It’s possible to measure the perimeter, area, surface area, and volume of various shapes. The perimeter is the distance around a shape. The perimeter is measured in inches, feet, yards, miles, centimeters, meters, and kilometers.
The area is the amount of flat space a flat shape encloses. Square units are used to measure area. Examples of square units are square inches, square feet, square yards, square miles, square centimeters, square meters, and square kilometers. Write square units by putting a small two over the units to indicate “square.”
Square inches = in2
Square feet = ft2
Square miles = mi2
Square meters = m2
Surface area is the outside surface of a solid shape. Surface area is measured in square units.
Volume is the space inside a three-dimensional shape. A liter of water and a cup of sugar are measured in volume. Volume is measured in cubic units. Think of each cubic unit as a little block or cube. Write cubic units by putting a small three over the units.
Cubic inches = in3
Cubic feet = ft3
Cubic miles = mi3
Cubic centimeters = cm3
Cubic meters = m3
Cubic kilometers = km3
It’s not hard to learn how to find the perimeter, area, surface area, and volume of common figures.
A triangle has three sides and three angles.
Perimeter
The perimeter of a triangle is the sum of the sides of a triangle.
EXAMPLE:
Find the perimeter of a triangle with sides 8, 10, and 12 feet.
Just add the three sides together.
8 + 10 + 12 = 30 feet
Area
The area of a triangle is the space inside the triangle. The area of a triangle is 
base × altitude. The altitude is the distance from a vertex of the triangle perpendicular to the opposite side. The altitude is also called the height.
Area of a Triangle = (base × height)
What is the area of a triangle with base 8 and altitude 10?
The area is (8)(10).
The area is 40.
1.Find the perimeter of an equilateral triangle with sides of 5 inches each.
2.Find the perimeter of an isosceles triangle with two sides of 4 inches each and a third side 1 inch long.
3.What’s the area of a triangle with base 4 inches and height 1 inch?
(Answers)
A rectangle is a parallelogram. It has four right angles and two pairs of parallel sides.
Perimeter
To find the perimeter of a rectangle, just add the length of all four sides together.
EXAMPLE:
Find the perimeter of a rectangle with sides 5 inches and 7 inches.
Just add all four sides of the rectangle together.
7 + 7 + 5 + 5 = 24
The perimeter is 24 inches.
Experiment
Learn how to determine the area of a rectangle by counting squares.
Materials
Graph paper
Pencil
Procedure
1.Draw the rectangles indicated on the chart on a piece of graph paper.
2.Count the number of small squares inside each drawn rectangle. Enter the results in the chart.
3.Compute the area of each rectangle by multiplying the length by the width of each rectangle. Enter the results in the chart.
4.Compare the areas you found in Steps 2 and 3.
Something to think about . . .
How could you determine the area of a rectangle with sides 10 inches × 12 inches?
Area
To determine the area of a rectangle, just multiply the length times the width.
Area of a Rectangle = lw
EXAMPLE:
To find the area of a rectangle 2 inches by 4 inches, just multiply 2 × 4.
The area of this rectangle is 8 square inches.
1.What is the perimeter of a rectangle with length 10 inches and height 6 inches?
2.What is the perimeter of a rectangle with sides 4 meters and 16 meters?
3.What is the area of a rectangle with length 10 inches and height 5 inches?
4.What is the area of a rectangle with length 4 centimeters and height 2 centimeters?
(Answers)
A square has four right angles and four equal sides. A square is a special type of rectangle. Since all four sides of a square are equal, the length and width of the square are called the sides of the square.
Perimeter
To find the perimeter of a square, just add all the sides together.
Perimeter of a Square = s + s + s + s or 4s
What is the perimeter of a square with sides 9 inches long?
Add all four sides of the square together.
9 + 9 + 9 + 9 = 36
Experiment
Discover the formula for finding the perimeter of a square.
Materials
Pencil
Paper
Procedure
1.Draw each of the squares listed in the chart.
2.Label each of the sides.
3.Add the sides of each square together to find its perimeter. Enter the results in the chart.
4.Multiply the length of one side of each square by the number of sides (4). Enter the results in the chart.
5.Did you get the same result each way? Which way was the easiest?
Something to think about . . .
Can you find a shortcut method to find the perimeter of a rectangle?
Perimeter of a Square = s + s + s + s
Perimeter of a Square = 4(s), which is 4 times s
Area
The area of a square is length times width. Since the length and width of a square are exactly the same, the area of a square is s × s or s2.
Area of a Square = s × s = s2
Experiment
Compute the area of different shapes.
Materials
Scissors
Magic marker
Pencil
Paper
1.Draw a square on a piece of paper.
2.Draw both diagonals of the square with magic marker.
3.Cut out the square.
4.Cut the square along the diagonals.
5.Create a new shape with these triangles. Trace the new shape on a piece of paper.
6.Next create another new shape. Trace this shape.
7.All these shapes have the same area.
Something to think about . . .
Is the perimeter of each shape you constructed the same?
1.What is the perimeter of a square with side 7?
2.What is the area of a square with side 5?
3.What is the area of a square with perimeter 12?
4.What is the perimeter of a square with area 100 square units?
(Answers)
A parallelogram is a quadrilateral with two pairs of parallel sides. All squares and rectangles are parallelograms, but not all parallelograms are squares and/or rectangles.
Perimeter
To find the perimeter of a parallelogram, add the length of all four sides.
Perimeter of a Parallelogram = l + w + l + w
EXAMPLE:
Find the perimeter of this parallelogram.
6 + 10 + 6 + 10 = 32
Area
To find the area of a parallelogram, multiply the base times the altitude. The altitude is a line segment drawn from one side of the parallelogram perpendicular to the opposite side.
Area of a parallelogram = l(a)
What is the area of a parallelogram with base 8, width 5, and altitude 4?
Multiply the base (8) by the altitude (4).
8 × 4 = 32
The area is 32 square units.
Change a parallelogram to a rectangle.
Materials
Pencil
Graph paper
Scissors
Tape
Procedure
1.Draw a parallelogram on a piece of graph paper. The parallelogram you draw should not be a rectangle.
2.Make the parallelogram 7 squares long and 4 squares tall.
3.Cut out the parallelogram.
4.Cut a triangle off one end of the parallelogram. Start at the inside corner and cut straight down. Look at the diagram.
5.Slide the triangle to the right and tape it to the other end of the parallelogram. The parallelogram is now a rectangle.
6.What is the area of the rectangle created? Multiply length times width.
7.What is the area of the original parallelogram? Multiply length times altitude.
8.How do the two areas compare?
Something to think about . . .
This rectangle is 7 squares long and 4 squares tall.
The total area is 28 square units. The area of the parallelogram is also 28 square units.
Use this diagram to solve problems 1 and 2.
1.What is the perimeter of this parallelogram?
2.What is the area of this parallelogram?
3.What is the area of a parallelogram with base 6 and height 4?
(Answers)
A trapezoid is a quadrilateral with only two parallel sides.
Perimeter
To find the perimeter of a trapezoid, just add all four sides.
Perimeter of a Trapezoid = Base 1 + Base 2 +
Side 1 + Side 2
EXAMPLE:
Find the perimeter of this trapezoid.
The perimeter of this trapezoid is 3 + 5 + 8 + 12 = 28 units.
Area
Area of a Trapezoid = (sum of bases) × a
To find the area of a trapezoid, follow these three painless steps:
Step 1:Add base 1 and base 2.
Step 2:Take of the sum of the bases.
Step 3:Multiply the result of Step 2 by the altitude.
Area of a trapezoid = (b1 + b2)a
EXAMPLE:
Find the area of this trapezoid.
Step 1:First add base 1 to base 2.
7 + 9 = 16
Step 2:Take of the sum of the bases.
(16) = 8
Step 3:Multiply the result of Step 2 by the altitude.
8(4) = 32
The area of the trapezoid is 32 square units.
Use the diagrams to solve the problems.
1.What is the perimeter of trapezoid ABCD?
2.What is the area of trapezoid ABCD?
3.What is the perimeter of trapezoid WXYZ?
4.What is the area of trapezoid WXYZ?
(Answers)
A rhombus is a parallelogram with four congruent sides. A square is a special rhombus since a square is a parallelogram with four congruent sides, but a square also has four right angles.
To find the perimeter of a rhombus, add all the sides.
Perimeter of a Rhombus = Side 1 + Side 2 + Side 3 + Side 4
EXAMPLE:
Find the perimeter of a rhombus if each of the sides of the rhombus is 10 inches.
The perimeter is 10 + 10 + 10 + 10 or 40 inches.
Area
To find the area of a rhombus, follow these two simple steps:
Step 1:Multiply the length of the diagonals together.
is a diagonal of the rhombus.
is a diagonal of the rhombus.
Step 2:Take one half of the answer.
Area of a Rhombus = (Diagonal 1 × Diagonal 2)
If one diagonal of a rhombus is 6 units long and the other diagonal is 5 units long, what is the area of the rhombus?
Area = (6 × 5) = (30) = 15 square units
Experiment
Transform a rhombus into another common shape.
Materials
Scissors
Paper
Pencil
Ruler
Procedure
1.Draw a rhombus. Draw the diagonals of the rhombus.
2.Cut the rhombus along its diagonals. Four small triangles are formed.
3.Rearrange the four small triangles into a new shape. What shape can you make?
Something to think about . . .
Is the area of a rhombus related to the area of any other quadrilateral?
1.What is the perimeter of a rhombus if each of the sides is 4 inches long?
2.What is the area of a rhombus with diagonals 6 and 8 inches long?
3.What is the area of a rhombus with diagonals 1 and 2 feet long?
(Answers)
A regular polygon is a polygon with equal sides and equal angles. An equilateral triangle and a square are both regular polygons. It is possible to construct a regular polygon with any number of sides.
Depending on the number of sides, polygons have different names.
•Three sides = Triangle
•Four sides = Quadrilateral
•Five sides = Pentagon
•Six sides = Hexagon
•Seven sides = Heptagon
•Eight sides = Octagon
•Nine sides = Nonagon
Angles of a polygon
The number of sides of a polygon determines the number of interior angles. A polygon has the same number of sides as interior angles.
The sum of the interior angles of a polygon is (n − 2)180.
•The sum of the interior angles of a triangle is (3 − 2)180 = 180 degrees.
•The sum of the interior angles of a square is (4 − 2)180 = 360 degrees.
•The sum of the interior angles of a pentagon is (5 − 2)180 = 540 degrees.
•The sum of the interior angles of a hexagon is (6 − 2)180 = 720 degrees.
To illustrate how this equation was determined, divide any polygon into triangles. The sum of the angles in each of the triangles formed is 180 degrees. Multiply the number of triangles by 180 to find the sum of the angles in a polygon.
Divide this square into two triangles by connecting vertices A and C. The sum of the angles of the two triangles is 180 + 180 or 360 degrees.
Divide a pentagon into three triangles by connecting point A to points C and D. The sum of the angles of the pentagon will be 180 + 180 + 180, which is 540 degrees.
Divide an octagon into six triangles by connecting point A to points C, D, E, F, and G. The sum of the angles of an octagon is 6(180), which is 1,080 degrees.
The sum of the exterior angles of a polygon is always 360 degrees.
•The sum of the exterior angles of a triangle is 360 degrees.
•The sum of the exterior angles of a decagon is 360 degrees.
•The sum of the exterior angles of a polygon with 100 sides is still 360 degrees.
1.What is the sum of the interior angles of a pentagon?
2.What is the sum of the exterior angles of a pentagon?
3.What is the sum of the interior angles of a heptagon?
4.What is the sum of the exterior angles of a heptagon?
5.For what shape is the sum of its interior angles equal to its exterior angles?
(Answers)
To find the perimeter of a regular polygon, just add the sides. Or multiply the length of one side by the number of sides. Remember, in a regular polygon, all the sides are the same length.
Perimeter of a regular polygon = n(s), where n is the number of sides and s is the length of one side
EXAMPLE:
Find the perimeter of a hexagon with side 2.
Add the sides of the hexagon together.
2 + 2 + 2 + 2 + 2 + 2 = 12
Or multiply the length of one side by the number of sides.
(6)2 = 12
Area
Finding the area of a regular polygon is easy.
Multiply the apothem by the perimeter.
The apothem is a line segment from the center of the polygon perpendicular to a side.
Area of a regular polygon = ap,
where a is the apothem and p is the perimeter
To find the area of a regular polygon, follow these painless steps:
Step 1:Find the length of the apothem.
Step 2:Find the perimeter of the regular polygon.
Step 3:Multiply the apothem by the perimeter.
Step 4:Multiply the answer by . The result is the area.
EXAMPLE:
Find the area of a regular hexagon with side 6 inches and apothem 
inches.
Step 1:Find the length of the apothem. The apothem is inches.
Step 2:Find the perimeter of the regular polygon. The perimeter is 6(6) or 36.
Step 3:Multiply the apothem by the perimeter.
Step 4:Multiply the answer by .
The area of the hexagon is 
square inches.
Experiment
Compare two methods for finding the area of a square.
Materials
Pencil
Graph paper
Procedure
1.Draw a square with side 6.
2.Draw an apothem. Draw a line segment from the center of the square perpendicular to the opposite side.
3.Measure the length of the apothem.
4.Find the perimeter of the square.
5.Multiply 
of the apothem of the square by the perimeter to find the area of the square.
6.Now find the area of the same square by multiplying s times s.
7.Compare the areas of the square that you found by the two different methods.
Something to think about . . .
What is the formula to find the perimeter and area of a regular octagon? An octagon is an eight-sided figure.
1.Find the perimeter of an octagon with sides 2 inches long.
2.Find the perimeter of a pentagon with sides 3 feet long.
3.Find the area of an octagon with sides 4 meters and an apothem 3 meters.
4.Find the area of a decagon with sides 3 inches long and apothem 5 inches. A decagon is a 10-sided figure.
(Answers)
Look at each pair of figures. Determine which perimeter is larger or if they are both equal.
1.A rhombus with sides 4 feet long. A pentagon with sides 4 feet long.
2.A parallelogram with length 5 inches and width 10 inches.
An equilateral triangle with sides 12 inches long.
3.A trapezoid with bases 4 and 8 centimeters and legs 3 and 9 centimeters.
A square with sides 6 centimeters long.
Look at each pair of figures. Determine which area is greater or if they are both the same.
4.A rhombus with diagonals 4 feet long and 8 feet long. A triangle with base 4 feet long and height 8 feet long.
5.A rectangle with sides 3 and 5 miles long. A square with sides 4 miles long.
6.A parallelogram with base 8 kilometers and height 4 kilometers long.
A square with sides 6 kilometers long.
(Answers)
To find the area of an unusual shape, add line segments to divide the shape into smaller known shapes. Find the area of each of these smaller shapes and add the result.
EXAMPLE:
Find the area of this shape. All the angles are right angles.
Add a line segment to change the shape into two rectangles.
Find the area of rectangle 1 and rectangle 2.
Rectangle 1 has a length of 6 and a width of 4. The area of rectangle 1 is 24 square units.
Rectangle 2 has a length of 4 and a width of 8. The area of rectangle 2 is 32 square units.
Add the area of rectangle 1 and rectangle 2 together to find the areas of the entire shape. The area of the entire shape is 24 + 32, or 56 square units.
Find the area of this shape.
Draw a line segment to divide this shape into a triangle and a square.
Find the area of the triangle.
Area of triangle is (Base) × (Height)
The length of the height is 2 units and the length of the base is 4 units. The area of the triangle is (2)(4) = 4 square units.
Find the area of the square.
Area of square = Side × Side
The length of a side of the square is 4 units. The area is 16 square units.
Add the area of the square and the area of the triangle together to find the area of the shape.
4 + 16 = 20 square units
The area of the shape is 20 square units.
Find the perimeter and area of this shape.
(Answers)
The volume of a solid figure is the capacity of the figure. The volume is the number of cubic units it contains. Cubes, boxes, cones, balls, and cylinders are all three-dimensional shapes. You can measure the volume of any of these shapes. The volume of three-dimensional shapes is measured in cubic units, such as cubic inches, cubic feet, cubic yards, cubic miles, cubic centimeters, cubic meters, or cubic kilometers. Cubic units are written by placing a small 3 over the units.
Construct a cubic inch.
Materials
Paper
Pencil
Ruler
Scissors
Scotch tape
Procedure
1.Copy the following diagram on a piece of paper. It is made out of six identical squares. Make each square 1 inch by 1 inch.
2.How many right angles are in the diagram?
3.Cut out the diagram along the outside border.
4.Fold the diagram along the other edges and make a cube.
5.Tape the cube into place. This is one cubic inch.
6.Count the number of right angles on the cube.
Something to think about . . .
How many inches are in a foot?
How many cubic inches are in a cubic foot?
Rectangular solids are everywhere. A book is a rectangular solid; so is a drawer, a cereal box, a shoebox, a videotape, and a CD case. To find the volume of a rectangular solid multiply the length times the width times the height of the solid.
Volume of a Rectangular Solid = l × w × h or lwh
EXAMPLE:
Find the volume of the rectangular solid with length 6, height 4, and width 4.
4 × 6 × 4 = 96 cubic units
A cube is a special type of a rectangular solid. The length, width, and height of a cube are exactly the same.
Volume of a Cube = s × s × s = s3
EXAMPLE:
Find the volume of a cube with side 3 inches.
3 × 3 × 3 = 27 cubic inches
Notice that the answer is in cubic inches.
Surface area
Surface area of a rectangular solid
The surface area is the area on the outside of a three-dimensional shape. Imagine if you had to cover the entire outside of a three-dimensional shape with a piece of paper, how large would the piece of paper be? How could you compute the surface area of a three-dimensional shape?
The surface area of a cube is six times the surface area of one side of the cube. Count the sides of a cube. There are six of them. The surface area of the cube is 6(s)(s).
Surface Area of a Cube = 6s2
EXAMPLE:
What is the surface area of a cube with side 4 units?
6(4)(4) = 96 square units
Find the surface area of a three-dimensional shape.
Materials
Empty cereal box
Ruler
Scissors
Paper
Pencil
Procedure
1.Cut an empty cereal box along the edges of the box. You should cut the cereal box into six pieces. Each side of the box should be a separate piece.
2.Each of the pieces will be a rectangle. Measure the length and width of each rectangle. Enter the results in the chart.
3.Find the area of each of these rectangles by multiplying the length of each rectangle by the width of each rectangle. Enter the results in the chart.
4.Add the areas of all six sides to find the total surface area of a cereal box.
Something to think about . . .
Do any of the sides have the same area?
How would you find the surface area of a pyramid?
To find the surface area of a rectangular solid, just follow these painless steps:
Step 1:Find the length, width, and height of the solid.
Step 2:Multiply 2 × length × width.
Step 3:Multiply 2 × width × height.
Step 4:Multiply 2 × length × height.
Step 5:Add the results of Steps 2, 3, and 4. The answer is the surface area of the rectangular solid.
EXAMPLE:
Find the surface area of a rectangular solid with sides 5 inches, 6 inches, and 7 inches.
Step 1:Find the length, width, and height of the solid. The length is 5 inches, the width 6 inches, and the height 7 inches.
Step 2:Multiply 2 × length × width.
2(5)(6) = 60 square inches
Step 3:Multiply 2 × width × height.
2(6)(7) = 84 square inches
Step 4:Multiply 2 × length × height.
2(5)(7) = 70 square inches
Step 5:Add the results of Steps 2, 3, and 4. The answer is the surface area of the rectangular solid.
The surface area is 214 square inches.
Surface Area of a Rectangular Solid =
2lw + 2lh + 2hw
1.What is the volume of a cube with side 5 inches?
2.What is the volume of a cube with side 1 inch?
3.What is the surface area of a cube with side 5 inches?
4.What is the surface area of a cube with side 1 inch?
5.What is the volume of a rectangular solid with sides 1, 2, and 3 inches?
6.What is the surface area of a rectangular solid with sides 1, 2, and 3 inches?
(Answers)
A cylinder is a common shape. A box of oatmeal is a cylinder, and so is a glass, a can of green beans, or a can of tuna.
To find the surface area of a cylinder, follow these painless steps:
Step 1:Find the height of the cylinder.
Step 2:Find the radius of the cylinder.
Step 3:Find the area of the circle that is the base of the cylinder using the equation A = πr2.
Step 4:Multiply the area found in Step 3 by 2 since there is a circle at both the top and the bottom of the cylinder.
Step 5:Find the circumference of the circle that forms the base of the cylinder. Use the formula 2πr.
Step 6:Multiply the circumference of the circle by the height of the cylinder.
Step 7:Add the answers to Steps 4 and 6.
Surface Area of a Cylinder = 2πr2 + 2πrh
EXAMPLE:
Find the surface area of a cylinder that is 12 inches high and has a diameter of 4 inches.
Step 1:Find the height of the cylinder. The height is given as 12 inches.
Step 2:Find the radius of the cylinder. Since the diameter of the cylinder is 4 inches, the radius of the cylinder is 2 inches.
Step 3:Find the area of the circle that is the base of the cylinder using the equation A = πr2.
A = π(2)2
A = 4π
Step 4:Multiply the area found in Step 3 by 2 since there is a circle at both the top and the bottom of the cylinder.
Area of both circles = 2(4π) = 8π
Step 5:Find the circumference of the circle that forms the base of the cylinder. Use the formula 2πr.
Circumference = 2π(2) = 4π
Step 6:Multiply the circumference of the circle (found in Step 5) by the height of the cylinder.
Area of sides of cylinder = (4π)(12) = 48π
Step 7:Add the answer to Step 4 to the answer to Step 6.
8π + 48π = 56π
The surface area of the cylinder is 56π square inches.
Lateral area of a cylinder
If a cylinder is pictured as a soda can, the lateral area of a cylinder is the curved portion of the can that is printed on. If you could peel the printed section off a soda can the result would be a rectangle. The length of the rectangle is the circumference of the can. The height of the rectangle is the height of the can.
To find the lateral area of a cylinder, follow these painless steps:
Step 1:Find the height of the cylinder.
Step 2:Find the radius of the cylinder.
Step 3:Use the radius to find the length of the rectangle which is the same as the circumference of the cylinder. The circumference of the cylinder is 2πr.
Step 4:Multiply the circumference of the cylinder by the height of the cylinder to find the lateral area.
Lateral Area of a Cylinder = 2πrh
EXAMPLE:
Find the lateral surface of a cylinder with height 7 and diameter 10.
Step 1:Find the height of the cylinder.
The height of the cylinder is 7.
Step 2:Find the radius of the cylinder.
The radius of the cylinder is half the diameter.
The radius of the cylinder is 10 ÷ 2 = 5.
Step 3:Find the circumference of the cylinder.
The circumference of the cylinder is 2πr = 10 π.
Step 4:Multiply the circumference of the cylinder by the height of the cylinder to find the lateral area.
The lateral area is (10π)7 = 70π.
To find the volume of a cylinder, follow these three painless steps:
Step 1:Square the radius.
Step 2:Multiply the result by π, which is 3.14.
Step 3:Multiply the answer by the height.
Volume of a Cylinder = πr2h
Find the volume of a cylinder with radius 5 and height 10.
Step 1:Square the radius.
5 × 5 = 25
Step 2:Multiply 25 by π.
25π
Step 3:Multiply 25π by 10.
25π × 10 = 250π
The volume of the cylinder is 250π cubic units.
Experiment
Compare the volume of three glasses.
Materials
3 glasses
Water
Pencil
Paper
Ruler
Calculator
Procedure
1.Find three glasses of three different sizes.
2.Using the ruler, measure the radius of each glass.
3.Using the ruler, measure the height of each glass.
4.Compute the volume of each glass using the formula πr2h.
5.Based on the volume you computed, rank order the glasses from smallest to largest.
6.Fill what you computed to be the smallest glass with water.
7.Pour the smallest glass of water into the next largest glass. Did all the water fit?
8.Fill this middle-size glass with water and pour it into the largest glass. Did all the water fit? Were all your calculations correct?
Something to think about . . .
How would you compare the volume of a box of cereal to a glass of water?
1.Find the volume of a cylinder with radius 4 and height 2.
2.Find the volume of a cylinder with radius 1 and height 10.
3.Find the volume of a cylinder with radius 10 and height 1.
(Answers)
The volume of a cone is 
πr2h.
To find the volume of a cone, follow these painless steps:
Step 1:Square the radius.
Step 2:Multiply it by π.
Step 3:Multiply the answer in Step 2 by the height.
Step 4:Multiply the answer in Step 3 by .
Notice that the formula for the volume of a cone is exactly one-third the volume of a cylinder of the same height.
Volume of a Cone = πr2h
EXAMPLE:
Find the volume of a cone with height 5 and diameter 6.
Step 1:Square the radius.
The diameter of the cone is 6.
The radius of the cone is half the diameter, or 3.
32 is 9.
Step 2:Multiply 9 by π.
9π
Step 3:Multiply 9π by the height of the cone, which is 5.
9π × 5 = 45π
Step 4:Multiply 45π by .
45π × = 15π
The volume of the cone is 15π cubic units.
1.Find the volume of a cone with radius 6 and height 3.
2.Which has a greater volume, a cone with radius 2 and height 6 or a cylinder with radius 2 and height 2?
(Answers)
A ball, an orange, and a globe are all spheres. A sphere is the set of all points equidistant from a given point.
To find the surface area of a sphere, just follow these painless steps:
Step 1:Find the radius of the sphere.
Step 3:Multiply the answer by 4.
Step 4:Multiply the answer by π. The answer is in square units.
Surface Area of a Sphere = 4πr2
EXAMPLE:
Find the surface area of sphere with radius 10 inches.
Step 1:Find the radius of the sphere.
The radius is given as 10.
Step 2:Square the radius.
102 = 100
Step 3:Multiply the answer by 4.
4(100) = 400
Step 4:Multiply the answer by π. The answer is in square units.
The surface area of the sphere is 400π in.2
All you need to know to find the volume of a sphere is the radius.
To find the volume of a sphere, follow these three painless steps:
Step 1:Cube the radius (r × r × r).
Step 2:Multiply the answer by 
.
Step 3:Multiply the answer by π.
Volume of a Sphere = πr3
Find the volume of a sphere with radius 6.
Step 1:Cube the radius.
6 × 6 × 6 = 216
Step 2:Multiply the answer by .
× 216 = 288
Step 3:Multiply the answer by π.
The volume of the sphere is 288π units3.
If you don’t want the answer in terms of π, multiply 288 × 3.14.
288 × 3.14 = 904.32 cubic units
1.Compute the surface area of a sphere with radius 3.
2.Compute the volume of a sphere with radius 3.
3.Compute the surface area of a sphere with radius 1.
4.Compute the volume of a sphere with radius 1.
(Answers)
Find the volume of these shapes. Use these formulas.
Cube = s3
Rectangular solid = l × w × h
Cylinder = πr2h
Cone = πr2h
Sphere = πr3
1.A sphere with radius 6.
2.A cone with height 4 and radius 2.
3.A sphere with diameter 10.
4.A cylinder with radius 4 and height 10.
5.A cube with side 8.
6.A rectangular solid with length 1, width 2, and height 3.
(Answers)
