Perimeters and areas
Definition and basic properties
Perimeter
The perimeter of a two dimensional closed figure is the total length of the sides of the figure.
The perimeter of the triangle is the sum of the lengths of the three sides of the triangle:
3+4+5=12
Unit square
A unit square is a square whose sides all have length 1.
Area
The area of a two dimensional object is the exact number of unit squares that fit into that object.
The following graphic contains a geometric object and each of the squares in the grid is one unit square.
Count the number of unit squares inside the figure to see that the area of the geometric figure is 10.
Areas of geometric figures do not have to be a whole number.
The following graphic contains a geometric figure and each square in the grid is a unit square.
Count the number of unit squares inside the figure to find that it has an area of 9.5.
Larger perimeters do not necessarily correspond to having larger areas.
The following graphic contains two figures labeled A and B.
Observe the following:
- Area\left(A\right)=5
- Perimeter\left(A\right)=12
- Area\left(B\right)=6
- Perimeter\left(B\right)=10
The area of A is less than the area of B but the perimeter of A is greater than that of B.
Perimeters and areas of basic polygons
Triangles
Area of a triangle
Consider a triangle as in the following graphic:
The area of the triangle is:
A=\dfrac{1}{2}bh
Consider the following triangle:
The area of the triangle is:
\dfrac{1}{2}8\left(3\right)=4\left(3\right)=12
Quadrilaterals
Perimeter of a square
The perimeter of a square whose sides are all of length b is:
P=4b
Consider the following square:
The perimeter of the square is:
4\left(5\right)=20
Area of a square
The area of a square whose sides are all of length b is:
A=b^2
Consider the following square:
The area of the square is:
5^2=25
Perimeter of a rectangle
The area of a rectangle with a base of length b and a height of h is:
P=2a+2b
Consider the following rectangle:
The area of the rectangle is:
8\left(3\right)=24
Area of a rectangle
The area of a rectangle with a base of length b and a height of h is:
A=bh
Consider the following rectangle:
The area of the rectangle is:
7\left(3\right)=21
Perimeter of a parallelogram
Consider a parallelogram as in the following graphic:
Then the perimeter of the parallelogram is:
P=2x+2y
Consider the following parallelogram:
The perimeter of the parallelogram is:
2\left(5\right)+2\left(8\right)=26
Area of a parallelogram
Consider a parallelogram as in the following graphic:
The area of the parallelogram is:
A=\ell h
Consider the following parallelogram:
The area of the parallelogram is:
9\left(7\right)=63
Area of a trapezoid
Consider a trapezoid as in the following graphic:
The area of the trapezoid is:
A=\dfrac{b_1+b_2}{2}h
Consider the following trapezoid:
The area of the trapezoid is:
\dfrac{8+2}{2}\left(4\right)=5\left(4\right)=20
Circles
Circumference
The circumference of a circle is the perimeter of the circle.
The number \pi
\pi is a real number which is approximately 3.14\ 159. It is used to measure angles and areas (or circumferences) of circles.
The circumference C of a circle with radius r is:
C=2\pi r
Consider the following circle:
The circumference of the circle is:
2\pi\left(3\right)=6\pi \approx 18.8\ 495
The area of a circle with radius r is:
A=\pi r^2
Consider the following circle:
The area of the circle is:
\pi\left(3\right)^2=9\pi\approx 28.27\ 431
Areas between two shapes and areas of compound figures
Compound figure
A compound figure is a geometric object which is made of two or more basic geometric objects.
The following graphic contains a compound figure.
The dashed line separates the figure into two basic geometric shapes, namely a rectangle and a triangle.
The area of a compound figure is found by summing the areas of the basic figures which form the compound figure.
The following graphic contains a compound figure:
The dashed line separates the compound figure into a rectangle and a triangle.
The rectangle has an area of:
8\left(5\right)=40
The triangle has a base length of 8-5=3 and a length of 9-5=4. The area of the triangle is:
\dfrac{1}{2}3\left(4\right)=6
Therefore, the area of the compound figure is:
40+6=46
To find the area between two geometric figures, find the area of the larger figure and subtract the area of the enclosed figure.
The following graphic contains two squares, one contained inside the other:
The larger square has an area of:
5^2=25
The smaller square has an area of:
2^2=4
Therefore, the area between the two squares is:
25-4=21
Surface areas and volumes
Definition and basic properties
Surface Area
The surface area of a three dimensional object is the sum of the areas of the sides of the surface.
Volume
The volume of a three dimensional object is the exact number of unit cubes that fit into the object.
Usual surface areas and volumes
Prisms
Prism
A prism is a three dimensional geometric objects whose bases are identical and whose sides are flat.
The following graphic contains a prism:
The bases of the prism are identical triangles.
Volume of a prism
The volume of a prism is equal to the product of the surface area of one of the bases of the prism by the height of the prism. If h is the height of the prism and S is the surface area of one of the sides, then the volume of the prism is:
V=h S
The following graphic contains a prism:
The prism has two identical ends which are rectangles that have a surface area of:
2\left(5\right)=10
The length of the prism is 7.
Therefore, the volume of the prism is:
10\left(7\right)=70
Surface area of a prism
The surface area of a prism is the sum of the surface areas of all the sides of the prism.
The above prism has six sides.
- Two of the sides have a surface area of 2\left(5\right)=10.
- Two of the sides have a surface area of 2\left(7\right)=14.
- The remaining two sides of the prism have a surface area of 5\left(7\right)=35.
Therefore the surface area of the prism is:
2\left(10\right)+2\left(14\right)+2\left(35\right)=20+28+70=118
Volume of a cube
If the length of an edge of a cube is a, then the volume of the cube is:
V=a^3
The volume of the above cube is:
V=4^3=64
Surface area of a cube
If the length of an edge of a cube is a, then the surface area of the cube is:
S=6a^2
The surface area of the cube in the above graphic is:
6\left(4^2\right)=96
Cylinders
Cylinder
A cylinder is a three dimensional object which has two parallel circular ends of the same size which are connected by a single curved side.
- The radius of a cylinder is the radius of either circle forming the ends of the cylinder.
- The height of a cylinder is the length of the side connecting the ends of the cylinder.
Volume of a cylinder
If r is the radius of cylinder and h is its height, then the volume of the cylinder is:
V=\pi r^2 h
The following graphic contains a cylinder with a radius of 2 and a height of 5:
The volume of the cylinder is:
\pi \left(2\right)^25=20\pi
Surface area of a cylinder
If r is the radius of cylinder and h is its height then the surface area of the cylinder is:
2 \pi r\left(h+r\right)
The following graphic contains a cylinder with a radius of 2 and a height of 5:
The surface area of the cylinder is:
2\pi \left(2\right)\left(5+2\right)=4\pi\left(7\right)=28\pi
Pyramids
Pyramid
A pyramid is a three dimensional geometric formed by connecting the vertices of a two dimensional polygon at a single point called the apex.
- h is the height of the pyramid.
- a is the area of the base of the pyramid.
Volume of a pyramid
Suppose a denotes the area of the base of a pyramid and h the height of the pyramid. Then the volume of the pyramid is:
V=\dfrac{ah}{3}
The following graphic contains a pyramid with a right triangle as the base:
The base of the pyramid is a right triangle whose legs are of length 3 and 2. The area of the base of the pyramid is:
\dfrac{1}{2}3\left(2\right)=3
The height of the pyramid is 7.
Therefore, the volume of the pyramid is,
\dfrac{3\left(7\right)}{3}=7
Cones
Cone
A cone is a pyramid which has a circle for its base.
Volume of a cone
If r is the radius of a cone and h its height, then the volume of the cone is:
V=\dfrac{\pi r^2 h}{3}
The following graphic contains a cone with a height of 7 and a radius of 3:
Therefore, the volume of the cone is:
\dfrac{\pi\left(3\right)^27}{3}=21\pi
Surface area of a cone
If r is the radius of a cone and h its height, then the surface area of the cone is:
\pi r\left(r+\sqrt{r^2+h^2}\right)
The following graphic contains a cone with a height of 7 and a radius of 3:
Therefore, the surface area of the cone is:
\pi\left(3\right)\left(3+\sqrt{3^2+7^2}\right)=3\pi\left(3+\sqrt{58}\right)
Spheres
Sphere
Let r be a nonnegative real number and C a point in three space. Then the sphere of radius r with center C is the collection of all points of distance r from the point C.
Volume of a sphere
The volume of a sphere with a radius of r is:
V=\dfrac{4\pi r^3}{3}
The volume of the above sphere is:
\dfrac{4\pi \left(2\right)^3}{3}=\dfrac{32\pi}{3}
Surface area of a sphere
The surface area of a sphere with a radius of r is:
4\pi r^2
The surface area of the above sphere is:
4\pi\left(2\right)^2=16\pi
Consequences of transformations on perimeters, areas, surface areas and volumes
Surface area under transformation
If two objects are similar and the sides of the larger object lengths are k times larger than the smaller object lengths, then the area of the larger object is k^2 times larger than the area of the smaller object.
The following graphic contains two squares.
The side lengths of the larger square are 2 times larger than the smaller square. Observe that:
- The area of the larger square is 4^2=16.
- The area of the smaller square is 2^2=4.
The area of the larger square is 2^2=4 times larger than the area of the smaller square.
Volume under transformation
If two three dimensional objects are similar and the lengths of the sides of the larger object are k times larger than the lengths of the sides of the smaller object, then the volume of the larger object is k^3 times larger than the volume of the smaller object.
The following graphic contains two cubes.
The side lengths of the larger cube are 2 times larger than the side lengths of the smaller cube. Observe that:
- The volume of the larger cube is 6^3=216.
- The volume of the smaller cube is 3^3=27.
The volume of the larger cube is 2^3=8 times larger than the volume of the smaller cube.