Question

# A star is made up of 4 equilateral triangles and a square. If the sides of the triangles are 8 units, what is the surface area of the star ?

Hint:

## The correct answer is: 174.85 sq. units

### It is given that side of equilateral triangle = 8 units

Area of equilateral triangle = a^{2}

= 8^{2} = 16 square units

Since there are 4 triangles, so area of four equilateral triangles

= 4 16 = 64 sq. units

= 110.85 sq. units

In the figure, length of the side of square = 8 units

Now, Area of square = (side)^{2} = 8^{2} = 64 sq. units

Surface Area of the star

= Area of 4 equilateral triangles + Area of the square

= 110.85 + 64 = 174.85 sq. units

### Related Questions to study

### If the side AC of a given triangle is 18 units and the height of the triangle is 10 units, what is the area of triangle ABC ?

Area of the triangle =

= = 90 sq. units

### If the side AC of a given triangle is 18 units and the height of the triangle is 10 units, what is the area of triangle ABC ?

Area of the triangle =

= = 90 sq. units

### A triangle has an area of 90 𝑚^{2} and a base of 12 m, find the height of such a triangle?

area of the triangle = 90 m

^{2}

= 90

= 90

### A triangle has an area of 90 𝑚^{2} and a base of 12 m, find the height of such a triangle?

area of the triangle = 90 m

^{2}

= 90

= 90

### If the height to base ratio of a triangle ABC is 3:4 and the area is 864 square units. Determine the height and base of this triangle.

⇒ Height, h = 3x and Base, b = 4x

Area of triangle =

864 =

864 = 6x

^{2}

144 = x

^{2}

12 = x

Height = 3x = 3(12) = 36 units and Base = 4x = 4(12) = 48 units

### If the height to base ratio of a triangle ABC is 3:4 and the area is 864 square units. Determine the height and base of this triangle.

⇒ Height, h = 3x and Base, b = 4x

Area of triangle =

864 =

864 = 6x

^{2}

144 = x

^{2}

12 = x

Height = 3x = 3(12) = 36 units and Base = 4x = 4(12) = 48 units

### If the side of an equilateral triangular park is 20 units. What will be half of its area ?

^{2}

= 20

^{2}= 100 cm

^{2}

Half of the area

^{2}

### If the side of an equilateral triangular park is 20 units. What will be half of its area ?

^{2}

= 20

^{2}= 100 cm

^{2}

Half of the area

^{2}

### If the base of a triangle becomes three times its height. What is the new area of a triangle?

It is given that base = 3 height

⇒ b = 3h

Area of the triangle =

=

=h

^{2}

### If the base of a triangle becomes three times its height. What is the new area of a triangle?

It is given that base = 3 height

⇒ b = 3h

Area of the triangle =

=

=h

^{2}

### The side of an equilateral triangle is 16 units. What will be the double of its area ?

Area of an equilateral triangle = a

^{2}= 16

^{2}

= 64 cm

^{2}

Double of the area = 2 64

= 128 = 221.7 cm

^{2}

### The side of an equilateral triangle is 16 units. What will be the double of its area ?

Area of an equilateral triangle = a

^{2}= 16

^{2}

= 64 cm

^{2}

Double of the area = 2 64

= 128 = 221.7 cm

^{2}

### In triangle ABC, AB = 8cm. If the altitudes corresponding to AB and BC are 4 cm and 5 cm respectively. Find the measure of BC.

Area of the triangle with altitude corresponding to AB

=

= = 16 cm

^{2}

With Base = BC , Height, h = AE = 5 cm

Area of the triangle with altitude corresponding to BC is

= 16 cm

^{2}

= 16

BC = 6.4 cm

### In triangle ABC, AB = 8cm. If the altitudes corresponding to AB and BC are 4 cm and 5 cm respectively. Find the measure of BC.

Area of the triangle with altitude corresponding to AB

=

= = 16 cm

^{2}

With Base = BC , Height, h = AE = 5 cm

Area of the triangle with altitude corresponding to BC is

= 16 cm

^{2}

= 16

BC = 6.4 cm

### The base and corresponding altitude of a parallelogram are 18 cm and 6 cm respectively. Find its area

Area of the parallelogram = b h

= 18 6 = 108 cm

^{2}

### The base and corresponding altitude of a parallelogram are 18 cm and 6 cm respectively. Find its area

Area of the parallelogram = b h

= 18 6 = 108 cm

^{2}

### The ratio of the bases of two triangles is a : b. If the ratio of their corresponding altitudes is c : d, find the ratio of their areas (in the same order).

⇒ Bases of the triangles = ax , bx

Similarly, it is given that ratio of altitudes is c : d

⇒ Altitudes of the triangle = cy , dy

Area of first triangle =

=

Area of second triangle =

Ratio of the areas of two triangle

=

=

Hence, ratio of area of two triangles is ac : bd

### The ratio of the bases of two triangles is a : b. If the ratio of their corresponding altitudes is c : d, find the ratio of their areas (in the same order).

⇒ Bases of the triangles = ax , bx

Similarly, it is given that ratio of altitudes is c : d

⇒ Altitudes of the triangle = cy , dy

Area of first triangle =

=

Area of second triangle =

Ratio of the areas of two triangle

=

=

Hence, ratio of area of two triangles is ac : bd

### The sides of triangle are 11 cm, 15 cm and 16 cm. What is the measure of altitude to the largest side?

⇒ Length of the side a = 3x , b = 4x and c = 5 x

Perimeter of the triangle = 144 m

3x + 4x + 5x = 144

12x = 144 ⇒ x = 12

Now Using Pythagoras theorem,

(5x)^{2} = (3x)^{2} + (4x)^{2}

25x^{2} = 9x^{2} + 16x^{2}

25x^{2} = 25x^{2} i.e. Pythagoras holds true

⇒ Given triangle is a right angled triangle

Base, b = 3x =3(12) = 36 m and Height, h = 4x =4(12) = 48 m

⇒ Area of the triangle =

### The sides of triangle are 11 cm, 15 cm and 16 cm. What is the measure of altitude to the largest side?

⇒ Length of the side a = 3x , b = 4x and c = 5 x

Perimeter of the triangle = 144 m

3x + 4x + 5x = 144

12x = 144 ⇒ x = 12

Now Using Pythagoras theorem,

(5x)^{2} = (3x)^{2} + (4x)^{2}

25x^{2} = 9x^{2} + 16x^{2}

25x^{2} = 25x^{2} i.e. Pythagoras holds true

⇒ Given triangle is a right angled triangle

Base, b = 3x =3(12) = 36 m and Height, h = 4x =4(12) = 48 m

⇒ Area of the triangle =

### The sides of triangle are 11 cm, 15 cm and 16 cm. What is the measure of altitude to the largest side?

Using Heron’s formula

Area of triangle = where s =

s = = 21

Area of triangle=

Since we have to find altitude to the largest side, base of the triangle = 16 cm

Also, area of triangle =

⇒

⇒ ( = 2.64)

Using Heron’s formula

Area of triangle = where s =

s = = 21

Area of triangle=

Since we have to find altitude to the largest side, base of the triangle = 16 cm

Also, area of triangle =

⇒

⇒ ( = 2.64)

### When a solid metal cube is completely submerged in a cylindrical vessel Containing milk with 30 cm diameter the level of milk rises by cm. Find the length of the edge of the metal cube.

We use principle of Archimedes to find the length of the cube.

Explanations:

Step 1 of 1:

Let the length of the edge of the metal cube be a.

Volume of cube =

*a*

^{3}

Given, r = 30/2 cm and h =

By the principle of Archimedes,

Volume of risen milk = volume of cube

, discarding the negative value since distance cannot be negative.

Final Answer:

The length of the edge of the cube is 10cm.

### When a solid metal cube is completely submerged in a cylindrical vessel Containing milk with 30 cm diameter the level of milk rises by cm. Find the length of the edge of the metal cube.

We use principle of Archimedes to find the length of the cube.

Explanations:

Step 1 of 1:

Let the length of the edge of the metal cube be a.

Volume of cube =

*a*

^{3}

Given, r = 30/2 cm and h =

By the principle of Archimedes,

Volume of risen milk = volume of cube

, discarding the negative value since distance cannot be negative.

Final Answer:

The length of the edge of the cube is 10cm.

### The circumference of the base of a cylindrical vessel is 132 cm and its height is 25 cm. Find the radius and volume of the cylinder.

We plug in the values in formulae and solve the problem.

Explanations:

Step 1 of 2:

Let the radius of vessel base be

We have, cm

Height h = 25cm

Step 2 of 2:

Volume of the vessel =

Final Answer:

The radius is 21cm and volume of the cylinder is 34650cm

^{3}.

### The circumference of the base of a cylindrical vessel is 132 cm and its height is 25 cm. Find the radius and volume of the cylinder.

We plug in the values in formulae and solve the problem.

Explanations:

Step 1 of 2:

Let the radius of vessel base be

We have, cm

Height h = 25cm

Step 2 of 2:

Volume of the vessel =

Final Answer:

The radius is 21cm and volume of the cylinder is 34650cm

^{3}.

### The volume of a metallic cylindrical pipe is 748 cubic.cm Its length is 14 cm., and its external radius is 9 cm. Find its thickness?

We find the internal radius and subtract it from external radius to get the thickness.

Explanations:

Step 1 of 1:

Let the internal radius of the pipe be r .

Given, external radius R = 9cm

Length = height of the pipe h = 14cm

We have volume of pipe = 748

Final Answer:

The thickness of the pipe is 8 cm.

### The volume of a metallic cylindrical pipe is 748 cubic.cm Its length is 14 cm., and its external radius is 9 cm. Find its thickness?

We find the internal radius and subtract it from external radius to get the thickness.

Explanations:

Step 1 of 1:

Let the internal radius of the pipe be r .

Given, external radius R = 9cm

Length = height of the pipe h = 14cm

We have volume of pipe = 748

Final Answer:

The thickness of the pipe is 8 cm.

### The difference between the outer and inner curved surface areas of a 14 cm long cylinder is 88 sq.cm Find the outer and inner radii of the cylinder given that the volume of the metal is 176 cubic cm

Forming the equations based on the given information, we will get two equations. Then we will find the radii by solving those two equations.

Explanations:

Step 1 of 3:

Given, outer CSA – inner CSA = 88

, where = length of cylinder, R = outer radius, r = inner radius

…(i)

Step 2 of 3:

Also, given volume with R – volume with r = 176

…(ii)

Step 3 of 3:

Adding (i) and (ii), we get

Putting R = 5/2 in equation (ii), we get

Final Answer:

The outer and inner radii of the cylinder are 2.5 cm and 1.5 cm

### The difference between the outer and inner curved surface areas of a 14 cm long cylinder is 88 sq.cm Find the outer and inner radii of the cylinder given that the volume of the metal is 176 cubic cm

Forming the equations based on the given information, we will get two equations. Then we will find the radii by solving those two equations.

Explanations:

Step 1 of 3:

Given, outer CSA – inner CSA = 88

, where = length of cylinder, R = outer radius, r = inner radius

…(i)

Step 2 of 3:

Also, given volume with R – volume with r = 176

…(ii)

Step 3 of 3:

Adding (i) and (ii), we get

Putting R = 5/2 in equation (ii), we get

Final Answer:

The outer and inner radii of the cylinder are 2.5 cm and 1.5 cm