes a 60° angle with the base edge. Therefore 2 of the angles are 60° each, since the sum of the interior angles of a triangle is 180°, the third angle too is 60°. this makes the triangle an equilateral triangle with equal angles, hence equal sides.
since lateral edge is 8 cm,base edge too is 8 cm.
since this is an equilateral triangle, the perpendicular line cuts the base edge at its midpoint, bisecting the line forming 2 right angled triangles.
in the right angled triangle, height of triangle is x slant height ,
base = 8 /2 = 4 cm
hypotenuse = 8 cm
We need to find x, use Pythogoras' theorem
4² + x² = 8²
16 + x² = 64
x² = 62 - 16
x = √48
x = √4x√4x√3
= 2x2√3
= 4√3 cm

Q3)
surface area of the square pyramid
surface area of the base + surface area of triangular faces
square area = length x length
= 6.2 x 6.2
= 38.44 yd²
triangular face area = 1/2 * length * height
since the angle between lateral edge and base edge is 60°, its an equilateral triangle where all sides are equal. in this case each side is 6.2 yd.
to find the perpendicular height, use pythogoras' theorem
the perpendicular line(slant height ) cuts the base edge at its midpoint, therefore length of the right angled triangle is = 6.2/2 = 3.1 yd

slant height - l
l² + 3.1² = 6.2²
l² = 38.44 -9.61
l² = 28.83
l = 5.37
area = 1/2 *length *height
= 1/2 * 6.2 * 5.37
= 16.64 yd²
there are 4 triangles = 4 * 16.64 = 66.58 yd²
total area = 38.44 + 66.58 = 105 yd²
correct answer is 3rd option - 105 yd²

Mathematics

Find the volume and the lateral area of a frustum of a right circular cone whose radii are 4 and 8 cm, and slant height is 6 cm. a chimney, 100 ft. high, is in the form of a frustum of a right circular cone with radii 4 ft. and 5 ft. find the lateral surface area of the chimney. the volume of a frustum of a right circular cone is 52π ft3. its altitude is 3 ft. and the measure of its lower radius is three times the measure of its upper radius. find the lateral area of the frustum. a frustum of a right circular cone has an altitude of 24 in. if its

Step-by-step answer

P Answered by PhD

14

Find the volume and the lateral area of a frustum of a right circular cone whose radii are 4 and 8 cm, and slant height is 6 cm.
$h= \sqrt{s^2-(R_1-R_2)^2} \\ = \sqrt{6^2-(4-8)^2} \\ = \sqrt{36-16} \\ = \sqrt{20}$
$Volume= \frac{1}{3} \pi h(R_1^2+R_1R_2+R_2^2) \\ = \frac{1}{3} \pi \times \sqrt{20} (4^2+4 \times 8+8^2) \\ = \frac{1}{3} \pi \sqrt{20} (16+32+64) \\ = \frac{1}{3} \pi \sqrt{20} (112) \\ =524.5cm^3$
Lateral area = Total surface area - area of base - area of top
$Lateral \ area= \pi (R_1+R_2)s \\ = \pi (4+8) \times 6 \\ =12 \pi \times 6 \\ =72 \pi \\ =226.2cm^2$

Mathematics

Find the volume and the lateral area of a frustum of a right circular cone whose radii are 4 and 8 cm, and slant height is 6 cm. a chimney, 100 ft. high, is in the form of a frustum of a right circular cone with radii 4 ft. and 5 ft. find the lateral surface area of the chimney. the volume of a frustum of a right circular cone is 52π ft3. its altitude is 3 ft. and the measure of its lower radius is three times the measure of its upper radius. find the lateral area of the frustum. a frustum of a right circular cone has an altitude of 24 in. if its

Step-by-step answer

P Answered by PhD

14

Find the volume and the lateral area of a frustum of a right circular cone whose radii are 4 and 8 cm, and slant height is 6 cm.
$h= \sqrt{s^2-(R_1-R_2)^2} \\ = \sqrt{6^2-(4-8)^2} \\ = \sqrt{36-16} \\ = \sqrt{20}$
$Volume= \frac{1}{3} \pi h(R_1^2+R_1R_2+R_2^2) \\ = \frac{1}{3} \pi \times \sqrt{20} (4^2+4 \times 8+8^2) \\ = \frac{1}{3} \pi \sqrt{20} (16+32+64) \\ = \frac{1}{3} \pi \sqrt{20} (112) \\ =524.5cm^3$
Lateral area = Total surface area - area of base - area of top
$Lateral \ area= \pi (R_1+R_2)s \\ = \pi (4+8) \times 6 \\ =12 \pi \times 6 \\ =72 \pi \\ =226.2cm^2$

Mathematics

This image shows a square pyramid. what is the surface area of this square pyramid? 25 ft² 100 ft² 125 ft² 200 ft² note: image not drawn to scale. the figure shows a square pyramid. the slant height is shown as a dashed line perpendicular to the base edge. the length of the base edge is 10 feet. the lateral edge makes a 45 degree angle with the base edge.

Step-by-step answer

P Answered by PhD

12

200 ft²

Step-by-step explanation:

Each face is an isosceles right triangle with a hypotenuse of length 10 ft. The area of each of those triangles is

A = 1/4·h² . . . . where the h in this formula is the hypotenuse length

So, the area of the four faces (the lateral area of the pyramid is 4 times this, or ...

A = 4·1/4·(10 ft)² = 100 ft²

Of course, the base area is simply the area of the square base, the square of its side length:

A = (10 ft)² = 100 ft²

So, the total area is the sum of the lateral area and the base area:

total area = 100 ft² +100 ft² = 200 ft²

If you think about this for a little bit, you will realize the pyramid must have zero height. That is, the slant height of a face is exactly the same as the distance from the center of an edge to the center of the base. "Not drawn to scale" is a good description.

Mathematics

This image shows a square pyramid. what is the surface area of this square pyramid? 25 ft² 100 ft² 125 ft² 200 ft² note: image not drawn to scale. the figure shows a square pyramid. the slant height is shown as a dashed line perpendicular to the base edge. the length of the base edge is 10 feet. the lateral edge makes a 45 degree angle with the base edge.

Step-by-step answer

P Answered by PhD

12

200 ft²

Step-by-step explanation:

Each face is an isosceles right triangle with a hypotenuse of length 10 ft. The area of each of those triangles is

A = 1/4·h² . . . . where the h in this formula is the hypotenuse length

So, the area of the four faces (the lateral area of the pyramid is 4 times this, or ...

A = 4·1/4·(10 ft)² = 100 ft²

Of course, the base area is simply the area of the square base, the square of its side length:

A = (10 ft)² = 100 ft²

So, the total area is the sum of the lateral area and the base area:

total area = 100 ft² +100 ft² = 200 ft²

If you think about this for a little bit, you will realize the pyramid must have zero height. That is, the slant height of a face is exactly the same as the distance from the center of an edge to the center of the base. "Not drawn to scale" is a good description.

Mathematics

Imran is paid ?286 per week Every week he puts 20% of his wage into a savings account How much does he put into his savings each week

Step-by-step answer

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The answer is in the image